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DEPARTMENT OF. APPLIED MATHEMATICS, . _ UNIVERSITY COLLEGE, UNIVERSITY: OF LONDON |
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.—XV. A. MATHEMATICAL | THEORY OF RANDOM. MIGRATION. —
BY KARL PEARSON, F, RS. WITH THE ASSISTANCE OF
JOHN. BLAKEMAN, M.Sc.
[With Seven Diagrams] |
| LONDON : PUBLISHED BY DULAU AND CO., 37, SOHO SQUARE, W. : 1906
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Il. MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.—XV. A MATHEMATICAL THEORY OF RANDOM MIGRATION.
BY
KARL PEARSON, F.R.S. WITH THE ASSISTANCE OF
JOHN BLAKEMAN, M.Sc.
LONDON : PUBLISHED BY DULAU AND CO., 37, SOHO SQUARE, W.
1906 mM.W:
Mathematical Contributions to the Theory of Evolution.
XV. A MATHEMATICAL THEORY OF RANDOM MIGRATION. By Karu Pearson, F.R.S., with the assistance of Jonn BLaAKkEMAN, M.Sc.
(1) Introductory. In dealing with any natural phenomenon,—especially one of a vital nature, with all the complexity of living organisms in type and habit,— the mathematician has to simplify the conditions until they reach the attenuated character which lies within the power of his analysis*. The problem of migration is one which is largely statistical, but it involves at the same time a close study of geographical and geological conditions, and of food and shelter supply peculiar to each species. Some years ago the late Professor Weldon started an extensive study as to the distribution of various species and local races of land snails, but he was struck by the absence in several cases of any definite change of environ- ment at the boundaries of the distribution of a definite race. It occurred to me in thinking over the matter that such boundaries, where they exist, may possibly not be permanent. To take a purely hypothetical illustration :.A species is pushed back to a certain limit by a change of environmental conditions—say, an ice age. Does it follow that if the environment again becomes favourable, that it will rapidly occupy possible country? What is the rate of infiltration of a species into a possible habitat? It depends, of course, on a whole series of most complex conditions, the rate of locomotion, the channels of communication, the distribution of food areas and breeding grounds in the new country, and the connecting links between all these. Every detail must be studied by the field naturalist in relation: to each species. All the mathematician can do is to make an idealised system, which may be dangerous, if applied dogmatically to any particular case, but which can hardly fail to be suggestive, if it be treated within the limits of reasonable application. The idealised system which I proposed to myself was of the following kind :
(i) Breeding grounds and food supply are supposed to have an average uniform distribution over the district under consideration. There is to be no special following of river beds or forest tracks.
* This is of course a perfectly familiar process to every mathematical physicist, but its unfamiliarity leads the biologist to suspect or even discard mathematical reasoning, instead of testing the result as the physicist does by experiment and observation.
1—2
4 KARL PEARSON
(ii) The species scattering from a centre is supposed to distribute itself uniformly in all directions. The average distance through which an individual of the species moves from habitat to habitat will be spoken of as a “flight,” and there may be » such “flights” from locus of origin to breeding ground, or again from breeding ground to breeding ground, if the species reproduces more than once. A flight is to be distinguished from a ‘‘flitter,” a mere two and fro motion associated with the quest for food or mate in the neighbourhood of the habitat.
(ii) Now taking a centre, reduced in the idealised system to a point, what would be the distribution after n random flights of NM individuals departing from this centre? This is the first problem. I will call it the Fundamental Problem of Random Migration.
(iv) Supposing the first problem solved, we have now to distribute such points over an area bounded by any contour, and mark the distribution on both sides of the contour after any number of breeding seasons. The shape of the contour and the number of seasons dealt with provide a series of problems which may be spoken of as Secondary Problems of Migration.
A little consideration of the Fundamental Problem showed me that it presented considerable analytical difficulties, and I was by no means clear that the series of hypotheses adopted would be sufficiently close to the natural conditions of any species to repay the labour involved in the investigation. At this stage the matter rested, until last year Major Ross put before me the same problem as being of essential importance for the infiltration of mosquitoes into cleared areas, and asked me if I could not provide the statistical solution of it. He considered that we might treat a district as approximately “equi-swampous,” and thus my conditions (i), (ii) above could be applied to obtain at any rate a first approximation to the solution.
Starting on the problem again I obtained the solution for the distribution after two flights, an integral expressing the distribution after three flights, which I carelessly failed to see could be at once reduced to an elliptic integral, and the general functional relation between the distribution after successive flights. At this point I failed to make further progress, and under the heading of “The Problem of the Random Walk” asked for the aid of fellow-mathematicians in Naturé*. The reply to my appeal was threefold. Mr Geoffrey T. Bennett sent me in terms of elliptic integrals the solution for three flights. Lord Rayleigh drew my attention to the fact that the “problem of the random walk” where the number of flights is very great becomes identical with a problem in the combination of sound ampli- tudes in the case of notes of the same period, which he has dealt with in several paperst. Lastly Professor J. C. Kluyver presented a paper to the Royal Academy
* July 27th, 1905. + Phil. Mag., August, 1880, p. 75; February, 1899, p. 246.
\ A MATHEMATICAL THEORY OF RANDOM MIGRATION 5
of Sciences of Amsterdam, entitled “A local probability problem.”* Professor Kluyver obtains the general solution in terms of the integral of a product of Bessel’s functions of the zero and first orders. He deduces Rayleigh’s solution for nm large, he shows that the Bessel function integral represents a series of different analytic functions in different intervals, and proves a number of special problems of very considerable interest. Referring to his general solution, he writes, however :
“From this result we infer that the probability sought for is of a rather intricate character. The n+1 functions J are oscillating functions, and have their signs altering in an irregular manner as the variable u increases. Hence even an approximation of the integral is not easily found, and as a solution of Pearson’s problem it is little apt to meet the requirements of the proposer.” t
Kluyver’s solution is of extreme suggestiveness for the analytical theory of discontinuous functions. In the endeavour to express it in a form suited to my special purposes I have come across a long series of Bessel function properties, some at least of which seem to me novel, but have unfortunately no bearing on the problem of migration. If we turn to Rayleigh’s solution for n large, I must confess at once to being unconvinced of the adequacy of the proofs used to deduce it, especially that in the Theory of Sound{. Kluyver's proof of Rayleigh’s solution § appears to me to also require much strengthening, and in neither case do we have any practical measure of what the number of flights must be before we have in practice a reasonable accordance between the discontinuous Bessel’s function integral expression and the Rayleigh solution of Gaussian frequency type.
After a good many failures I have succeeded in obtaining a solution in series of the Bessel function integral, but this not of a character to be of service for Srequent arithmetical calculations. It serves, however, to test the approximation of the Rayleigh solution and the accuracy of the solutions for few flights obtained by other processes. At this stage I realised that the functional equation between the distributions for and n + 1 flights could be solved graphically, and that starting with the known distributions for n= 2 or n=8, we could by very great labour, but absolutely straightforward graphical work and the use of mechanical integrators, build up in succession the solutions for n = 4, 5, 6, 7, ete. I proposed that this process should be continued until the graphically found distribution coincided with the plotted values obtained from the above solution in series. This was achieved for n=7. For n=6 and n=7, the solution in series approaches to the Rayleigh solution, with which for all practical purposes it may be asserted to coincide for n=10. We have thus reached a continuous graphical illustration of the transition of a series of discontinuous and, in many respects, remarkable analytical functions, step by step with the increase of x into a normal curve of errors. The relation-
* Koninklijke Akademie van Vetenschappen te Amsterdam. Proceedings, Oct. 25, 1905, pp. 341—50. t loc. cit. p. 343. f{ 2nd Edition, § 42a. § Kluyver, doc. cit. p. 345.
6 KARL PEARSON
ship is a noteworthy one, and not without suggestion for other branches of investigation.
The exact method of graphical solution will be described later ; the whole labour of it, involving many weeks’ work, was due to my assistant, Mr John Blakeman, M. Se.
(2) General Analytical Solution of the Fundamental Problem. Let the origin be taken at the centre of dispersion and 7 he the distance of any small elementary area a from the centre of dispersion. Let ¢,(7°). a be the frequency of individuals on a after the nth flight, and ¢,,,(7°)a their frequency on the same element after the (n+1)th flight. Let / be the length of the flight. Then only those individuals who were on a circle of radius 7 round the centre of a after the nth flight can reach a with the (n+1)th flight, and only those individuals of these who take their flight in one definite direction. Let O be the centre of dispersion, C the centre of a, P a point on the circle of radius 7 round C, 4 PCO=6, then the frequency per unit area at P is ¢,(7?+0—27lcos 6), and the amount which goes in directions between @ and +66 is d@/27. Hence the frequency per unit area at C’ after the {n+1)th flight is given by:
Qn bras (t*) = 5 $e (024 E 2786080) dO cisesyssssssvenceones (i). 0
This is the equation, which I shall speak of as the general functional relation between.
the densities at successive flights. Now assume: ¢,(7°)=C, J, (ur), where C,, is any
undetermined function of n, 7 and uw, and u is at present an undetermined variable. Then by Neumann’s Theorem *:
J, (uJ? +2 —2rl cos 6) = J, (ur) J, (ul) +2 > J, (ur) J, (ul) cos t6. 1
Hence : x |, G J, (uJ/P +P —2ri cos 8) dd=C,, J, (ur) J, (ul)
=C,4,J5, (ur), by (i). Therefore Cal, tk) G. It follows that C,, =D {J, (wl)}", where D may be any function of J, but not of n. Thus we have: dn (1°) =D J, (ur) {J, (ul)}",
where we may sum for any or all values of 1.
Now when n=1, ¢, (7’) must be zero, for all values of r except r=1 to 1+7, and it then equals N/(27/r), + being very small and N the total number issuing from the centre of dispersion. We know, however, that tT:
| as | * upf (p) Jn (up) Jn (ur) dp=f(r), if q<r<p; 0 ; =0, ifr >por <q.
* ©. Neumann, Theorie der Besselschen Functionen, 8. 65. + Gray and Mathews, Treatise on Bessel’s Functions, p. 80.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 7
N Now take n=0, g=l, p=l+7 and JP ae © od then we have : [, ule J, (ul) J, (ur) rdu =, (2”), or je. | wh Gay a (ii ? , 1 27} 0 0 CMU) AU cenecccccccerervereseees -
This determines the form of D and the summation of u; for, if we take
blr?) = | AST Ur) {Ta (WYP Ae cesesareseaseeerneee (ii),
we satisfy the general functional relation (i) and further the initial equation (ii). Let P,, (r) be the probability that an individual after n flights will be a distance or less from the centre of dispersion. Then clearly
P,, (vr) =2n { "1 digas)
=W J's dr [,w J, (ur) {J, (ul) }" du.
But* wipe. Oe eee hence P, (r)=N i da i vd (ur) abs oe ur} {Jo (ul i -N | ” J, (ur) (J, (ul) Pd, aaa =N i es () J, (2) Le ee (ie),
(iv) is Kluyver’s fundamental solution, which he reaches by a very different and more general analysis. (iii) is the form of it which best suits my present investigation.
(8) On an expansion in series of the expression for >, (7°). By straight- forward but somewhat laborious multiplication it can be shown that:
(6n—11)n ,
+
(50n — 57) Me (1892 —2125n +270n’) n
1800 y’, ete.
103,680 :
* Gray and Mathews, loc. cit. p. 13.
8 KARL PEARSON
Hence putting 2 /y=z,
x
(6n—11)n 2
J. n — jn? a ee >, (6n—11)n Wale [ a Gre Iga 266 (50n — 57)” ge _ (1892 — 2125n + 270n’) n ge i (v) 1800 1024 103,680 4096 CLG. PH cx stgars Vv
— ane 4 6 8 10 19 = {1 —ay2*— a2 — 4,2 — A)" — 4.2" — ete. }, let us write, for brevity. The a’s are then known coefficients. Now by (iii)
bn (7) = x i “a J, (ur) {J, (ul) }" du.
But we know that*:
[, ae eT, (ur) du = i RO fia Did klicres sce iak (vi) Write : HE SO aw vl vp ates ey cmaiencere ees vee wake vil). 2 Thus: er ey ee eet tad (viii). 0 o
Differentiate (viii) s times with regard to o”:
_1)\s i Qs+1—F420? = ds 23 —}7°/0? ( ay e Js (ur) du =e | 5¢ \.
Hence, if B= —20’/r’, we have: 1,~ | uth e288 T (ur) du 0 6) 26+ Ss Ll Bigg rap \B
2841
= — Tari te say. We have therefore : NV dis NSO 67 . dn = 5" eh Ties ae eee e cence ee eenee (ix),
3 : : ae wal where it remains to evaluate i= a (5 em).
