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A model for genetic relationship in areally continuous plant populations
THEORETICAL
POPULATION
BIOLOGY
8,
331-345
(1975)
A Model for Genetic Relationship in Areally Continuous Plant Populations HANS-R• Lehrstuhl
fiir
Forstgenetik
LF GREGORIUS
und Forstpjlanzenziichtung Received
June
der
UniversitCt
Giittingen
27, 1975
Representations are based on plant populations, continuously distributed over their habitats according to specified density patterns. Migration of genetic material takes place via pollen and seed dispersal. Monoecious plants with arbitrary rates of self-fertilization and dioecious plants are considered. The model was constructed with the intention of determining coefficients of inbreeding and kinship for all locations within the seed population after its dispersal over the habitat, assuming the respective genetic relationships of the parental generation to be known. To display the consequences of single components hidden in the general result, the following specifications have been treated: finite population size combined with random dispersal of seed, equilibrium states for hypothetically infinite population size with “limited” dispersal of pollen and seed, random dispersal of pollen, and random dispersal of seed.
INTRODUCTION
An important component of the mating system of seed plants is determined by the mobility of male garnets (pollen) as well as zygotes (seed) and the immobility of individuals (plants), including the female garnets (egg cells) they produce. In many cases, this mobility is such that pollen and seed are distributed continuously over a specified area and therefore, rise to spatially continuous dispersal of genetic material. Consequently, we are concerned with a more or less special type of continuous migration, a field that Wright (1943, 1946) and Malecot (1948, 1950, 1967) pioneered. The most direct method of measuring the effect of this sort of migration on the genetical structures of populations uses coefficients of inbreeding resp. kinship, where as Wright concentrated on the coefficient of inbreeding, having in mind calculations of the inbreeding-effective population size. Both authors avoided further specification of mating systems underlying the construction of their models, merely applying the results they obtained from their model assumptions to several situations. Since even more recent restatements of Malecot’s model, as, e.g., given by Maruyama (1972) do not refer to concrete migration modes, we thought it worthwhile, using mainly Malecot’s ideas, to build up a model that takes into account the particularities introduced by the above mentioned migration mode of seed plants. This model 331 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
332
HANS-ROLF GREGORIUS
should be suitable for determining coefficients of inbreeding as well as kinship becausethere might be a nontrivial relationship between them. Furthermore, the results should be stated in a form that makesit feasibleto decide whether they are identical to those given by Malecot. 1. THE MODEL For simplicity and becausedetails can be distinguishedmore clearly, we start with the discontinuous version of the model, i.e., we assumeany decomposition of the population’s habitat (which may be finite or not and one, two, or threedimensional) into disjoint colonie#. With decreasing size of these colonies, the continuous casewill be approximated at a limit. Organismsare assumedto be diploid. The following assumptionsare made. Each pollen arriving at a colony (originating from outsideor insidethis colony) pollinates all the egg cells present in this colony randomly, except seif-pollination, which may be included at a given rate. All membersof a colony produce pollen resp. egg cells to the sameextent. The over all pollen production is large enough to pollinate all egg cells. Each fertilized egg-cell constitutes a seed. All movementsof pollen aswell asseedoccur independently of each other. There is no variation of flowering times of the single plants. With theseassumptions,we derive at a first step the coefficients of inbreeding and kinship at all placesfor the seedpopulation after its distribution over the area, knowing the coefficients for the parental population. This will be done for two different mating types: Monoecism, including self-fertilization or excluding self-fertilization in form of pollenelimination (for definition see Finney, 1952), and dioecism. The probability laws governing migration (movement) of pollen and seedwill be stated in a way that makesthem directly applicable to the essentialparts of the derivations. Furthermore, since we aim at a continuous representation of the results, keeping the more illuminating discontinuouscasein mind, all parameters will be given in their continuous version. These parametersare (:= should be read as“is defined to be”): p(x) := Population density at placex. ~(x 1y) := Probability density that a pollen, which cameto fertilization at placey, originates from place x. s(x 1y) := Probability density that a seedthat landed at place y originates from placex. r(x ) y) : = Probability density that the male gamete(pollen) contained in a seedthat landed at placey originates from place x. g(x 1y) := Probability density that a seedthat landed at place y contains a gamete(male or female) originating from place x.