We find: . 1 /r\? 72/262 r ir 4= 5 (Ze (2 25475):
* Gray and Mathews, loc. cit. Eqn. (162), p, 78
A MATHEMATICAL THEORY OF RANDOM MIGRATION 9
bog = a (Ey eee (120 — 300 c+ 150 r-25 5+ +5 x = 5 =). ie — (2) e-"" (720 ~2160 o+ 1350 = 300 7 4. a ~ : (2) + o (2). Remembering that by (vii) /?=20°/n, we have from (ix) ea (24 ~ 48 O+18 m2! ae i 5) + Frome (120- 300 +150 "— 25 stip 57 3g o) = 18522124310 ag argo 1350 “- 300754 a= sot aes)
This is the general expansion for the distribution of the individuals emerging “from a centre of dispersion after n random flights. Clearly if we want to go as
1 : : far as ni We must retain terms up to (7°/o*)*, and the convergence is small for n
small, Thus for the first two or three flights, (x) as far as I have calculated the terms gives poor results, even if they are notwithstanding better than the Rayleigh solution. The arithmetical work required to calculate the ordinates is also severe. ‘Tf we put n=, we have
Lord Rayleigh’s expression. Now o*=4nl’, hence unless / becomes indefinitely small as n becomes indefinitely large the population becomes widely scattered. If the - single flight 7 be very small, but the total flight n/ be finite, then 4n/? tends to become vanishingly small, or the population remains close to the centre of dispersion. This is really the ‘‘flitter” as distinct from the flight.
Examining the solution found it is clear that it may be looked upon as the sum of products of two factors, one series of factors not involving r/o but only n and the other not involving 1 but only r/o. Thus we may write
dn (7?) =N (V) Wy + V,@, + V4.0, + Y,05 + «. ny
where a= me en y,=1, ear (,_ 1 = = 5 aoe (1 553)? v,=0,
10 KARL PEARSON
1 _ 4p 2/g2 rn ok 1 8 Oro oe (2-2 BT i) 6 En? __1 ,-trjer(e_ ym 9 _1n a Se Oe Ono (6 : ge Aw 8°’ Ys On? 2) RL a EERO ER Tn eee ee en ere se (xii)
The o-functions form a series of such special interest that a few of their remarkable properties will be stated in the next section.
(4) Properties of the w-functions.
Let us consider the pth moment round the origin of the 2sth w-function taken over all plane space. We will denote it by m,,,. Then
re “ao | VAT ay.7? 0 0
=i i bgt Glaser ccemastateneaeaccauind iat, 0 a = same! Stl a 1 2) . Now Wy. = Snot | B) ap (3° Saaienrarpesieasaunes (xiv), and by B= —20°/r’ we have dp=*F dr
Hence writing p=2q we find
-o ds 1 =(—])ets-1 2\q ad aoe | = glip Mag as = (—1)"** (20°) i Br as (Fe )ag eapaceel (xv). Integrate by parts and we have
Manse (1) (20% [ 18 Fees (pelt) (aya) {err (2 a) a].
The part in curled brackets vanishes at the limits and thus
rae (—Uyr(eerneg—nf “pr 2 (Lou) a
= Mag, 95-2 (s—¢q =i},
Repeating this process we find Mg,20= (8— 1 —g) (8—2—g) (8-3-9)... (—4)
x (—1)"? x (20°)? x oe GIN aecieie Doane ca (xvi).
The integral is finite and known; hence if y be less than s we find for integer values
Moors = Os PES” ahcnednt os od ieee ain gota (xvii). Now consider w,, as made up of two parts,
ag Lge
A MATHEMATICAL THEORY OF RANDOM MIGRATION 11
Then it is clear that y.., if g be less than s, consists of powers of 7° less than s, and therefore
i Wes Xoqg Ar = 0. 0
Accordingly a remarkable property holds for the y-function part of the w-function, namely, if x. and x.. be two such functions, then it follows that
Qe wg Xardr=0, if g and @’ be different, ............ xix). 5 Xeq X29 q | Returning to (xvi), let us put g=s, then
Mag, o9 = — (20°)' s[ “areemag ~J -0
2¢°)8 wa 2 a a) E(-ay | atte da, 0
or, Mega = C— LPP 2" (sp cesestens sehantena vy capanne ees cnmman ees (xx).
Let us now consider the integral over the plane {= 2a | WogXosV AT. 0
Except for the last term in y,,., it will consist of a number of terms having for factors My, with q<s and these all vanish. The last term in y,, is
(-13(2)",
re] —1)y re) i= 2a | Wop XosT AT = cia oa Be | w,,7 dr, 0
and accordingly
or by (xx) Pes (|S) -peacdunes ateneesarnweviaonpenves (xxi).
Hence we have the following properties :
(a) The integral all over the plane of distribution of one product of a x-function into an @-function of a different order is zero.
(b) The integral all over the plane of distribution of the product of a x-function into an o-function of the same. order is, if 2s be the order, equal to (js).
These properties enable us—as in the case of Bessel’s functions or Legendre’s functions—to expand any function symmetrical round a centre and a function only of the square of the distance from that centre in w-functions.
Thus let F (r’) = (b,5 os),
2—2
12 KARL PEARSON
multiply by y,, and integrate all over the plane,
20” F (1°) xosrdr = b,, Qar i Woe Xog VO = by, {|S}. 0
0
Hence Ts if, EPO? Gucenndaen erpananseretee (xxii).
Now x5 consists of an algebraic series in (2). Thus the discovery of the
value of the integral ( “FP (7*) %..7dr depends solely on the determination of the 0
odd moments of F'(r’) between 0 and o. We conclude therefore that an expansion in w-functions involves merely the determination of moments, such as every statistician has been accustomed for years to calculate. This is not the proper occasion to deal fully with the properties of the w-functions, nor to generalise them for odd powers of 7, and to consider the convergency of expansions in terms of w-functions. They will be discussed on another occasion, but the present writer believes that they will be found of not inconsiderable service, not only in statistical problems, but. in certain physical problems where intensity round an axis diminishes with the distance.
(5) Two further problems are of service for the theory of dispersion. Suppose
F(r')= 8 (bye). s=0 Integrate over the plane and remember that y,=1,
2a | “P() Pee s 2 |” Dos @op Xo? AP 0
Thus the first coefficient is merely the total volume of the surface z= F'(7), taken over the plane. Next consider the second moment
2m |e P(r ‘yrdr= 8 an |" Bag: Gags Fh. POM, 0
Every term of the summation vanishes except for s=0 and s=1, and the left- hand side is the second moment of the function about the axis perpendicular to the plane through the centre = volume x (swing-radius)’ = b, x A®, say. Thus:
bx Kat i he RO ale i be He" (1 ae =) Pay a /0 oO Jo 20
= 2b,0° +b, (2-4) 0? =2(b,—b,) &, bx De LO er esa asatis Wa md eee ces OG onsvaantomeras (xxiv).
A MATHEMATICAL THEORY OF RANDOM MIGRATION 13
Thus far no choice has been made of o% If we take o=tk’, we have b,=0, or if o* be taken half the square of the swing-radius about the axis of the solid of revolution z=F (7°), that is if o be the swing-radius of the solid about any plane through its axis, then the second term in the expansion of F'(r’) in w-functions disappears.
We are now able, I think, to grasp the relation of the Rayleigh solution to the complete solution of the random scatter round a centre of dispersion. If ¢, (7°) be the function giving the distribution after n flights, then 4, (7°) can be expanded in a series of w-functions, 7.e.
Pn (7°) = byw, + b,0, + ba, + tee + Dy, 0.5 + aseie's
By choosing the o° of the w-functions = 44°, this becomes, since b, the volume = JN,
2 N —$77/g? $, (7) =5—se Br) {1 + Bye oes + Das Xap te fe
Lord Rayleigh’s solution provides the first term of this series, or is the correct solution to two terms in the expansion by o-functions. It possesses the properties (a) that its volume is the same as that of the complete solution, and (b) the mean square deviation from the centre of dispersion is the same, ue. 20°, as for the complete solution.
The latter depends upon the fundamental property of the w-functions that
i w,rdr=0, if s be >1. 0
The expansion in w-functions shows us at once that, whatever be the magnitude of n, the mean square deviation from the centre of dispersion is Jnl, and this gives us readily a rough measure of the range of habitat of any species for which n and J are approximately known.
Another point may be noted here as to the Rayleigh solution. That solution is the best fitting Gaussian error surface to the distribution, ze. its volume and its standard deviation are the same as those of the actual distribution, whatever nm may be. If we take the section, however, of the distribution through its axis the standard deviation of this according to the Rayleigh solution is o=J/}nl, but this is not the standard deviation of the section of the actual distribution, ae. the Rayleigh solution does not give the best fitting normal curve to the section. It gives only the standard deviation corresponding to ». It is of some value to note what .are the standard deviations of other component
w,, terms. To obtain this we must determine the area and even moments corresponding to any w,, term. Let = boa rae 1 1 sth is, s—4 ds 1 vn) Ay= |" ode = 2 5a(-itt{ “et oe, (Be) dB,
14 KARL PEARSON
whence integrating by parts: 3 1 1 3 5 1f-2 oss Avmzg ga—Ut(2-3) (0-5) (0-5) 5] Semana eat _t Pe _é Lf? gat =a (8 5) (s 5) (8 3), eda
ee eee = Jong 9 9 9, 1 rrreee ee )
S= or >l. If s=
I now take: -|°
0 and find, reducing in the same manner
Loo 1 3 { 1 se Pee Fae (s-p-5) (s-p-5)...(-p+3) pa ar ees a (2p — 1)... (xxvii).
Clearly : Moy 1,05 = QT | * at? dr by (xiii), hence :
Map -1,93 =V 2a?) (s-p- 5] (sp -5) ooh (-»+5)
x1.3.5(2p-1) Thus the odd moments of the w,, functions are known*
For the particular case when p=1:
1 = (s *)(s 4 ( : ae os === (8s - = care | Med l e |reayaelaiidlened woken XXX11), Boe Pao 2\ 2 a 2 a
if k,, be the swing-radius round the axis of the function «,,
1 a” 71 a op ( sti _~ [= plB a ce one U ( B) ( e€ ) dr,
Or,
Hence by (xxv)
Fe a (xxxiii).
* If x=r/o the following finite difference and differential equations are fundamental in the theory of the w-functions:
2 (542) — (28 + 3 —
ibumranmenae ag ated tana: (xxix), d Wo(s43) = (8 +1) w.5+ Bx ae seals Sia speemnai Wai caitnaee bacamesneies (xxx), = 1L\ dw. w+ (w+ me ae +2 (s+ 1) W., = 0
sno nh ea cera oa eae as (xxxi). But the fuller treatment of the w-functions must -be deferred
A MATHEMATICAL THEORY OF RANDOM MIGRATION 15
This is also true for s=0, as well as any integer value. It follows accordingly that while the total area of any w-function from 0 to © is positive, its & is negative for values of s>1. In other words the negative parts of w are on the
whole furthest from the axis. Again the absolute value of &,, decreases as
i i = ° . . é pa when s increases, or the higher the w-function the less it contributes relative
to its area to the total mean square deviation of the curve. Applying these results to the curve of scatter given by (x), we.
bn (*)=N (o,— Fogo, 4 BS 50n — 57
4n*~ One 192° “** T800n* °* 1892 —2125n+4 2707 :
- 103.6802! Fin GbE.) |e sane eaxateie (xxxiv),
we have if A be the whole area and & the radius, -A iho) 41 85 6n—11, 21 50n—57 J 2Qare 2 16” 247? 1024 7 1280 =n
77) =—1892—2125n+270n
=e - ORG.) vsvvedieddes (xxxv),
oe N fits Ld 1 5 6n—11 7 50n—57
J 2m 16” 24m? 1024 n® 3840 ni 7 1892—2125n+270n° + 79152 ne +ete.! oleae Gaee (xxxvi)
Hence if we even neglect terms of order s we see that the Rayleigh solution
gives too large an area for the curve of section and too small a swing-radius ; these values are
Rayleigh area, ee , Rayleigh swing-radius, o, 2/200
True area to ue : Te ae ‘) . True swing-radius to - , C (1 +~) : n 2/2no n 8n
Accordingly for n small the graph of the Rayleigh solution tends to exaggerate the concentration, 7.€. using it as an approximation we shall somewhat reduce the extreme parts of the curve at the expense of exaggerating those near the centre of dispersion.
While there is no difficulty about determining the curve of distribution when nm is large from (xxxiv), beyond the great labour of dealing with hitherto untabled functions, the investigation becomes very troublesome when » is small. The functions w are suited in this case to represent the discontinuous functions which actually form the values of ¢,(7*), but the extreme discontinuity of ¢, (7°) for n
16 KARL PEARSON
small, compels us to use a very great number of w-functions, and the convergency
of (xxxiv) is then small. Another method of determining the distribution of the dispersed population has
then to be applied to the case of ” small.