CONTINUOUS
333
MIGRATION
l$(x,y) := c oefficient of kinship of two individuals from the parental population located at placex andy. f(x) := Coefficient of inbreeding of an individual from the parental population located at place x. &(x, y) := Coefficient of kinship of two seedsthat landed at place x and y. f’(x) := Coefficient of inbreeding of a seedthat landed at place x. Obviously, r(x 1y) = sp(x / x) . S(Z/ y) dZ, integrating over the population’s habitat and g(x /y) = $(r(x /y) + s(x 1y)). location of 5 parents (pollen source)
location of 9 parents (fertilization place)
location of offspring (seed lands)
Some further remarks on the properties of P(y / x) have to be made concerning the rate of self-fertilization with monoecisousplants. In probabilistic terms, this rate is the probability that a pollen that came to fertilization at a given plant originates from this plant. If we denote this probability for a plant at place x by q(x), then 1- q(x) is the rate of cross-fertilization and we may define a probability density p’(y 1x) analogousto p(y 1x), which accounts only for cross-fertilizations. With thesenotations, ~(y ( x) may be given the representation P(Y
I4
=
(1 -
4(x))
.P’(Y
I4
+
S(Y
-
4
.4(x),
where S(y - x) is the Dirac function. Thus, ‘(Y
I 4 = J (1 - ~(4) P’(Y I 4 - 42 I 4 dx +
q(y)
. S(Y I 4.
Note that 4(x, x) andf(x) have to be distinguished.Since the influence of mutation onf’ and 4’ can be taken into account easily at any stage,derivations will be given without considering this parameter. Definitions of q$f, #‘, and f’ refer to one locus only and are usedin agreementwith Malecot (1948). a. Monoecism Computation off I. To obtain the probability that two alleles at one locus contained in a seedthat landed at place x are identical by descent,it is necessary to identify the male as well as the female parent of this seed, their coefficient of kinship, and their probability of contributing an offspring (seed)at place X. In caseself-fertilization is allowed, these parents may be the sameindividual, if they are assumedto come from the same infinitesimal area (colony) centered around a specifiedplace. First, let the maleand femaleparent come from disjoint
334
HANS-ROLF
GREGORIUS
areas dv and du centered around places v and u; then the probability that the male and female parents of an offspring at place x come from area dv resp. du and pass on identical genes to this offspring is p(v 1u) dv . s(u 1x) du . +(v, u) = (1 - q(u)) p’(v / u) dv . s(u j x) du . d(z), u). If both parents are the same individual, i.e., u = v and du = dv, they pass on identical genes to an offspring with probability 8(1 +f(u)), and thus, the above probability now is p(u / u) du * s(u / x) du * +(I +f(u))
= q(u) * s(u j x) du * &(I +f(u)).
Summing these terms, we get f’(x)
= 1 (1 - Q(U)) P’(V I 4 4~ I 4 NV> 4 dv du
+ j- ~(4 4~ I 4 HI +fM> da Computation of 4’. The procedure is essentiallythe sameasbefore, except that insteadof two allelesin one seed,we now have to consider two allelesfrom two different seedsthat landed at places x and y, say, and thus have probability g(u 1x) du resp. g(v Iy) dv of originating from a parent (male or female) from areadu centered around place u resp. from area dv centered around place v. The assumptionof equal pollen as well as seed production of all individuals within each colony implies, in caseu = v and du = dv, that all the p(u) du individuals within area du are equally probable asparents. Therefore, if two parents of two given offspring come from the samearea du, they are the sameindividual with probability l/(p(u) du) and passon identical genesto those two offspring with probability (l/2)( 1 + f(u)); on the other hand, the probability of both parents being different individuals and passing on identical genes to each of their offspring is (1 - l/(p(u) du)) *+(u, u). Th us, the probability that two parents of two offspring at placesx resp. y comefrom areadu and passon identical genes to these offspring is [ &-;(I