(6) Graphical Solution of the Fundamental Problem for n small.
Let us consider the general functional relation (1) Qa te | $, (1° +2 —2rl cos 8) dO. Qa 0
Suppose the graph of ¢, from 0 to nl known. This may be any discontinuous function. From nl to , it will be zero. Let ABD be the graph of ¢, and OA the axis.
a B
OP=r. Round P describe a circle of radius 7, take the radius PQ, so that the angle OPQ=0; then clearly, OQ?=7° +l —2rl cos 0; rotate OQ round O down into line OD, as ON; draw the ordinate of the graph Nq, then we have
Nq= $y (7? +2 — 271 cos 6) Qar and bans (OP!) = 5 | Ny a6.
Hence if we divide the circle up into a number of equal parts, and determine the ordinates Ng, corresponding to each of them, we can plot a curve to the base 27, of which the mean ordinate will be ¢,,,(OP’), or the ordinate at r of the new curve of dispersion for n+1 flights. This can be done for a series of values of r from 0 to n+1 2 and thus ¢,,, (7°) will be determined as a new graph. The area
A MATHEMATICAL THEORY OF RANDOM MIGRATION 17
of the plotted curve which gives any new ordinate can be found mechanically. It will be seen that the process is theoretically straightforward, but very laborious. Thus for the dispersion curve after the fourth flight some 43 points had to be found, and this involved the construction of 43 subsidiary curves and their integration.
There were, of course, graphical difficulties in the construction of the subsidiary curves in the neighbourhood of the asymptotes and various devices had to be used, but
at almost every point there were tests of the accuracy of the work. Some of these T shall now notice.
Case (i). The solution for two flights is:
a a = nape <2! | apmaueeNaaceol (xxxvii). =0 r>al }
The reader will find no difficulty in deducing this directly from the case of n=1, which corresponds to a narrow zone of radius r=/, the rest of the plane being unoccupied. Thus:
$= 5 from r=l—}e to r=l+}e
=0 from r=0 tol—de and r=l1+4e to « e being taken indefinitely small. By distributing each element of ¢, on the zone round a circle of radius J we obtain (xxxvii). The result may be obtained also from (iii) by putting n=2, we.
$2 (7") = El ud, (ur) {J, (ul)? du,
_ N[(2l+7) 7 (2l—r) ry? On Vr 2 TE (—4) =0 from r= 2/1 to 0, from a theorem of de Sonin by putting a=r, b=c=/. Compare Gray and Mathews, p. 239, Ex. 52. Case (ii). The solution for three flights may be obtained from that for two, by distributing analytically the density given by ¢, round circles of radius / about
each point. The resulting double integral is then expressible in elliptic integrals”. We find:
from r=0 to 21,
y-N 1 p(t $s (7*) = onl J rl cP ( x) : where «= 16l'r/{(r +1) (31—r)},
r>Oand <J; } N i T } ee en ee Qa Jr (.*),
where «= (r+) (3l—r)/(16?r), r>land <3; = 0 r>3l to r=oo
* This solution, or its equivalent, was first sent me by Mr Geoffrey T. Bennett.
18 KARL PEARSON
We have here at r=/ a typical instance of the discontinuity.
In Table I. columns (i) and (ii) the calculated ordinates of $, and ¢, are given, the latter having been determined by the use of Legendre’s Tables of the Elliptic Integral F. In these cases, as in the later values of the ordinates of the dispersion curves, NV is taken as unity. The dispersion curves are plotted in Diagrams I. and II. The Rayleigh solution is given in broken line; it will be noticed how very far it is from representing the facts at this early stage of the number of flights. One of the most interesting features of the investigation is to mark the gradual approximation of the discontinuous series of functions to the Gaussian normal curve of errors as the value of n increases.
The first test of the graphical method of dealing with the problem was to start from the curve for n=2 and construct the graph of ¢, The result was found to be extremely close to the elliptic integral solution obtained by analysis and calculated from Legendre, and this gave us every confidence in the correctness within reasonable limits of the graphical solution, where no such direct verification was possible. After the ordinates of any graph had been found their differences were plotted, and these difference curves submitted to most careful inspection. Larger irregularities led to a reinvestigation of the points, smaller irregularities were smoothed with the spline, and from the final smoothed difference curve the ordinates were corrected.
Another test was now possible. In every case 27] ¢, (7’) rdr ought to be unity. 0
Each ordinate was now multiplied by its 7 and a quadrature formula used to find the integral. The integral would usually differ very slightly from unity. Its reciprocal was then used as a factor to each ordinate and the ordinates so modified were the final corrected ordinates of the corresponding graph. The graphs were made on a large scale, and the accompanying Table I., columns (iii)—(vi), gives the ordinates of the dispersion curves from four to seven flights.
Additional tests were as follows:
Since Pra (P) = = ds (+P —2lr cos 6) dé,
Qa it follows that baa (0)= 5 | $, (2) d0 = 4, (2), 0
or: The axial ordinate of the n+1th dispersion curve is the ordinate at a distance J, or a flight, from the centre of the nth dispersion curve. Table IV. illustrates the degree of accuracy reached here.
The ordinate at r=/ of the seventh curve given by the expansion in w-functions is ‘0375, and this is precisely the value of the central ordinate of the eighth curve given by the same expansion. Thus the graphical method runs with surprising accuracy into the analytical. The Rayleigh solution gives °0398 for the central ordinate of the eighth curve as against the ‘0375 of the w-expansion, or the ‘0378 of the
19
A MATHEMATICAL THEORY OF RANDOM MIGRATION
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20 KARL PEARSON Tasie II. Values of the w-functions. r/o a7 W9 ow, ow, a We o* we a ang ow. 0 +°159,1550 | +-159,1550 | + -318,3100 | +-954,9300 | + 3:819,7200 | +19-098,6000 |+114:591,6000 ‘1 +°158,3611 | +°157,5693 | + -313,5589 | + -935,9497 | + 3-724,9378 | +18-530,6202 |+110-620,7235 2 | +°156,0035 | + -152,8834 | + -299,5891 | + -880,4201 | +3-449,0302 | + 16-885,5699 |+ 99-177,8011 3 | +°152,1517 | +°145,3049 | + -277,2242 | + -792,4264 | +3:016,3079 | +14°332,2150 (+ 81-601,7164 4 | +°146,9185 | +-135,1650 | + -247,7634 | +°678,3356 | + 2-464,2124 | +11:127,4046 |+ 59-905,9470 ‘| +°140,4537 | + -122,8970 | + -212,8751 | +-546,1784 | + 1-839,0999 | + 7:583,1582 |+ 36:489,3471 6 | +°132,9374 | +:109,0087 | +-174,4670 | + -404,8965 | +1:191,1906 | + 4:027,9574 |+ 13-802,7339 7A + °124,5713 | +-094,0513 | + -134,5401 | +-263,5330 | + °569,3039 | + -767,7288 |-— 5:975,6743 8 | +°115,5702 | +-078,5877 | + -095,0449 | +-130,4593 | + -016,0640 | — 1°947,9139 |- 21:205,3207 9 | +°106,1526 | + -063,1608 | + -057,7497 | +-012,7166 | — -435,8815 | — 3-949,8656 |- 30°951,7904 1:0 + -096,5323 | + -048,2662 | + -024,1331 | —-084,4658 | — -766,2251 | — 5-161,4614 |- 35-039,7166 1:2 + °077,4690 | + -021,6913 | — -028,0128 | —+206,6600 | —1-:045,7098 | — 5°351,9170 |— 28-874,9613 14 + °059,7326 | + :001,1947 | — -057,3194.| —-235,2026 | — -900,0451 | — 3°455,1198 |-— 12°119,1731 16 + 044,2510 | — -012,3903 | — -065,5623 | —-194,3306 | — -521,5103 | — -916,7705 |+ 4:126,7483 1:8 + 031,4966 | —-019,5279 | —-058,4451 | —-119,4328 | — -116,5429 | + 1:050,8391 |+ 12°770,4428 2-0 + 021,5393 021,5393 | — -043,0786 | — -043,0786 | + °172,3144 | + 1-895,4584 |+ 12-751,2656 2-2 + °014,1523 020,0963 | —-025,8081 | +-013,8001 | + +295,4776 | + 1:723,4411 |+ 7:400,1858 24 + :008,9341 016,7961 | —-010,9496 | + -043,9712 | + -279,7081 | + 1:008,2741 |+ 1:194,4811 26 + :005,4188 012,8967 | —-000,5180 | + -050,7478 | + -188,3692 | + °246,6709 |-— 2°829,6050 2°8 + 003,1578 009,2208 | + °005,3253 | + -042,6344 | + -083,3863 | — -:258,5489 |-— 3-915,1831 30 + :001,7680 006,1880 | + -007,5140 | +-028,5090 | + :003,6465 ; — -439,7348 |-— 2-949,4384 3:2 + °000,9511 | —-003,9185 | + -007,3562 | +-014,7914 | — :038,3979 | — -385,6461 |— 1:308,1481 3°4 + 000,4916 | — -002,3498 | + -006,0410 | +-004,6874 | — -048,6501 | — -231,6523 |+ :007,0283
I owe this preliminary table of w-functions to the kindness of Dr Alice Lee. Much more elaborate tables will have to be calculated, if as I anticipate the w-functions are found valuable for
other purposes. The present table suffices to indicate their general numerical character, and enables
one to calculate some of the quantities needed in the present memoir.
graphical construction. The fact that the central ordinate of the sixth curve is almost identical with the ordinate of the fourth curve at r=/, seems conclusive as to the general accuracy of the process.
The above test of the general accuracy of Mr Blakeman’s graphical work is only a part of the still more sufficient test that in the seventh curve the graph and the w-expansion practically coincide. See Diagram VI. After r=5/ the two curves cannot be distinguished, and between r=0 and 3/ the deviation is probably as much due to the neglect of higher w-functions as to errors in the graphical treatment.
_ Another method adopted by Mr Blakeman for testing the accuracy of his graphical work, especially at the end of the range, was to obtain expansions to ¢, (7°), when r does not differ much from nl, =nl—& say, where € is supposed small. If f, (€) =¢, ((nl— €)’), then generally for € small :
_ N ‘é\e-on ; Srl(Q= ra —=7e( ; ) BBG 00 Ae ene ee (xxxix),
1,= [cos 6d6 = (4 (n+ 1)) 1 (3)/T(
wl —~ 3 + bo ~— =
where
A MATHEMATICAL THEORY OF RANDOM MIGRATION
TaBLE IIT. Table of the v constants. m=1 n=6 n=7 n=8 Vy — 041,666,667 — 035,714,286 — 031,250,000 V% — 003,086,420 — 002,267,574 — 001,736,111 Vg 000,602,816 000,470,724 000,376,383 “Yo 000,104,167 000,067,796 000,046,522 Vig + °000,001,412 — 000,000,142 — °000,000,639 Table of the N constants. m=5 n=6 n=7 n=8 N, — 008,333,333 — 007,142,857 — :006, 250,000 N, — 000,123,457 — 000,090,703 — 000,069,444 N, 000,032,600 000,024,174 000,018,636 Ny 000,000,990 000,000,627 000,000,422 Ms, — ‘000,000,078 — ‘000,000,047 — ‘000,000,033 m=10 n=6 n=7 n=8 N, — 004,166,667 — 003,571,429 — 003,125,000 NV, — 000,030,864 — 000,022,676 — 000,017,361 N, 000,008,415 000,006,211 000,004,771 Ny 000,000,126 000,000,082 000,000,053 Ny — 000,000,011 — 000,000,007 — 000,000,006 m= 20 n=6 n=7 n=8 N, — 002,083,333 — 001,785,714 — 001,562,500 N, — 000,007,716 — 000,005,669 — ‘000,004,340 NV, 000,002,137 000,001,574 000,001,207 No 000,000,016 000,000,010 "000,000,007 Ni, — ‘000,000,0014 | — -000,000,0009 | — -000,000,0006
TasBLe IV. Central ordinates and ordinates at r=.
No. of Flights
Central ordinate Ordinate at r=1
0 rr) 00 0585
0585 oe) 00 0537 -
0537 0537
0538 0415
0415 0378
21
22 KARL PEARSON
N (E/D2O- —3) (/2n)"" 2/n 2"-* T (4 (n—1)) This can be proved by induction.
ase atime eee (xxxix bis).
Hence : RLE\=
For most of the cases more approximate formulae still were deduced. Thus:
A(é= ie (1 - + 2) che kp pion aeeismnaAN eo aaGDeS Rs (xl), A(O= vi ae (45 = +S) poieie aioe: (xli), f(é)= ie (+) (1+ Se sas) Leauniteseeeed ean, hO==a5($)- (1+ - a) ee (sliii),
after which the first term only as given by (xxxix) is sufficient. It will be observed that after ¢, (7°), the curve touches at r=nl or €=0, and the contact becomes higher and higher as 7 increases. Thus, although short of n=, there is no real asymptot- ing to the axis, still ¢, (7’) for n>5 not only vanishes for r=nl, but has increasingly higher contact as n increases. This explains how the Gaussian curve can fairly well represent the state of affairs towards the end of the dispersal range, if n is> 5.
Mr Blakeman found that the ends of the range for the various cases ran closely into the curves (xl) to (xliii), and they were tested and, if needful, corrected by these formulae.
Thus the whole graphical work was kept in check, and, I think, we may be confident that the true forms of the dispersal curves for n=4 to 7 are really given by our diagrams and tables.
(7) We may note a few features of these curves.
Dispersal Curve for Two Flights (Diagram I.).
There is no discontinuity in the solution from r=0 to 2/, the range within which all individuals fall. The curve asymptotes to the vertical at the axis and at r=2I. Of course, while the density becomes infinite, the number on any small area near 7=0 or r=2i, is finite. Thus the number between the circles of radii 7, and 1, is
2N mak Te — 1 71 3)
= sin 5] ~ Sin” 57). If r,=0 and r,=«, where e, is small, the number », within the small circle of radius e, at the centre of ene If r,=7,+.6, the number lying on the zone of
€,
-3 breadth e, is Ns ( 1 - Fi :} , and this if 7, =2/—e, is vy, = — T° At the position of
A MATHEMATICAL THEORY OF RANDOM MIGRATION 23
minimum density 7, =,/2I, and the number on the zone 1, to 7,+¢ is »,= e,|[(al/ 1/2). Hence it follows that the numbers on narrow zones «, ¢,, €, in breadth, of equal areas me) = wAle, =72/2le,, are. given by
Ne|(al), Nve/(rVl), and NewJ/2/(z/),
or in the ratio
Ne,/(nl) :4.Ne,/(rl), $Ne,/(nl) x -
Thus the total population on a small area at the centre of dispersion is twice that on an equal area at the periphery of the distribution, and at both indefinitely greater than on an equal belt at the distance of minimum density. The same point can be indicated in another way. From r=0 to r=4H is 7y of the total area occupied after dispersion, it contains 16 or about 4+ of the total population; from r= 3/ to r=2l is zy of the total area, it contains ‘54.N. In other words the half of the area nearest and farthest from the centre of dispersion contains 7% of the dispersed population; the “middle” half of the area contains only 3, of the population. The nature of the distribution is thus extremely different from that given by the rotation of the Gaussian curve about its axis for this small number of flights. For in the Gaussian case if the central area 7e2=2zr,e,, the area of the zone at distance 7,, the population on the centre patch is $Ne?/o’ and on the zone is
4 Ne}/o° x ele", which is always less and diminishes continuously with -increase of 7,. Thus the Rayleigh solution fails in this, as in the next three cases, not only to give the
form of the curve at dispersion, but to indicate that the dispersed populations on zones of equal area round the centre do not decrease uniformly in number.
Dispersal Curve for Three Flights (Diagram II.).
The solution is discontinuous at r=/. The density is here infinite, but has become finite at the origin. There is no discontinuity at r=21, but at the end of the range the density drops suddenly from a finite value to zero. Thus the in- tegral of the Bessel function product (see Eqn. (iii)) is discontinuous at two points. The Rayleigh solution is still widely divergent from the true curve of dispersal.
Dispersal Curve for Four Flights (Diagram III.).
By the rule already referred to (p. 18) the infinite density has returned to the origin. There are only two points of discontinuity, ae. at r=land r=4l the end of the range, at both of which there is an abrupt change in the slope of the curve. The density at the end of the range is now zero and will remain so, but the dispersal curve rises at right angles to the axis. The true dispersal curve is bending round somewhat to the Rayleigh curve, but the latter is not even yet a rough approximation to the facts.
24 KARL PEARSON
Dispersal Curve for Five Flights (Diagram IV.).
All infinite densities have now finally disappeared. The density vanishes at the end of the range, but the dispersal curve makes a finite angle with the horizontal axis. There is a marked discontinuity of slope at r=/; a still more noteworthy feature is that from r=0 to r=/ the graphical construction, however carefully reinvestigated, did not permit of our considering the curve to be anything but a straight line. If this could be verified from the analytical expression
B.C) =F wa ur) (Fe (ul) au
by showing that the integral is independent of r from 0 to / it would be of much interest. Even if it be not absolutely true, it exemplifies the extraordinary power of such integrals of J products to give extremely close approximations to such simple forms as horizontal lines.
The approach of the Rayleigh curve to the result is now more noticeable.
Dispersal Curve for Six Flights (Diagram V.).
There is contact now of the first order at the end of the range. From r=0 to r=l the curve of dispersal appears to be a sloping straight line tangential to the continuous curve from r=/ to r=6l. No other discontinuity of a low order is now visible. The curve, except for the finite slope at r=0, is becoming much more of the Gaussian form. It runs fairly closely to the solution in o-functions up to @,, in fact is not separable at the extreme part of the range, where the Rayleigh curve still gives finite ordinates beyond the possible range.
Dispersal Curve for Seven Flights (Diagram VI.).
All sign of discontinuity has gone, the curve is horizontal at the centre of dispersion and might be easily mistaken for a normal curve of errors. The expansion in w-functions represents the result within the limits almost of constructional error. It was not thought necessary to continue the graphical work beyond this stage. We may conclude that:
The deviation of the Rayleigh solution for seven and more flights from the true dispersal curve is practically the same as its deviation from the solution in w-functions when five terms of that series are retained.
This I think completes the full solution of the fundamental problem. The dispersal curves for the cases of 2 to 7 flights are given in the Table I. of ordinates and the Diagrams I. to VI. For higher values the w-function series gives the solution. This solution could be applied to calculate the ordinates of the dispersal curve for fewer flights than 6 or 7, but several more w-functions would have to be used and the arithmetical work—especially while these functions are as yet untabled*—then becomes somewhat severe.
* Table II. provides a preliminary series of values of the w-functions.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 25
(8) Secondary Migration Problems.
Problem I. On one side of a straight line there is supposed to be a uniform distribution of habitats; on the other at starting no habitats. To investigate the distribution in the unoccupied area after one migration. Each individual is supposed to take n-flights to the new habitat.
Y A P Su ee B |< ¢ ——30 Cc Y
Let YY be the straight line and O a point at distance ¢ from it on the unoccupied side of it. Let N be the average density per unit of area on the occupied side. Then after an n-flight migration, the contribution from P (co-ordinates r, x) at O will be NrSy8r ¢, (7°), and integrating this all round a cirele of radius 7 from A to C within the occupied area, we have for the quantity F,(c) at O
© eos” tefr P,(c)= 2Nv| I dt nae ¢ JO
= an{ cos”*c/r ¢,, (7°) r dr.
Hence dB (2) = —2N i pa alc de c al -—¢
ON | AXPRGGP wom iv). a i dn (Ce +y’) dy (xliv)
The evaluation of this integral needs a further consideration of the w-functions. By (xiv)
= 1 = wi fl i" ee 2 Oy= — 55 ( B) ap \ Be , where B= si dad)
26 KARL PEARSON
Transfer the differentiations from 8 to o* and we have:
(-1)° = dé (See) o
O88 OF ( ) Toy =(—1) (0) ry tree ents (sini
or, all the w-functions can be found by differentiating the first w-function with regard to the standard-deviation squared. Then a (xii) we have
dn (r)=41 +, (0°) ey vy, ( oa Tey +... +(-1) rx (a's oy fs }o,
Thus, if we put o’=t:
dF, Cc a : ad 2 : ds o : ; na )_ = 2 (1 +0" 5 — vg! Se + ot( 1p aye ae ) x | ele +y’) dy.
ole) dy= 2 [ e-@+' gy = 1 Q- toa Jon But : [, 0 (ey) dy= 525 |e dy == e 1/2 o
2 om $C?/07,
lane
c ele Now i ent” dose i e *” da, and this integral vanishes when c=. Hence
N 2 3 ds 2 _ 12 F, (0)= om (14 — uf Ft (-1 nt SZ +) [oe tay
Since F,(c) clearly vanishes for ¢ infinite, it is not needful to introduce a constant.
It remains accordingly to determine the successive differentials of the integral with regard to ¢. Call the integral 7; then, if n=c/o=c/,/t, di —t7 dy le _jar _¢ a —¢é. a ~ 3 8° = oy = 7eu//t. By (xlv) we know that d*w,/dt*=(—1)*,,/t%. Hence differentiating s—1 times we have:
di, me /d**w, (s—1)(s—2) dw, 1.3 1 ae it (a =e 1) ae we Sra) aI ar pF)
ayes s—1)(s—2 158
A MATHEMATICAL THEORY OF RANDOM MIGRATION 27
dl d% = s—1)(s—2). , Thus # 5% =(-1)" 1° (cag. ae) = eae {ee ae ne a "= ae = 8) 1.3.5 Wass_4 + ete.) rr (xlviii). Substituting in (xlvii) we have, if ()=—/ | P pNP Te 22 chiacane (xlix), Qn ;
— /¢e = &, ()=N| a (<)-,/2% — {o* Wy (4v,4+ 2,4 42v, + 408 Vay + % en LB Vy») +o°o, (y, ++ LY, — IP 1% 5 525-y,.) +o° W, (v, + $5 + 30+ Thr.) + o’a, (v, + 2) + 4% r.)
+.0°O, (Vp + $Y») + PO Vy ++. 3] Swsdniaws abeg eu auoonde see (1),
as far as coefficients of the order v,, and functions of order w,.
This is the solution in w-functions. Table III, p. 21, gives the values of the v’s for certain values of n, and Table II., p. 20, is a preliminary table of the w-functions. These will enable us to readily find the values of F,(c). I have done this for the case of n=6 and n=7, which will suffice to illustrate the character of these curves. (c/o) can be found at once from Tables of the probability integral. It is drawn with a broken line in Diagram VII. and is the Rayleigh solution for this case. I term F,,(¢c) an “infiltration curve” of the first order.
Substituting the values of the v’s from Table III., we — for n=6: F,(c\/N=% (<) +2 © {026,712,414 (oe,) + 053,325,539 (o*w,)
+ 002,114,303 (o7w,) — 001,029, 898 (o%e,) — 000,134,978 (o%«,) — 000,001,770 (c°w,) + «.-}s and for n=7:
F,(c)/N=$ (¢) + <{-022,850,925 (o*«,) + 045,644,347 (o%,)
+ 001,578,008 (o°w,) — 000,758,570 (c°w,) — 000,084,525 (o°a,) + 000,000,178 (o°w,.) + ---}.
The first term (=) is the ogive curve already drawn corresponding to the
Rayleigh solution. We see at once that the term o’w, will not affect the
fourth place of decimals. 4—2,
28 KARL PEARSON
TaBLe V. Ordinates of Infiltration Curve over straight Boundary.
+elo n=6 n=7 n=o0 -¢elo n=6 n=7 n=O 0 5000 5000 5000 — — — —_ ‘1 4614 4612 4602 - il 5386 5388 5398 2 4231 4228 4207 - 2 5769 ‘5772 5793 3 3854 3850 3821 - °3 6146 6150 6179 4 3488 3483 3446 - 4 6512 6517 6554 D 3135 3128 3085 - 5 6865 6872 6915 6 2797 2790 2743 - 6 7203 7210 7257 7 2478 2469 2420 - 7 7522 ‘7531 7580 8 2177 2169 2119 - 8 7823 ‘7831 7881 9 1898 1889 1841 - 9 8102 8111 8159 1-0 1640 1632 1587 -1:0 8360 8368 8413 1:2 1193 1186 1151 -12 8807 8814 8849 1-4 0834 0830 0808 -14 9166 9170 9192 16 0558 0557 0548 -16 9442 9443 9452 1:8 0356 0356 0359 -1°8 9644 9644 9641 2:0 0215 0214 0228 - 2-0 9785 9786 9772 2:2 0121 0124 0139 — 2-2 9879 9876 9861 2-4 0064 0066 0082 — 2:4 9936 9934 9918 2°6 0030 0033 0047 —2°6 9970 9967 9953 28 0015 0013 0026 — 2:8 9985 9987 9974 3:0 00046 00060 00135 -30 99954 99940 99865 3°2 00012 00020 00069 —3:2 99988 99980 99931 34 “00000 00004 00034 -3-4 1-00000 99996 99966
n= oo is used to denote the Rayleigh solution.