+fW)+(l
=-- &)
-&)+(uij
[; (1 + f (u)) - %4
*g(uIx)du*g(uIy)du u)] du
I 4 g(u I Y) du
+ C(u, 4 Au I 4 & I Y> du da Adding the obvious result for parents from disjoint areas,we have
4% Y) = I g(u I 4 cdv I Y>4h 4 du dv +s “‘“%pl
[; (1 + f (u)) - c$(u, u)] du
(lb)
CONTINUOUS
335
MIGRATION
Formula (1 b) looks very much like Malecot’s representation; nevertheless, it is a different result because the coefficient of inbreeding f’ in general cannot be calculated from 4(x, x), which is just limy+Z+‘(x,y). This difference is due to the special migration mode of seed plants which implies that offspring generally do not grow up at the place where their parents mate. Furthermore, note that the influence of the rate of self-fertilization explicitly appears in (la), while in (lb) it is hidden behind the g’s. The great importance of this influence is shown as we consider the two extremes for the values of 4, namely, Q(U) = 0 and q(u) = I. The expression q(u) = 0 means self-fertilization is prevented and changes (la) into f’(x) = j p’(v 1 u) s(u 1 x) +(v, u) dv du. (2) Here, q(u) = 1 of courseimpliesf(u) = fandf’(u) = f’, and thus,f’ = t(l +f), which is the well-known result for complete self-fertilization, i.e., crossfertilization prevented. b. Dioecism
By CL’resp. I-L”we denote the population density of the male resp. female population. Clearly, f’ again is given by Eq. (2). Computation of 4(x, y). As bef ore, the probability that two alleles, each contained in a seed that landed at place x resp. y, are derived from parents occupying disjoint areasdu resp. dv and are identical is
Au I 4 du - g(v I Y>dv . d(u, 4. For u = o, the above two allelesmay be derived from the samemaleor female parent and in all other cases, necessarily originate from different parents. Consequently, all the possiblecombinationslead to the following sum of probabilities: $. T(U j x) du * Y(U j y) du
1 .[ P’(U) du - ; (I +fk))
+ (l - T&d;)
d@, ‘11
-f ; * s(u I x) du . s(u 1y) du
z g(u 1 x) g(u [ y) +(u, u) du du + ; * [ ‘(’ . [; (1 +f(u))
- 4(u* u)] du.
I ;?(;I”
I y> + w;!;;;
’ ‘) ]
336
HANS-ROLF GREGORIUS
Putting all this together, we finally get
+‘4.s1 p’(u)
y(u I 4 y(u IY) + s(u I 4 4u I Y) P”(U)
1
*[;(1+f(u>) - Wu,]du.
(3)
Thus, even for p’ = p”, this result clearly differs from that obtained for mating type a.
2. HOMOGENEOUS AND TRANSLATION-INVARIANT
MIGRATION
The term homogeneousis used to describea situation where the populationdensity is the sameeverywhere in the habitat, i.e., p(x) = p (constant). Translation-invariant migration occurs if distribution of pollen as well as seed takes place independently of the location of the producing plants, i.e., p(x Iy) = p(x + a 1y + a) and s(x 1y) = s(x + a 1y + a), which implies p(x 1y) = p(x-y/O) =:P(x--y) ands(xIy) = s(x-~10) =: s(x-y). It might be difficult to realize a situation like this with finite size of habitat exactly, but it will be a good approximation in case migration distances are small compared to the whole habitat of the population. Naturally, translation-invariance is preserved with T and g. Furthermore, the validity of the relations 4(x, y) =: 4(x - y) = qS(y- x), which also holds for 4 and f(x) = f, f’(x) = f ‘, and q(x) E q is obvious. With these assumptions, the previous results obtained for two mating-types now appearto be: Mating-type (a). f’ = (1 - 4) - j ~‘(4 4(4 du + PU/‘U~ + f>
(44
4'(x) = j g(u)g(v) +(a - v + 4 du du +
(UP>[WW
+ f 1 - #91 * j & - 4 m da
(4)
If self-fertilization is prevented, (4a) changesto
f’ = j- P’(U)4(u) du
(5)
CONTINUOUS
337
MIGRATION
Muting-type (b). f’ = 1 P(U) co4 du
$‘(x) = J-g(u)g(w)4(u - w+ x>du de, + %(l +f) -C(O) . s
4P’P”
[(r(u)
(6)
r(u - x) + p’s(u) s(u -
x)] du.
Comparing Eq. (4) with the correspondingresult of Malecot (1967, Eq. 2.13), it now can be stated that they are even identical, but that specification of the inbreeding coefficient again differs greatly. One of the main questions in population genetics is whether there exists a state in which a population doesnot changewith respectto certain parametersin the course of time. In our case,this so-called stationary or equilibrium state is reached if +’ = + andf’ = f. To arrive at reasonablesolutionsof this systemof equations, we now include the possibility of mutation at a rate c, say, which implies, ascan be easily seenby the initial derivations, that the right-hand sides of Eqs. (la)-(6) must be multiplied by (1 - c)“. We will confine ourselvesto solutions for which j+(z) d x is finite. Thus, the Fourier-transforms of all functions appearing in Eqs. (4a)-(6) will exist and will be characterized by double points, e.g., for a function g given, its Fourier transform will be g(y)
= 1 eizyg(x) dx,
where i = - 11/2and yx is the scalarproduct of vectors x and y. Muting-type
(a). Equating 4 to + andf’ tofin (4) and (4a), we get
#J”(x) = g(x) g(-x)
. $.(X)(1 - c)” + h . g(-x)&c)
* (1 - c)2,
where
The inversion of this Fourier transform resultsin
Ye9 = p$
I dY)12 . (1 - cl2 * s e-zsy . 1 _ 1g(y)12 (1 - c)” dy,
where n = dimension of the habitat, and especially4(O) = h . G, where “‘+ 653/8/3-7
I B(Y)12 s
(1 - cl2
1 - 1g(y)12 * (1 - c)” dy
338
HANS-ROLF
GREGORIUS
and therefore
From that, Jp’(x) 4(x) dx = h . R, where R = &
.
s
I dY)12 (1 - cJ2
li;‘(-JJ)
I - j g(y)12 (1 - c)”
dy,
Furthermore,
f=
(1 -c12.4
+
2 - (1 - c)” . q from which, considering rearrangements:
2(1 - ‘j2 ’ (’ - 9) , $@) #@) du , I’ 2 - (1 - c)” * q
the previous results,
f
can be computed after some
-1 f=[
(74
((1 - q)(R/(~ + & + 4) . (1 - 4" - ' I
From this, again A=
(2-qq(1
-c)‘)(p+G;-(1
-q).(1-c)2R
*
Thus, finally
d(x> = .i e-ixu. [(I ~Y)I” . (1 - ~)~)/(l- Ia(r
. (1 - cJ2)14
(27~)~ [(2 - q(l - c)~)(~ + G) - (1 - q) . (1 - c)~ R] . For the case of pure cross-fertilization, imply
f=($$k
(7)
i.e., q = 0, the preceding formulae
__ - 1)-l
(84
and
w =
s e-izu . [(I gW12 (1 - c)“)/U - I &TY)/~ (1 (27+ [2(G
+ p)- R(1- c)“]
Equation (7a) shows that for pure self-fertilization, to be.
4”)l 4
(8)
(q = 1) f = 1, as it has
Muting-type (6). Apart from a few steps, all essential derivations again follow the above pattern and altogether are straightforward. The results can be stated as
f=(
2(G1 + ~&CL”) _ 1)-l R,(l - c)”
(9a)
CONTINUOUS
339
MIGRATION
and
G>
= -
s
e-&” . I-1” I +(Y)l” * (1 - CT + P’ I s’(YV (1 - 4” 1 - I ‘f(Y)12 (1 - cJ2 (2~)” [2(G, + 4p’p”) - I?,(1 - c)“]
I RI :=&-
(jr Y
(9)
w I +(Y)l” * (1 - de + CL’I s’(Y)12(1 - q dY and , 1 - I dY)12 (1 - 4” P” I f(Y>l” . (1 - cj2 + CL’I f(Y)12 (1 - 4” dy
I PC-Y)
1-
I a(Y
(1 - CY
Note that the properties of the Fourier transform imply i-‘(x) = #(LX) . s’(x) and because of that, g(x) = $(l + l;(x)) . s’(x), which makes it possible to give a representation of the “steady state” equations just by using the Fourier transforms of the initial probability densities p and s. Furthermore, the existence of solutions (7a)-(9) of course requires the finiteness of the integral of the functions
I ec(YN2(1 - 4” 1-
/g(y)12 (1 - c)2
resp.
CL” I L’(y)? (1 - c)” + CL’ I i(
(1 - C)2.