This table suggests some interesting points. The curves for n=6 and n=7 are fairly close together, but differ sensibly from the Rayleigh solution, perhaps 4 or 5 per cent., where the density is at all material. For many practical purposes this might be close enough, and we see that for infiltration as distinct from dispersal curves, the Rayleigh solution—owing to integration over an area —gives fairly close results. The greatest percentage deviations from the Rayleigh solution are to be found in the tail. Now no individual can be found beyond
the range ni from the boundary, and o=J4nl; thus the maximum range is /2no, or, for n=6 and 7, the maximum range is 3'460 and 3°740 respectively. The w-function expansion brings this out well. For n=6 at 3'4o0 there is not one in 100,000 individuals, while the Rayleigh solution gives 34. For n=7 there are still 4 in the 100,000, because we are a little distance still from the limit of
A MATHEMATICAL THEORY OF RANDOM MIGRATION 29
the range. The Rayleigh solution continues to give sensible densities beyond the range, although they may be sufficiently small to be neglected in practice.
For rough purposes a first approximation to the infiltration curves may be found from the Rayleigh solution, they will err on the side of safety if we are con- sidering the effect of a clearance at a considerable distance from the boundary. But with the aid of the tables of the w-functions and the v-coefficients, it is not difficult to obtain the actual form of the infiltration curves as I have done in the present case. Diagram VII. compares the Rayleigh approximation and the infiltration curve for n=7.
It will be seen that an infiltration curve of the first order gives not only the density of the population after a first migration into cleared or unoccupied area across a straight boundary, but also the diminution of density on the populated side of the area, when we put ¢ negative, 1.e. it gives both the ‘depopulation’ and ‘repopulation.’ The reduced density at the boundary is $N, and if we take the point where the infiltration curve cuts the vertical through the boundary as origin, we see that it is centrally symmetrical; or the loss of population at a given distance from the boundary is exactly equal to the gain at the same distance on the opposite side of the boundary.
If we require an infiltration curve of the second order, we must now multiply the ordinates of the curve of the first order by (i) the average fertility of the species, say p, and (ii) the survival rate A. If the environment be the same on either side of the boundary, and neither w nor A affected by the density of the population, then wA may be treated as a constant and the infiltration curves of higher orders can be found with moderate ease for simple cases. We thus have the distributions after two, three or more migrations accompanied by reproduction and death. On the other hand both » and A may be functions of the density of the population, and in this case the ordinates of the infiltration curves of the second and higher orders can only be determined when the nature of m and A is known. On the whole it is probable that the average fertility depending on the mating frequency will be highest where the density is greatest, as mating opportunities will then be most frequent, but in such cases the survival rate A may be lower, as more enemies are likely to be present and the food supply is also likely to be less, where the population is densest. Thus pA as a whole may not be very different on the depopulated and repopulated sides of the boundary. We shall only consider in this memoir cases in which this product is (i) supposed constant throughout, or (ii) constant for each migration season ; but supposing uniform environment on both sides of the boundary, it is conceivable that pA will be correlated with the population density and this will modify the basis of the distribution from which the second and later migrations start.
30 KARL PEARSON
(9) Problem II. To investigate the distribution after m migrations from uni- formly densely occupied space across a straight boundary into unoccupied space.
Let the axis of « be taken perpendicular to the boundary and the axis of y be the boundary. Let us consider the density at w=c, on the originally unoccupied side of the boundary. Then the density at a distance x from the boundary is given by (xlvii), or if we write the operator as Q,, we have
F, (2)=NQ | Re a a cee (li).
Here Q, involves only n and o and not a. Now the distance r from the point x, y to the point c, 0 at which we want the density after the next migration is given by: Payt(e—ey, and wA being the fertility-survival factor, we have for the density at c,
y= i a pAu®d,, (7) dady.
Now bs (7) = Qan= gage.
To mark that this Q, operates only on this part of the expression, write it @/ and suppose it to operate on o’ written for o. After the operations are com- plete we can put o’ again =o. Let
1 ots d. ae = ime a Then if pA be constant (see p. 29):
ee EAN a” aie v, pe Mele” dandy,
—-@
Completing the — with regard to y we have:
+0 ji Uy ee OO) ite v, eB” de,
Differentiate with regard to c: 2 , prdN sacar er Jie diie! U. -1— Q,Q/ [. uaa (-e $(a—c)/ "ae Integrate by . and notice that the part between limits vanishes at both
of them and we have: du, = praAN
te de, 1-4 (a-)jo" oo JO Q:0/ 3 dae” dx, dy, _ LL a ajoy, But dx coe see a é ’
+O ae (a—c)? hence : ee Q.0; pa i e (G+ a” ) dx.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 31
This is integrable and gives:
du, ae 40 Io? +0°2)
> MAN OO RE Integrate with oe to c, and remember that u,=0 if c=oo; thus:
m= aN QO! 5 | en bette) a
y a+ se =pAN eo 3” pANQ,Q/ Jon = ee Comparing this with (li) we see that u, differs from 1, by (a) the introduction of
the factors Q/ and wA and (6) the replacement of o in the lower limit by Jo? +o”. The process can therefore be repeated as often as we please, and we have for Um the value : 2 1 —Ly? Um = (wA)”*NO,O/O,"... to m terms —— et ae, (uA) Q:0/ OQ; 0 erm: Jan i
where YVa=o0t+o"’+o'7+... to m terms.
After the operations indicated by the Q’s are completed, we are to put sion So" == a. Now it is clear that a differentiation with regard to any o° is precisely the
same as one with regard to 2% We can therefore write for all the Q’s the simple expression ds 1+, (0°) ic —v, (0° Pi, pice Col (oo a (St understanding that d/d(=’) operates only on = and that after the operation is
completed we can put 2=/mo. Thus the complete solution is:
tg = (uasy* (1-4 (0 sep” (o*)'5 yt we t(—1) at a)
This is true for ¢ positive or negative, 7.e. whether the density be considered at a point on the originally occupied or originally unoccupied side of the boundary.
Up to terms of order 1/n*’ we have for the operator the value
1 +m (vg? — vg? + %69' — Pg’ + Png’) m(m—1) 1.2
ge a (no z vq, where g stands for o°d/d (2").
+ (v,2q4 — 2v,y,q° + 2v,v,9')
32 KARL PEARSON
Now exactly as on p. 26 we may show that:
1” 4-40 g (TV? 1 =) 6-4 S Fa [gat = 8) Jeg ae en
— 1 (=1)? 4? Tan 2m? Nine Woce—1) (Mm)>
;—1 —1)(s—2 where : Wale (71m) = X2\s-2) (71m) + ae + X2(e—2) (71m) oe 1.3. X2(s—-2) (Im)
s—1 —2)(s-—3 + Ms- it 11.3.3 ea (Mm) Fees
1m=0/(/mo), and x, is defined on p. 10, Equation (xviii). Thus 1
tig = N (ws )" 1 (rn) = 57pm toe (7 veh) + 20)
ln tem (m— 1)»;
i)
mv,,+m (m—1) vy,
a me We (7m) 4 Mn tm (m—1) vv, tom (m — 1) (m—2) vf hy (10))} puna (liv).
We see that this expression converges much more rapidly than that for ¢, (7), if m be at all large.
The result (liv) might have been reached in a different manner. We might have supposed the (uA)”""'Na individuals to have started from any element a on the populated side of the boundary and taken mn flights without multiplying to their final resting-place. The effect of this would be that o?=4mzml’, and that in the values of the v’s we must write mn for n. But doing this gives us precisely the coefficients of the ws in (liv). Thus (liv) is deduced directly from (xlix). The proof becomes then much shorter, but it is more artificial ; the fact that we may suppose all the unborn individuals to scatter from the original centre is not so easily realised, and further it does not in the process picture what takes place until the final arrangement after the mth breeding cycle is attained. In the method I have adopted we see the exact process of each breeding multiplication, its increase of the operating factor by an additional Q,, and its increase of the square of the standard deviation by an additional o”. Lastly the final form (liv) enables us, without recalculating the v’s for each breeding cycle, to see very easily the effect in the case of any n-flight species, of taking any number of breeding cycles.
So long as we keep wA constant of course our result for m breeding cycles with n flights will be the same as for a simple scattering for mn flights of a larger number of individuals. If A varies, however, we must adopt the method indicated in the above proof, and work out each migration successively, The same method must be adopted if a patch be rendered permanently sterile, because
33
A MATHEMATICAL THEORY OF RANDOM MIGRATION
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‘nsuagq haopunog fo sua, ur aw wounibuuwy ay) of paunspau huppunog oy) wouf sarmsuagr uaarb fo saounisig “[A WAV],
34 KARL PEARSON
in such a case » is not constant for all parts of the integrated area, and we cannot suppose the whole final population to scatter from the original centres.
If we neglect the y,, W,... terms in (liv) we have the value which would follow from the Rayleigh solution of the fundamental problem, and this can be very readily expressed in geometrical terms. For we mark at once that u, and u, are in type identical curves. Take wu, and stretch it vertically in the uniform ratio
of wA™-* to 1, and horizontally in the ratio of J/m to 1, and it becomes tp. In other words the broken line on Diagram VII. represents the approximate solution in this case after m migrations provided we read N(A)" for NV on the vertical scale and Y=/mo for o on the horizontal scale. The Table on p. 33 gives the chief results.
The unit of this table is the length / of “flight.” It will be desirable to illustrate its application. Any such application can be of course only a suggestion, and on this account the above Table has been calculated to only a few places of decimals. But such suggestions may not be without value. They will become more than suggestions when our knowledge is greater of the migratory habits of different species. At present only rough approximations can be made as to the values of n and J, and these admittedly are of small weight.
Illustration I. In captivity I have noted that H. aspersa will live for five years. For two years it does not usually lay eggs, and then it will generally, but not invariably, reproduce twice in the year. This is of course subject to claustral conditions, and while these seem in some cases unfavourable, in others they may be advantageous both in matter of longevity and—owing to the constant food supply—in number of broods. This snail, as far as my observation goes, appears to return to the same shelter after seeking its food. Leaving such “flitters” on one side, I think we might look upon thirty to forty yards as a maximum “flight” for such a snail and regard seven or eight such flights between its egg layings as on the average an exaggeration.
We might therefore take /=40 yards, n=8, and an average during life of one brood a year as being quite possible approximations in the case of some snails.
This indicates that the progress across a boundary into unoccupied country would be such that 1 per cent. of the density at the boundary and, therefore, possibly 4 per cent. of the density in the fully-occupied country, would only be reached at 2061 yards from the boundary after 100 migrations. In other words, such a species would only progress a mile or two at most in a century. Such progress would hardly be noted in any studies hitherto made of distribution; the limits of a species a hundred years ago were certainly not closely defined to a mile or two, even if they have been recently. Of course there are many other ways in which a slow moving species can be transported than by its own “ flights,” and further no special stress is laid on the above case, but a study of Table VI. shows
A MATHEMATICAL THEORY OF RANDOM MIGRATION 35
that the advance of a slow scattering species* may be comparatively small, The inference can accordingly be made that the existing boundaries of the geographical distribution of certain forms of animal and plant life which are not marked by natural barriers, and which do not correspond to obviously changing environ- mental conditions, need not after all be associated with subtle physical differences which have escaped the observation of the naturalist. The species may be pro- gressing into an unoccupied area, but at a rate hardly observable in the time during which accurate distribution observations are available. If this view be correct we should expect such boundaries with no apparent environmental change in the case of species for which we might reasonably predict a small n and J.
Illustration IIT. JI have endeavoured to apply the above theory to the im- migration of mosquitoes into a cleared area. We will suppose in the present treatment that the area bounded by a straight line (some attempt to allow for curvature of the boundary will be considered later) has been cleared but is not kept sterile to the species. I shall speak of a district as rendered sterile to a species when it is made impossible for it to breed there, and kept sterile when the breeding possibilities are persistently destroyed. The distinction is an important one, especially in the mosquito case. For in the latter case all mos- quitoes are immigrants, and in the former case we have not only immigrants, but their produce.
Major Ronald Ross, who has most kindly provided me with information as to mosquito habits, makes the following remarks :
(a) That the number of mosquitoes produced varies roughly (ceteris paribus) as the extent of surface breeding area.
(b) That the breeding area can be taken as consisting of numerous isolated small pools or vessels of water scattered fairly uniformly over the country.
(c) That the feeding places (houses, stables, birds, etc.) may be taken as scattered pretty uniformly between the breeding pools.
(d) That abundance or scarcity of food can scarcely influence the question much. A single man or bird will yield enough food for many mosquitoes, and if they starve it is not because the food is, not there, but because they cannot reach it. They are therefore not likely to be drawn in general by special abun- dance of food in any special direction. Wind tends to make mosquitoes “sit tight,” rather than allow themselves to be scattered.
It would thus appear that on the average an “equi-swampous” condition of the environment and random “flights” of the mosquito will not be very wide of the truth. The difficulty is to form some estimate of n and /. On these points again Major Ross came to my help, but naturally the statements he made were with great reservation.
* Of course any more quickly moving species that depends on this for food would have the same
boundary, but in its case the boundary would be environmentally defined. 5—2
36 KARL PEARSON
(a) From egg to egg (7.e from laying of eggs, hatching, larval and pupal stages, to laying of eggs again) takes roughly about a fortnight in hot countries with most mosquitoes. In England, gnats may have only one generation or two in a summer, but in the tropics they may go on breeding throughout the year. In cool countries the egg to egg cycle may be prolonged to a month or two. In certain very hot and dry countries, breeding may be checked entirely except during the rainy season. I have accordingly taken 20, 10 and 5 breedings to the year to represent roughly these conditions.
(8) Major Ross-distinguishes between “minor vicissitudes,” which an insect makes when it hovers round its victim or mate, and “major vicissitudes” which it makes when it passes from feeding place to pool for egg laying. These cor- respond to my ‘‘flitters” and “flights.” He considers that they go back to water every four or five days, so that a ‘‘major vicissitude” occurs every two days or so. We might therefore take, excluding flitters, the average number of flights to be six or seven. Of course this is the roughest approximation, but still not an unreasonable estimate of what probably takes place in the mosquito’s life.
(y) As to the magnitude of 7 we have less definite data. Mosquitoes of a rare kind have been said to have been found two or three miles from their breeding place. Major Ross thinks that Anopheles will exceptionally, when no houses are near, probably travel 4 mile for their food, or perhaps further, but he supposes the average distance scarcely to exceed 4 mile, and it may, as houses and suitable pools often abound not more than 50 yards apart, be not greater, perhaps, than 100 yards.
I have accordingly taken 100 yards and 500 yards as likely values for /, and considering 1 per cent. of the boundary value of the mosquitoes’ density as a limit to their existence and 5 per cent. as objectionable, we have the following table :
TaBLE VII. Distantes from the Boundary of a cleared but not sterile area at which 1 per cent. and 5 per cent. of the boundary density of Mosquitoes will be found in the course of a Year.
Supposed number of Breeding Cycles in Year
5 10 20
Number of Flights............ 6 7 6 7 6 7
Density 1 per cent. | = 100 998 1078 1411 1524 1995 2155
” ” » ‘|
Density 5 per cent. | = 100 759 820 1074 1160 1518 1640
A MATHEMATICAL THEORY OF RANDOM MIGRATION 37
The distances are all given in yards.
Thus we see that the least of these distances for 1 per cent. is greater than half a mile, or, if an area be cleared but not rendered sterile, we might expect within a year the mosquitoes to reappear within half a mile of the boundary, and to reach an objectionable frequency even at this distance for most of the cases considered.
_As far then as these rough numbers can be taken to indicate the state of affairs, it is needful not only to clear an area but to maintain it sterile. The clearance radius may be only 4 mile and is hardly likely to exceed a mile, and the above results only mark the progress of immigration in the course of one year after the clearance. Further the results. would be accentuated if the boundary were curved or an approximately circular clearance made.
It does not appear to me that any substantial difference would be made in the main result by reducing n to 8 or 4, although some difference would occur if / were reduced to 20 or 30 yards.
(10) Problem III. To determine the distribution after m n-flight migrations starting with a centre of population Na,
The previous two problems indicate the nature of the general solutions to which I now proceed. I shall adopt the longer process of proof in this first case as being the more suggestive.
By (xii) and (xlvi), calling the operator as before Q,, we have for the distribution at X, Y due to a centre at the origin:
(X,Y) =32 Q, (ae Per a) Queer (lv).
Hence the distribution at (fh, &) after a second migration of n flights is +o to Gg te ee a Pn (h, k)=pA [i [60% so > AXAY. Call the Q, in this Q,, and write the o* on which it operates o,'; call the Q, in io, (X, Y), Q, and the o° on which it operates o,’, we have: eles ~he+(Y— He aan
o) oe? a1? sb, (h, by =" 0.0, [- ie sae dX aY. The integrations can be performed and give us AN —h(BPLB (oto? ‘ aha (hy bt) =" 0,0, mam BUN oa) ace (Ivi).
eak?
This only differs from ,¢, (X, Y i" by the introduction of o,’+0, for o* and of the factor pAQ,,.
38 KARL PEARSON
We can accordingly repeat the process as often as we like and we have:
A)y™ON 24 fe re mPn (h, k) = a QV, + Que 5 se ee nee (lvii), where Maelo t of + w.ton,
and after the operations have been performed we are to put all the o's equal to o or }’=mo*. But no operator Q, affects any o*® in any other operator, and
d d Ag Sey ‘ doi = as" Thus (c/) d(oay may be put =(o”) qSy and this makes all the operators identical in form and we may write
2\2 dl 2\ 3 ad ns : Q, Qe Qa= {1 + v,(0") Ts %(") d(sy trl") d (=*)'
+... 4+(— 1), (0°) Test vf
d? 2\3 a 2\4 a aay Ne(o") aay t No(o’) d (3°)
=1+4+N,(o°)
d’ = SAT 2\s +... +(—1)*N,, (o’) dey In this form of the operator we can now write at once ale and call
the expression Q,”.
2\2 a? 2 a 2 dt Thus =14+N,(% ter aay Ns N, (2) A aay +N he Va di (3) F Sas a +...+(—-1)N,, (=?) a(syt ee (Iviii),
where ae Vs; n= Vsy - m m
yc ee) aa fear acl (ned) YP
m m N., _ Mv + mm Sle tom cm ee) a4 CGC. gaeiasun (lix).
These values of the V’s rapidly converge and their values are given in Table ITI. on p. 21 of this paper with those of the v’s for a few values of n and m. As we have seen on p. 32, they are the v’s obtained by using values of nm for n.
We now have the general solution of distribution from a centre: pis m-1 Na m 1 — 3 (+) /s? mPn (hy k) =(pA) = Q: xe es
= (wA)™"*Na (Q,+.N,0,+.N,O,+ ... + Nog Qee +...) ...(ix).
This is absolutely identical with (xii), except that the constants v are replaced
A MATHEMATICAL THEORY OF RANDOM MIGRATION 39
by other constants N of known value, and in every w-function we are to replace o by mo’ or >, that is to say a uniform stretch in the ratio of /m to / is given to any surface z=w,, parallel to the axes of « and y. This is denoted by writing ©,, for o,,.
If we confine our attention to the Rayleigh part of the solution—which will be more and more nearly exact as m increases, for the N’s rapidly decrease in value—
then we have Pn (Ui, Jes (pes PUN a) coated a Us seadae eyes de (1x1),
and we see that every density gradient curve for the successive migrations is to be obtained by a stretch from the first migration density curve.
In general, however, this result is not absolutely true because the different components of the true solution are mixed in different proportions, the N’s being functions of m. We see, however, that the stretching rule becomes more and more accurate, as we increase either the number of flights or the number of migrations.
(11) Problem IV. To find the form of the general solution for the distribution into surrounding space after m migrations of any population initially spread uniformly over any given patch with density N.
The density at h, k, after m migrations due to a centre Ndady, is by (Ivii) above
_ 7 a\m-1 Ndady ml )-4 (eh +(y— by 3? = nay Te OF se :
To give the patch let x be integrated from v, to v,, where v, and v, will usually be functions of y, and then let y be integrated from u, to %,. We find:
mfg (by B) = (way? AOE a nf [Ee He MOP dad... (lati).
This is the general form of the om when the population spreads from a uniform patch into non-sterile surrounding country.
If on the other hand we want the distribution after m migrations starting with a cleared patch, which is not kept sterile, we have
ll, Uh, B) SGIAYOIN Sy Fig Wie BY saseoiveneanvevennes (xiii),
for the whole district would have had a uniform density of (uA)”""\V had there been no clearance. Hence
F,(h, b)=(wayr"NQ? (1-5 ah ro G=he> "9 daedy).. (xiv).
Now F(t, #) = NQ/ (1-x i Le : ete Re NM pdy).
40 KARL PEARSON
Hence the rule: If the solution can be found for a single migration, replace o° by mo’, and each pv by the proper N, multiply by the factor (uA)”, and the solution for m migrations is deduced.
It will thus be clear that, if the solution can in any case be found for one migration fully, we can at once extend it to the case of any number of migrations, with constant fertility-survival factor.
(12) Problem V. To determine the distribution after a first migration into a cleared rectangular area,
Let the area be the rectangle 2a x 2b, and the origin be taken at its centre and axes of « and y parallel respectively to the sides 2a and 2b. Then the density at any point A, k, after a single migration F’,(h, k) is given by the principle of
the last problem by Feliy B= NSB nd) aastiiccovsiasdarveniess (lxv),
where F,, (hf, k) is the distribution from a uniformly occupied rectangular area into surrounding unoccupied space.
But F,(h, k= N u i le ba {(e— hy + (y— 8) dady
+b ~9 = x) vas" Hie dedy
+b ny No: | g-ble- Mot gy. i Me ee
Let P,(e) stand for the ae integral
u [; e 2 da. Joa
1 TO ep RP? 1 ene — aoe Then : | gre fea e 2 da JV 20 J -a Q7 bee
1 (a-h)le ((athle\ 4a ee d JQa a 7 [ } i c neesAyan (8 Thus:
F(h, E)=NQP, (=) +P, (=) {p, ("—*) +P, C2*)\. Kiev:
Now consider the differentiation of P, (3) with regard to o°.
d U d 1 _1,y? lu 1 liye P, s)he {r 2" dae 2a pee) | (“ ede Jon 20/27
A MATHEMATICAL THEORY OF RANDOM MIGRATION Al
Writing o’=t as before, we find
7 {P, ()} ee ia eR devi Hence ae {P(5)} = —3 vB dom (5) = : r(— yu (-» ig 1) 4-9), -e eS) is = a(s—3) = co eer) i! . {P, ()} - ; re Eire" Wie) oe aenml ae (Ixviii),
the expression being the same as that on p. 32.
Now let us write the following for brevity where n=u/o:
1 1 SU ai et
(Vth (7) a3 Ve, (7) Tees + 25 Wy (s-1) (7) + ves
1 il _1,2 Daly) =) age ¥ (2 Bohl) on rade (a) bo)
1
L, (7) =; jee (30+ Tae) oie
1.2
Lies (n) + i) ‘
L,(n) = 575 c= ne *” (a, + 109% (9) +--+ +e) Vos ois-4) (9) + 1.) (lain),
and so on. All these functions are directly expressible in w-functions as
on p. 27.
ds 1 1 a Further let P,(n)=(— 1 5 P.(n)=5 = 70 Peat) () avnies secon (Ixx).
Then we have, if m=(a—-h)/o, n=(ath)/o, «=(a—h)/o, «=(ath)/o, F,, (h, k) = N[{Po (1) + Po (n:)} {Po (€:) + Po (€2)} + {Ls (1s) + La (ms) } {Po (€:) + Po (es) } + {Po (m) + Po (m2)} {Lr (42) + La (&2) + {P, (6) +P, (€)} {Ze (m) +L, (n:)} +... +{P,(e) +P, (e)} {Leo (tm) + Leer (Mm) Hee] ceceeeeeeees (Ixxi).
The Z-functions involve the rapidly converging v-coefficients, and the first few terms will suffice to get an idea of the distribution. If we retain only the Rayleigh terms we find:
F,,(h, ky) =N[1 —{Po (m) + Po (2) {Po (4) + Po (e)}] -..- eee (lxxii), which can be ascertained for given values of a, b, h, k and o from the ordinary
tables of the probability integral. 6
42 KARL PEARSON
If we make b infinite, then P,(e,) and P,(e,)=0 for s>0, and L,(¢,) and L, (e) =0, Py («) =P, (€) =%, and __ F(h, k)=N {1 —Py (1m) — Po (9.) — LZ, (m,) — La (M2) } vee eee eee (lxxiii).
This could be deduced directly from (xlix) and it represents the first migration distribution into an indefinitely long cleared strip or belt. This is a result of some interest as it might approximately apply to the migration into a zone cleared by a flood or a fire of certain types of animal or vegetable life.
(13) Problem VI. To determine the distribution after m migrations into a cleared but not sterile rectangular area.