1 - I dY>l” (1 - 4”
Disregarding the type of the densities p and s, the steady state values of +, as given in (7)-(g), have the remarkable property that, because 1e-imy / = 1, necessarily 4(x) < 4(O), which by no means is a trivial statement if we consider, for instance, a situation in which all pollen coming to fertilization at a given plant originate from outside a certain circle around this plant and all the seed it produces again goes down outside a certain circle. Furthermore, the possibility of a population of seed plants attaining a state in which all individuals are equally related, i.e., in which the coefficient of kinship does not depend on distances, is to be excluded; a fact that can be deduced directly from formulae (4) resp. (6) by putting +(x) = 4 == +’ z b’(x), which leads to a contradiction. Analytical representations of the above equilibrium values for specific pollen and seed distributions require immense mathematical expense, even if most simple assumptions, such as one-dimensional habitat and uniform distributions of gametes, are chosen. On the other hand, formulae (7a)-(9) in many cases may be applied to obtain reasonable results by means of numerical integration. As an example for gamete dispersal in the two-dimensional plane, we choose an isotropic (radially-symmetric) negative-exponential distribution, which is widely accepted as realistic for most seed plant species (see for instance Bateman, 1947), i.e., for monoecious individuals
p’(x) = g
* exP(--ol,
II x II)>
resp.
s(x) = g
. exd--ol,
I! x IIL
340
HANS-ROLF
GREGORIUS
where 0 < 01~, 0~~, and Ij x /j is the Euclidean length of the two-dimensional vector x. In this case, the Fourier transform is known to be equal to the Hankel transform of order zero and thus, given by &‘(a2 + I/ x l12)-a/2(see, e.g., Sneddon, 1972, pp. 79, 315). From this g(x) = $(x) . (1 + 4 + (1 - 4) B’(4) = gag3(01: + I/ x 11y’2 * [I + Q + (1 - Q) %3(%2 +
II32ll2)-3’2l1
which enters into the calculations of G, R, and the integral part of c$,asgiven in (7). The complete representationis given in the Appendix. Finally, it should be stated that at least with radially symmetric probability densitiesp’ and s, the coefficient of inbreeding asgiven in (7a) may be regardedas an increasingfunction of the rate of self-fertilization q and a, decreasingfunction of the population density CL.The latter, of course, also applies to nonradiallysymmetric distributions.
3.
SOME
SPECIAL
CASES
The following specificationswere chosenwith the intention of giving an idea of how the singlecomponentsof the model affect genetic relationship. (1) Finite Population Size N = sp(x) dx. Each plant of a given generation results from a seed, randomly drawn from the whole seed production of the preceding generation; the plants are distributed over the area according to a population density given by CL. In terms of the present model this situation can be describedby assumingthe dispersalof seedto be at random, i.e., s(x / y) = p
=: c(x).
Thus, g(x I y) = $[J(l - q(u)) P’(X I 4 44 and therefore, from formulae (la) and (lb):
du + (1 + q(x)) 441 =: f(x),
f’ = + . j- (1 - q(u)) p’(v I u) 4~) dv du + Hl + f) * j- qO4 44 du = 4 * (1 - q) + &Cl +f)(r, where Q = J’q(u) c(u) d u is the averagerate of self-fertilization in the population, and 4’ = 4 + ((1/2W +A
- 4 . M,
with
jti’ = s (E(42/~(4) dx-
CONTINUOUS
341
MIGRATION
The influence of varying rates of self-fertilization clearly if, additionally, we assume cross-fertilization $‘(x 1y) = ~(x)/(N - 1). In this case, M = (1 /N)(l + where V = s(q(x) - q)s . C(X) dx is the variance of the (obviously I’ < a(1 - @).
can be displayed more to be at random, i.e., (1/4)(N/(N - 1))2 . V), rates of self-fertilization
(2) Homogeneous and Translation-invariant Migration; Cross Fertilization at Random. As has been stated before, this case requires a hypothetically infinite population size (resp. size of habitat), which may be regarded as the limit of increasing, finite size S, of habitat H. If we assume S, large enough, so that (4a) and (4) hold approximately, then p’(x) has to be replaced by S;;” and g(x) by a[(1 - 4) S;’ + (1 + a) s(x)]. Letting S, approach infinity and assuming the existence of lim++, S;;’ . s,, C(X) dx =: 4 (which then is the average coefficient of kinship), consequently, (4a) and (4) change to f’ = (1 - 4) - Q, + 4 . U/2)(1 +a
resp.
C’(x) = B . [; (1 - q2)+ + (1 - q)2] + (9)’
- / s(u) s(v) +(u - v + x) du dv
+ TT
[; (1 + f) - 4(O)] -1 s(u - x) s(u) du.