By the general proposition on p. 39 we have only to write 2=mo for o, and the N’s for the v’s in the L’s. Let us put
mn =(a—h)/Z=n//m, n/=y,//m, «'=e/,/m, & =e,//m. Let feo A L (m!) ar Jon and so forth, then we have for the full solution: mln (hy k) =(pA)" ON [Po (m') + Po (n2!)} (Po (a) + Po (ert + {LZ (ni!) + Li! (mV {Po (6) + Po (er) + {Po (m’) + Po (m)} A (6) + Lr (e)} {Pi (el) +P, (6) (LE (ny!) + Le (ne) iced Pele 42 le el) pag Oe sce || Gs ati cicae (lxxiv). The terms here will very rapidly converge for any fairly large value of m, so that for many purposes we may write the solution:
min th, y= (pas nN {Pit + Polo Ps (eo) + Pale))} vies: (Ixxv), which can be found at once from the usual tables of the probability integral.
me 2” {Nhs (ni!) + Neb, (1) + + + Mesa (mn!) + +f
Illustration I. A rectangular patch 2 miles long and 1 mile broad is cleared of mosquitoes, but not retained sterile. What would be the central density at the end of the year? Suppose 10 breeding cycles with their scatter migrations, each of 6 flights, to take place in the year. Then if we take 200 yards as a possible round value for the flight we have:
m=10, n=6, 1=200 yds, oo =$nl?=120,000 or o=346°41 yds.
a= 880 yds., b=1760 yds. », =n, =2°540, €, =e, = 5'081, Z=VJ100 = 109544 yds., ni! =n! = °803, e/ =e! =1°607. Hence wl; (0, 0) = (wA)4P, (°803) P, (1°607) N, or, using Sheppard’s Tables : »F, (0, 0) =(wA)'4 x 2890 x -4460N,
= (pA) x 5156N.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 43
Thus Fs (0, 0) = (WA)? (1 —"5156) N = (pA) 48,
We see accordingly that if the fertility and the death-rate were the same in the clearance and in the populated district outside, the density at the centre of the cleared patch would at the end of the year be almost 50 per cent. of that in uncleared country. It is thus obvious that clearance can be of small use, unless it is followed by permanent preservation of sterility. Even if one annual clearance were made it is very unlikely—if the actual values of the constants are at all near those assumed—that the mosquitoes would not by the 9th or 10th breeding cycle within the year before the annual clearance was repeated have reached a very substantial density even at the centre of the patch. We have thus an additional argument in favour of rendering a district not only sterile, but keeping it so. In such a case since », and »%, yy, Wy, are negative we shall have a density somewhat less than :
F,(0, 0)=N {1 —4P, (2°540) P, (5°081)}=_N (1 —-9889) about. Thus : if,(0, 0)="01N approximately. It follows that in the centre of such a rectangular patch, there would roughly be only about 1 mosquito for every 100 in uncleared country. But while this shows that such a sterile patch would be a great improvement
for a denizen at the centre it is well to enquire what happens in such patches some way from the centre. I accordingly add the following illustration.
Illustration II. A square area of one mile side is cleared and kept permanently sterile. What will be the density at the centre and a quarter of a mile from the centre on the same assumption as before ?
Here a =b=880 yds. At the centre y,=y,=«=¢=2'54 and: F,(0, 0) =N[1—4 {P, (2°54)! ] = {1 — (9889) = 022N ; or, we find one mosquito for every fifty in uncleared country. Taking our quarter of a mile directly towards one of the boundaries, we have h= 440, k=0, and: y,=1°27, y,=3'81, e=e,=2°54. Thus: LF, (440, 0)=N[1 —{P, (1:27) + P, (3°81)} {2P, (2°54)}] = N {1 — (3980 + -4999) (-9889)} = "112.
Thus at + mile from the centre (or from the edge) of the clearance, the density is 11 per cent. of that in uncleared country. It may be doubted whether this is a sufficient reduction, and, supposing the above assumptions to be any- thing like roughly correct, it may be needful to render more than a square
mile permanently sterile to protect a patch of one square half-mile. 6—2
44 KARL PEARSON
On the other hand a cleared but not permanently sterile square mile would after a year have a density at the same point—i mile from the centre—of:
wf (440, 0) =(uA)V[1 —{P, (402) + P, (1'205)} {2P, (*803)}] = (uA)? 69.N, or of 69 per cent. of that in uncleared country.
Another point seems of some interest. What is the density at the boundary after the first migration ?
At the middle point of the edge it is iF, (880, 0)=N[1—{P, (0) + P, (5°08)} {2P, (2°54) t] = N (1—*5000 x *9889) = 506N. This is almost the $.N of an indefinitely long straight boundary. At the corner it is iF, (880, 880) =N[1—{P, (0) + Py (5°08)? ]= "75 nearly, or, as we should expect, has risen much beyond the 4M value.
There is no difficulty in tracing the contour lines of the population density in this case.
If we consider a cycle of 10 breedings in a non-sterile patch we have:
F< (880, 0) =(wA)'N'[1 — {Py (0) + Pa (1°607)} {2P, ('808)}] = ‘742. (pA), and Fs (880, 880) =(A)N [1 —{P, (0) +P, (1°607)}] = '801N (uA).
Thus if the patch were not sterile, the effect of the clearance would at the boundary after the lapse of a year be marked by a 20 to 25 per cent. reduction. The illustrations I have given are of course dependent on the values of the constants selected. Such constants have at present been little studied, and accordingly small weight can be laid on the actual numerical results. But the theory appears to indicate useful lines of inquiry, even if its results will of course need to be controlled everywhere by local facts. In a general way there can be little doubt that a theory like the present will not only lead to a more systematic classification of local facts and to fuller observation of the habits of
local species, but that this knowledge itself will in its turn test the applicability of the theory, or suggest the directions in which it may need modification.
(14) Problem VII. To determine the distribution after a first migration into a cleared circular area.
Let the radius of the cleared area be a. Then at distance c from the centre, inside or outside the circle of radius a, the distribution F,(c) is given by:
A MATHEMATICAL THEORY OF RANDOM MIGRATION 45
foo) Qar F,(c)=N | I, bn (C7? = 216 C08 0) PHOT ee ceseeccssssesessecceseeeenees (Ixxvi) -* Qf le. Fo BEM 9 ~ (02008 yd gety
ae ao [(@r 72/2 o | r\™ Cc m : i) in e BN le r (2) (2) cos” bdbdr. 0 m! o
Now [ee cos”0d0=0, if m be odd, and= i cos™ 6dé
0
jee per) =o 2s! 2s (2s—2) .. (2° s!)?’ if m be even and =2s. Hence : r ()=Q eR le? ° bile? R (2) (2) ty dp (eesti) - = t a ‘ se = oe (2° 81)? eco eeeees * Now [oe ele = d (2) =|" eo 2 y8+1 gy a o fey alo
eg eee ee (Ixxviii).
M,,,,(a/0) is thus the 2s+1th moment of the ‘tail’ of a normal or Gaussian curve of errors (multiplied by /27) about its axis. Its values have been tabled for s=1, 2, 3 and 4.
Thus we have:
F, (c)=NQ e218 § (5) Ae le) gRaanonen (Ixxix).
But it is easy to see that:
=o ste~tHlet fy 1 @ 54 (5) a =) M,4, (4/0) 2 sle +s Steg oi) to tart (sa) f° Accordingly : -k(e+ar7e2e (e\* la 1 oy 1 ie
F, (c)=N Qe 8 (5) {1455454 i(3 ra S| bey (eee,
The successive differentiations of this expression with regard to t=o°, involved in the operator Q;, which are needful if we wish to give the corrections to the Rayleigh solution, are straightforward but extremely laborious. We can
throw the solution into other forms. Write : 6=s¢c/o’, 4 =4a'/oc’,
then we have:
F,(c)=NQe RS (1tq+ St... 45) ee ce’ -% =NQ,e°°S GH. tia | bean Cee ere (Ixxxi)
46 KARL PEARSON
Here | xe-*dzx is the incomplete T-function for an integer value of s. This
can be found fairly easily from the above series, or may be determined from tables of the incomplete I-function which it is hoped may be shortly published.
Again : J, (20/2) = 8 0 hence we have:
F,,(c)=NQ,e7 i J, (2in/€,a) BN etrnsinneneegisieles (lxxxii),
a very concise form, which does not, however, simplify the calculations. Integrate by parts and we have:
F,(c)=NQ.e- 18 © 67, (ave) 0 2 -(e\ +e q s dl? Na oem = NQe (a+ Se, d(««,)° {J, (2i/e,€,)}. Uy Si (BiVe) Due dz’ Ja (2i/2) oe V(qts)! (aJz)e ’ or: F,, (ce) =NQe- #98 {« Z.0h/sa)\ 0 (ive, €,)* = NQ,e- «+ § ( A =) Oe ames (Ixxxiii), 0 2
This is the solution in Bessel’s functions, and inside the cleared area, where e, is greater than e¢, would give fairly good results if tables of the higher Bessel’s functions for imaginary values of the argument were available.
We can also express the solution in terms of w-functions as follows:
= (° (LP) -geje8 rar Write I, (a)=| E ) e as oly ide ft ey EKe)= 6 Then F,, (c) = Qs + 1,(a) B,(e): 1 372/52 {1 7\8 a Now E,(r)=5e%" (5 =) = 210° Sb yy,
the b’s being undetermined constants, for dividing by the exponential factor we have an integer algebraic expression in 7*/o° on both sides. Multiply both sides by x 7dr and integrate between 0 and «, p being = or <s. Then:
Ll (% ~272/¢2 1 7r\s F 2
=f, el Xap 6 3) rdr = 210° b,, ’ Xop Ora,
9 2 (0 pstida : ; ar i Ow Boar 27 b (p!), by (xix) and (xxi).
0
A MATHEMATICAL THEORY OF RANDOM MIGRATION 47
Therefore by a :
by = app (P— 1-8) (P=2=5)...(=3) (=a | “Brena
(ai s(s—1). Jeter [ ate*da, if x= —1/8,
ae aa s(s—1) (s—2)
Thus E, (r) =210° 4 @,— 8 +e o,— (31) ot of (lav)
= 270°, (r)*, say.
Now consider:
Gia Bs ues vstapana you ebsew i nededat eeemean comes (Ixxxv) We can now express J,(r) in terms of w-functions. We have: ={" gi ror I, (7) _ St E, (7) = _ Sates s(s—1) — s(s—1)(s—2) rdr = 8! 200 : {0,— so, + (21 o,— (31) Web vee 6 sw, _s(s—-1) — s(s—1)(s—2) = 8! 20°" (s+ 1) {0-134 aTar Mn 314! Ot... = 8! 2Qro*(s+1)V, (7). Thus F,, (c) = NQ,4ntot S ((s + 1) U, (6) V; (at) .cseseseeeseee (Ixxxvi), 0 ; s s(s—1) s(s—1) (s—2) where: Lae (2) O- (3!) Ogt..., s s(s—1) s(s—1)(s—2) =O grag OO pe
a result which allows of fairly rapid determination from tables of o°,,.
There is, perhaps, less difficulty in this form in allowing for the first term or two of the operator Q,, for U,(r) and V,(r) can be at once differentiated with regard to o°, but even then the final result has considerable complexity.
* This result involves the expression of any power of r® in x-functions.
48 KARL PEARSON
The Rayleigh solution value is easily found by putting Q@,=1 in any of the forms of (lxxix), (lxxx), (Ixxxi), (Ixxxiti) or (1xxxvi).
A case of peculiar interest arises when c=0, or we take the density at the centre of the clearance. In this instance we have:
F,,(0)=NQ,e 8,
Now 2\2 d? 2\3 dl? s 2\s dl? = 1 +y, (o°) Toy —”(o") d (ot cae +(-1) V5 (o°) d(oy* tee and etl? _ ongte,, dé 1 g2)o2 ara) dsl th rs o\s $a2/o = 2 0 7 0 $ erefore (o’) 7 (a? )=2r0 {3 (oy a roa (o’)
= 270? {(— 1)*@,, +8 (—1)**@, (5-1
—~ta2/o2 ne ae 8 (1) fsa:
Thus F,, (0) = NeW 271 (1 — 2,4 + (v4 — 80%) Xe + (Vp — 4%) Xe+ (Ys — BPs) Xe «+ = 2ara°N (w, — 2v,0, + (v, — 8v,) @, + (v, — 4.) @, + (Y, — 5), + ...)...(Lxxxvil). We are also able to consider the secondary problem:
What 1s the distribution into unoccupied space surrounding a uniformly occupied circular area due to a first migration ?
Let the radius of the area be @ and let the density at any distance ¢ be F,(e) after the first migration. Then clearly, if all space were uniformly filled, we should have uniformity after the first migration, or:
F,, (¢) + F, (c)=N, hence : D(a aPC): eaihasnermiedene vase (Ixxxviii).
The solution is thus thrown back on the solution obtained for the previous problem. In particular at the centre of the populated area we have:
FAM uy Fo WON siansosrreesibaala bere aeons (laxxix),;
We are thus able to calculate the reduced central density due to a migration from the area to the surrounding unoccupied district, 1.e. the effect on population of the spread outwards of a colony.
(15) Problem VIII. Indirect solution of the General Problem of the Random Walk.
It may not be without interest to put on record the distribution density after n flights in the case of a cleared circular area, if it be expressed in Kluyver’s manner by the integral of a Bessel’s function product.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 49 We have: a (ar Ff, (c)\=N f $n (+7 — 2re cos 6) rdédr, o Jo
and
F,(c)=N—-F, (c)
fa (2%r (oe =W {1 a i I | dCi poe ey (ul)*dwraar|. T JOJO 0 by (iii), =N Fi -|° [ous (ur) Jy (uc) {Jy (ul)}” durdr | : L 0 Jo
by Neumann’s Theorem (see p. 6)
= iN E -|; Wo(ui)y" { |’ (ur) urd (ur) J, (uc) du
=w[i- | i eos | “dd, (ur) wr} J, (ue) de |,
by the theorem cited on p. 7, =N/1- | peacls urJ, (ur) | (uc) du L 0
0 U
=N E _ [er (ul)|}" aJ, (ua) J, (uc) du ,
Or, writing v=au, we have:
F,(c)=N E -[y, (v) Jy (v <) {7 (v Ale dv | Bini aansoes (xc).
This expression is concise. The integral expresses the probability that if an individual start from the origin and take (n+1) flights, the first of magnitude ce and the remaining n of magnitude /, at random, he will find himself within a distance a of his starting point. But there does not seem any convenient method of evaluating the integral. Comparing with (lxxxiil) we have the curious identity :
i l\)" —~(ai+e)/n2® /. e\s : ‘ [, J,(v) J (v <| {7 (v A dv=1—Qze ie eein 8 (i <) J, (21 or) (xe).
0
/
Write c=l, a=r and n—1 for n, then
: NY 1 HO. etn - DIRS (BV F (oe 7 [20 14, (v:)} ca fen ee (i) J, (26 wom):
where Q,_, 18 the operator,
d d d 1+v,(n—1P 75 —v(m— 1) Fate +(— 1) (nm — 1)? ae
50 KARL PEARSON
or, by (iv), P, (7), the chance that an individual taking n flights from a centre should be found within a distance r from that centre, is:
(P22 fin—1)12 &/. b\* ee 5 P,()=N {1 ~ Qe (7? +12)/{( DAs ae (2i atim)} ask (xcii).
" : 1 d Since d, (7°) = aan dr {P, (r)}, we have here the complete analytical solution in known functions—ze. the Bessel’s functions with imaginary arguments—of my original problem of the random walk. But this formal solution provides no better method for shortly determining the dispersal curves than that already indicated in these pages.
(16) Problem IX. To find the distribution after m migrations each of n Jlights, there being originally a circular clearance which is not kept sterile.
The solution is found by writing mo® for o°, putting the N’s for the »’s in @, which becomes @Q/”, and multiplying by the factor (wA)”"* assumed to be constant. This can be done to any of the forms (lxxix)—(lxxxiil), or (1xxxvi). If we write:
é,=e/m and ¢€,=«/m
we find:
Fa (0) = (uay" Qe" S (a i “ate-*de) eee: (xciii), ne Fa (0) = (way Qre- i ACRES a eee (xciv). Or again : ,
ny (6) = N (way Qine 648 (f =8) 4. (NSE) veeiseaanices (xev),
sei N (ua) Qaatnt oS ( U, (sa) vy, (sa) (s+ 1)) ner (xcvi).
Of these, I have found the first quite as convenient as any other to obtain numerical results from. I shall now illustrate the circular patch formulae.
Iilustration I. A circular patch 4 mile radius is cleared of mosquitoes but not kept sterile. To find the density at the centre, at 4 mile from the centre, and at the margin after ten breeding cycles.
We shall suppose as before /=200 yards, n=6, and therefore
‘ ell o’ = 120,000 square yards. = Soi 8227. The second term in Q,” will be of the order =, of the first and I shall neglect it. Accordingly the solution may be taken
A MATHEMATICAL THEORY OF RANDOM MIGRATION 5}
af. n(C) =e" &*®) (WA) W(1+e(1+8) +5 +=, (Lat <)
e, €. €,° ey, (lta § +5) + Hes The successive bracketted terms in €, are
13227, 1°3748, —«:13804, —«1°3809 and 1°3809,
which is equal to e*® to our number of decimal places. Hence we may put
ats (c) =(uA)'Nen +4) {1 + €, (e* — 0582) +55 (6 (e
* —-0061)
3! = (uA) Ne-@t® (1 — e% -+ et — -0582€, — 0030E, — 0001€,) = (pA) N {1 —e-% +e-G* (1 — 0582€, — -0030e2 — 0001€,)}.
ee oer. + © (é —-0005) 45 et 4 $s!
At centre F,(0) = (uA) fe (1)} = (ua )'N-724. We can test the accuracy of this result by using Equation (lxxvii) which, if we put v,=N,, gives: oF. (0) = (wA)*Ne“* (1 — 2.x. + «.+) 6773
ey, ee = 9 AT "3227 ee? and X.= 1—€,=(wA)*Ne (1+ 130 + ° .)
= (wA)N'730. The agreement is accordingly good enough for practical purposes, and we
may say that within a year the mosquitoes would at the centre of the patch have a density 73 per cent. of what they would have in uncleared country.
I now consider the density at a quarter of a mile from the centre, €,='0807, and using the above formula we find: F, (440) = (pA)N (1 — 67" + 67 x +9958) = (wA)N'75, or, we see that at a quarter mile, midway between centre and boundary of the patch, the density is only 2 per cent. more than at the centre. Finally, at the boundary itself, €,=°3227 =e, wf (880) =(uA)N (1 — 7” + e~ ™ x 9809) = (uA)N'79, Thus the cleared patch would within the year have filled up with a population of mosquitoes varying in density from 73 per cent. at the centre to about 80 per cent. at the boundary, or the clearance without permanent sterility would have
been quite ineffectual with the assumed values of the constants. 7—2
ay KARL PEARSON
Lilustration II. Let us assume precisely the same conditions as in the previous illustration, except that the area shall be supposed sterile, and we will consider what happens at the end of the first migration.
At the centre we have by Equation (lxxxvii): 1 F,(0) = Ne? {1 — 21, y+ (, — 85) Xa + (Y5 — 405) Xe + +f But —2v,= 083,333, x (e)=1—e@= —2°227,000, v,—3v,= —°032,407, y,(e) =2—4e, +e, = —°494,471, v,—4%,= — 005,498, y%,(¢,) =6 — 18¢, + 9e? —€? = 8-031,303, V,— 5v,, = '000,082, Xs (&) = 24 — 96, + 726, — 16e, + «4,
&, = 3'227, = 34°752,347, Hence: —_, F, (0) = Ne~*™” (1 — 185,583 + 016,024 — ‘044,156 + 002,850) =O3LN.
This three per cent. of the density in uncleared area might possibly prove a trouble and on our assumptions it may be doubted whether the half-mile radius is sufficient. If we take the first term only, we find ‘040N, or four per cent., not an important practical difference.
The introduction of even the first modifying term when ¢ is not zero appears to lead to such complexity that I content myself with calculating the approximate value given by the Rayleigh solution for distances of + and 4 mile from the centre of the clearance. In this case ¢,=3'227, ¢,='807 and 3°227 respectively half-way to and at the boundary. I proceed just as before and deduce the following approximate value for ,f,(c), 2.2.
F,(c) =(pA)N {1 —e7* + e-©*#) (1 — 20°9769e, —7°8851¢2 — '2355¢! — 0228e,— 0016e, — 0001.e,")}. Hence
if, (440) ="179 (uA), corresponding to ¢,='807 and
LF, (880) ='709 (uA)N, corresponding to ¢,= 3227.
Thus the density at 4 of a mile from the centre of the cleared patch would be some 18 per cent. of the density in uncleared country. In other words on our assumptions a clearance of one mile diameter, if kept sterile, would hardly suffice to keep an area of } mile diameter free of mosquitoes.
Compared with a straight boundary, where the density falls to about one half that of uncleared country at the boundary, we see that the bending of the boundary has a most marked effect in its neighbourhood, the curvature raising the boundary density from about 50 to 71 per cent. of the uncleared density. In fact the density is almost equal to the 75 per cent. in the boundary angle of a square clearance.
A MATHEMATICAL THEORY OF RANDOM MIGRATION 53
The differences between a square and a circular patch inscribed in it are noteworthy, indicating the marked influence of the area at the angles. Thus at the centre we have only 2 per cent. as against 3 per cent., and at 4+ mile from the centre 11 per cent. as against 18 per cent.
As far as the above numerical investigations are to be looked upon as anything but illustrations of the nature of the calculations requisite to apply the theory of random migration to the mosquito clearance problem, they must be taken:
(i) As merely an incentive to further study of the manner in which mosquitoes scatter from the breeding ponds. It would seem possible, if difficult, to experimentally test this by in some way marking a large number of insects, and determining the nature and extent of the flight.
(ii) As indicating that permanent sterility of the protection belt is almost certainly needful. The 4 to 3 per cent. of mosquitoes at the centre of the clearance amounting to 6 to 18 per cent. at 4 mile distance may or may not be serious, but they certainly would very soon be if they were able to breed.
(iii) As showing that on the rough numbers taken, that a clearance belt of probably 4 mile round a settlement would be the minimum desirable sterile zone. But it is quite possible that, when the requisite constants are better known, it will be found that smaller belts will suffice. It is possibly rather an exaggerated view to suppose a mosquito to make six random flights of 200 yards between breeding spot and breeding spot. But certainly many insects I have noted will fly with great rapidity in one flight 50, 100 or 200 yards, and these flights are quite distinct from “ flitters.”
. (17) Conclusions. The present memoir suffers of course from all the defects which must accompany a first attempt to develop a mathematical theory of phenomena which have hitherto not been studied with this development in view. The theory itself suggests hypotheses and constants which have never yet been considered. How far with a broad average of environment in relation to food supply, breeding places, shelter, foes, etc. is the spread of a species random? Are any of the geographical limits to plant or insect or animal life non-environmental and in course of change? If so, statistical studies of the density gradients of such species for a few miles either side of the supposed boundary would form most interesting work for biometricians. But, apart from this observational work, a good deal of experimental inquiry might be usefully attempted with regard to the constants of random scatter or flight in the cases of both seeds and insects.
On the theoretical side there are many problems left untouched. The present memoir has only opened up the outskirts of a very big field. It would be of value to investigate the number of terms in the expansion in @-functions requisite to practically reproduce the graphically constructed density distributions for migrations of 3, 4 or 5 flights. Our expansion to 6 terms is hardly close enough
54 KARL PEARSON
for practical work until n=6 or 7. Many other shapes of populated or of cleared areas would provide problems of some interest, especially when the spread of the colony was limited in one or more directions by environmental barriers, such as sea, river or mountain range. The problem of sterile areas has been by no means exhausted, for in such cases I have only dealt with a result of the first migration, but actually there will be a second and later migrations in which not only new immigrants will appear but a portion of the first immigrants will be emigrants and again able to breed when they reach uncleared country. Our solution thus gives only a minimum limit to the percentages if the immigrants do not die at the end of the first breeding cycle. Much interest attaches also to cases in which the fertility and the death-rate are correlated with the density, z.e. pA is not to be considered a constant. But in these as in other problems which suggest themselves, a further preliminary knowledge of some of the ecological constants suggested by the present enquiry would be an extremely valuable guide to the direction that research should take.
On the purely mathematical side the problem of the “random walk” may now be considered as fairly completely solved. The distribution curves have been determined until they pass into an analytical solution expressed by a new type of function. The expansion in these functions shows the limits to the accuracy of Lord Rayleigh’s solution of a certain allied problem in the theory of sound. But the -functions which have arisen in the enquiry have most interesting properties, and have led me to a whole series of allied functions of one and two variables which I propose to discuss on another occasion. The expansion in w-functions will I venture to think be found ultimately to have considerable importance for mathematical physics, especially in the evaluation of certain definite integrals which arise there. The possibility of practically carrying out such expansions depends on the determination of the successive moments (and products) of the original function, a process with which every statistician is now fairly familiar. But applied to definite mathematical functions it loses the disadvantage with which it is burdened in statistical practice—the high relative probable error of very high moments—and becomes closely allied to the process of deter- mining the integral of the product of any function and a Legendre’s coefficient (or solid harmonic). Should the generalised w-functions prove, as I anticipate, of some mathematical interest, it will be another illustration of how the need of the applied mathematician has thrust him, almost unawares, into the path of a novel functional development.
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Scale of density of Population.
Original density. DIAGRAM VII. PLATE VI.
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Area of Emigration Area of Immigration
————Infiltration Curve n=7. ------Approximate Solution, exact for an infinity of indefinitely small flights.
N.B. The density =1 when r/o=-3°74, and =0 when r/o=+3°74,' for n=7
Boundary of original Populated District
“30 28 -26-24 -22-2-:0 18 -16 14 -12 10 “8 -6 -4 -2 0 2 4 6 8 10 12 14 16 16 20 22 24 26 28 30 Scale of r/o, where o=,/$nl.
First order infiltration curve across a straight boundary.
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