It can be shown easily that $’ = 4, which implies 6 = 0, in case the population started with an average genetic relationship equal to zero. Thus, the most apparent difference in our case compared to nonrandom cross-fertilization is that the coefficient of inbreeding does not depend on the coefficients of kinship; whereas conversely, there is a dependence. Because this case allows a comparatively simple analytical solution for the equilibrium value of 4(O), if again a radially symmetric two-dimensional negativeexponential distribution is applied (see Appendix, Eq. (i)), we shall present some numerical examples that refer to monoecious populations. An important property of this specification is that the equilibrium value of 4(O) just can be increased if nonrandom cross-fertilization is added (i.e., (Ye > 0; see Appendix, inequality (ii)). Our intention is to investigate the influence of the range of seed dispersal on 4(O). To give the term “range” a precise meaning, it is defined to be the radius D of a circle CD for which the probability SC, s(x) dx (i.e., the probability that a seed, which landed at a given place, is produced by a plant located at a distance not exceeding size D) has a specific value. In our case lc, s(x) dx = (or,2/2rr) * 1 exp(--(Ys // x 11)dx = cc,2 . iD t * exp(-a$) CD = 1 - (1 + CQD). exp(--or,D),
dt
342
HANS-ROLF
GREGORIUS
which adopts the specific value 0.96, if D is chosen as equal to 5101,. Keeping the probability 0.96 fixed, 01, can be calculated for arbitrary D according to CY,= 5/D and thus, C(O) may b e regarded as a function of D. The following graphic representations are based on these preassumptions and on $(O) = G/((P + GN2 - dl - 4”) ((f rom (7)), where G is given by Eq. (i) in the Appendix. The order of magnitude chosen for the parameters in question may be applied to several conifer species up to a certain extent.
07 p= G B
003[m-21,c~10~5
05
02 01 10
0
20
FICURG 1.0
70
50 D i-m1
100
1
t-
Ic
D Cm1
FIG. 2. The equilibrium value of the coefficient of kinship 4(O) of two very closely D of the seed dispersal. 9 = rate of self situated plants as a function of the “range” fertilization, c = mutation rate, and p = population density [Ind./m2].
CONTINUOUS
343
MIGRATION
Figure 1 demonstrates that severe differences in $(O), according to different rates of self-fertilization, may be expected for comparatively low ranges of seed dispersal only (D < 15 m), while the+(O) va 1ues for different population densities grow more and more alike with decreasing size of D, after having passeda certain range of greater diversity (5 m < D < 30 m), as shown in Fig. 2. In all cases,the quick decreaseif+(O) is striking. (3) Homogeneous and translation-invariant migration; random dispersal of seed. If we again apply the method of regarding the results as limiting values of increasing finite size of habitat, two ways of proceeding can be chosen: First, with help of specification (1) by taking q(u) = q, C(U)= S;;l, and p’(~ / U) = p’(~ - u); and second, with help of specification (2), by replacing s(x) with S;;‘. Both procedureslead to the sameresult, namely, $‘(x)
= (1 -
q
.$ = (1 -
42 * 4 = 4’ = $‘,
f’ = (1 - c)2(1- q) YJ + (1 - cyq. $(l +f). Thus, the equilibrium valuesare
+c4 = $7
f = (1 - 4) +4 + (q/2) 1 - (d2) ’
d(x) = 0,
f = 2 2 (y”2c;2 q ,
in case c = 0
and in case c # 0.
Specifications (2) and (3) show that random cross-fertilization as well as random seeddispersalproduce the sameamount of inbreeding, which depends solely on the rate of mutation and self-fertilization.
APPENDIX
The explicit representation of g(y), when using two-dimensional radially symmetric negative-exponential probability laws for seed and pollen distributions, has been stated before. Applying this result to
I aY)12
1-
1g(y)12
(1 - 4” (1 -
c)2
:=
W),
with
r = 11 y //,
344
HANS-ROLF
GREGORIUS
we get (l/4) %y@,2 + Yy 1 - (l/4) (Y:(oL: +
T(r)=
G = &
* [ 1 + q + (1 - q) (Y93(ay92+ y2)--31212 . [l + q + (1 - q) 019s(0192 +y2)-3/2](l R+j”
dr,
- j-@ rT(r)
(1
-
c)" -
y2)--3
42
9
YOL~~(~~~ + Y~)-~/~ . T(Y) dr,
0
sr yT(*)
d(x) =
. JoPo- II x II) dy
27~[(2- q(l - c)“)& + G) - (1 - q)(l - c)” R] ’
where Jo(.**) is the Besselfunction of order 0, defined as Jo(t) =
f
(--1)‘;~~~~2)t)2’
= $
. 6’
cos(t . cos r) dy
?&=O
For computation of G, R, and 4 by means of numerical integration, it is convenient to get an impressionof how fast the respective integrals converge to zero if the lower limit of integration approachesinfinity. This can be taken from the fact that LY~~(cQ,~ + y2)e3i2 < 1, 1Jo(t)/ < 1, and T(y) ’
a,6(01,2+ y2p-3 (1 - c)2 1 - a,8(Ly,6+ y2)-3 (1 - c)”
(1 - c)2 =
(1
+
(Y/c@)3
-
(17
*
Thus, 2
Ia
m y . T(y)
dy
G
=
s.y
(1+
;r;ayf8)J!(1
_
(1 - c)2 a,2 .
m s1+(ata,P
2
+ &
[ (1 -
dy 2
1 P-(1-c)”
dt = 2(laB 4”
1
*[ -+F-6 * In
c)2
((I - c)2 - 1 - (a/01s)2)2 q + (1 - cj2 (1 + (&“) + (1 + (4%)2)2 I
. arctan
[
2 + q+?d2 + (1(1 - cy 3112
4” II ’
where we used the substitution t = 1 + (r/01J2. In casecross-fertilization is assumedto take place at random, ashasbeen done in specification (2) of Section 3, this is equivalent to o/== 0. Employing the samereasoningasbefore and putting b = a(1 + q)2( 1 - c)~, we are led to .arctan [+-$I
-&.ln
[ ,a(iilJl]
-&. (9
CONTINUOUS
345
MIGRATION
Now, since R = 0, we get from Eq. (7), $(O) = G/((p + G)(2 - q(l - c)~)), where G has to be substituted by the expression (i). Furthermore, if we denote the right-hand side of(i) by B, in case 6 = i(l + f#(l
- c)”
and by B, in case 6 = (1 - c)~, it can be easily shown that in general (i.e., for arbitrary (YJ, the following inequality for G holds: B, < G < B, . From this and from the fact that R < G, it can be deduced that B2
(p + B,)(2:
q(l - c)~) ’ ‘(O) ’
p(2 - q( 1 - c)~) + 442 - (1 - c)“) ’ (ii)
A. J. 1947. Contamination in seed crops. III. Relation with isolation distance, Heredity 1, 303-336. FINNEY, D. J. 1952. The equilibrium of a self-incompatible polymorphic species, Genetica 26, 33-64. MALBCOT, G. 1948. “Les mathematiques de l’h.%CditC,” vol. 1, Masson et Cie, Paris. MALBCOT, G. 1950. Quelques schemas probabilistes sur la variabilitt des populations naturelles, Ann. Univ. Lyon, Sci. Sec. A 13, 37-60. MALBCOT, G. 1967. Identical loci and relationship, Proc. 5th Berkeley Symp. Math. Stat. Prob. 4, 317-332. MARUYAMA, T. 1972. Rate of decrease of genetic variability in a two-dimensional continuous population of finite size, Genetics 70, 639-651. SNEDDON, IAN N. 1972. “The use of integral transforms,” McGraw-Hill, New York. WRIGHT, S. 1943. Isolation by distance, Genetics 23, 114-138. WRIGHT, S. 1946. Isolation by distance under diverse systems of mating, Genetics 31, 39-59. BATEMAN,
POPULATION
BIOLOGY
8,
331-345
(1975)
A Model for Genetic Relationship in Areally Continuous Plant Populations HANS-R• Lehrstuhl
fiir
Forstgenetik
LF GREGORIUS
und Forstpjlanzenziichtung Received
June
der
UniversitCt
Giittingen
27, 1975
Representations are based on plant populations, continuously distributed over their habitats according to specified density patterns. Migration of genetic material takes place via pollen and seed dispersal. Monoecious plants with arbitrary rates of self-fertilization and dioecious plants are considered. The model was constructed with the intention of determining coefficients of inbreeding and kinship for all locations within the seed population after its dispersal over the habitat, assuming the respective genetic relationships of the parental generation to be known. To display the consequences of single components hidden in the general result, the following specifications have been treated: finite population size combined with random dispersal of seed, equilibrium states for hypothetically infinite population size with “limited” dispersal of pollen and seed, random dispersal of pollen, and random dispersal of seed.
INTRODUCTION
An important component of the mating system of seed plants is determined by the mobility of male garnets (pollen) as well as zygotes (seed) and the immobility of individuals (plants), including the female garnets (egg cells) they produce. In many cases, this mobility is such that pollen and seed are distributed continuously over a specified area and therefore, rise to spatially continuous dispersal of genetic material. Consequently, we are concerned with a more or less special type of continuous migration, a field that Wright (1943, 1946) and Malecot (1948, 1950, 1967) pioneered. The most direct method of measuring the effect of this sort of migration on the genetical structures of populations uses coefficients of inbreeding resp. kinship, where as Wright concentrated on the coefficient of inbreeding, having in mind calculations of the inbreeding-effective population size. Both authors avoided further specification of mating systems underlying the construction of their models, merely applying the results they obtained from their model assumptions to several situations. Since even more recent restatements of Malecot’s model, as, e.g., given by Maruyama (1972) do not refer to concrete migration modes, we thought it worthwhile, using mainly Malecot’s ideas, to build up a model that takes into account the particularities introduced by the above mentioned migration mode of seed plants. This model 331 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
332
HANS-ROLF GREGORIUS
should be suitable for determining coefficients of inbreeding as well as kinship becausethere might be a nontrivial relationship between them. Furthermore, the results should be stated in a form that makesit feasibleto decide whether they are identical to those given by Malecot. 1. THE MODEL For simplicity and becausedetails can be distinguishedmore clearly, we start with the discontinuous version of the model, i.e., we assumeany decomposition of the population’s habitat (which may be finite or not and one, two, or threedimensional) into disjoint colonie#. With decreasing size of these colonies, the continuous casewill be approximated at a limit. Organismsare assumedto be diploid. The following assumptionsare made. Each pollen arriving at a colony (originating from outsideor insidethis colony) pollinates all the egg cells present in this colony randomly, except seif-pollination, which may be included at a given rate. All membersof a colony produce pollen resp. egg cells to the sameextent. The over all pollen production is large enough to pollinate all egg cells. Each fertilized egg-cell constitutes a seed. All movementsof pollen aswell asseedoccur independently of each other. There is no variation of flowering times of the single plants. With theseassumptions,we derive at a first step the coefficients of inbreeding and kinship at all placesfor the seedpopulation after its distribution over the area, knowing the coefficients for the parental population. This will be done for two different mating types: Monoecism, including self-fertilization or excluding self-fertilization in form of pollenelimination (for definition see Finney, 1952), and dioecism. The probability laws governing migration (movement) of pollen and seedwill be stated in a way that makesthem directly applicable to the essentialparts of the derivations. Furthermore, since we aim at a continuous representation of the results, keeping the more illuminating discontinuouscasein mind, all parameters will be given in their continuous version. These parametersare (:= should be read as“is defined to be”): p(x) := Population density at placex. ~(x 1y) := Probability density that a pollen, which cameto fertilization at placey, originates from place x. s(x 1y) := Probability density that a seedthat landed at place y originates from placex. r(x ) y) : = Probability density that the male gamete(pollen) contained in a seedthat landed at placey originates from place x. g(x 1y) := Probability density that a seedthat landed at place y contains a gamete(male or female) originating from place x.
CONTINUOUS
333
MIGRATION
l$(x,y) := c oefficient of kinship of two individuals from the parental population located at placex andy. f(x) := Coefficient of inbreeding of an individual from the parental population located at place x. &(x, y) := Coefficient of kinship of two seedsthat landed at place x and y. f’(x) := Coefficient of inbreeding of a seedthat landed at place x. Obviously, r(x 1y) = sp(x / x) . S(Z/ y) dZ, integrating over the population’s habitat and g(x /y) = $(r(x /y) + s(x 1y)). location of 5 parents (pollen source)
location of 9 parents (fertilization place)
location of offspring (seed lands)
Some further remarks on the properties of P(y / x) have to be made concerning the rate of self-fertilization with monoecisousplants. In probabilistic terms, this rate is the probability that a pollen that came to fertilization at a given plant originates from this plant. If we denote this probability for a plant at place x by q(x), then 1- q(x) is the rate of cross-fertilization and we may define a probability density p’(y 1x) analogousto p(y 1x), which accounts only for cross-fertilizations. With thesenotations, ~(y ( x) may be given the representation P(Y
I4
=
(1 -
4(x))
.P’(Y
I4
+
S(Y
-
4
.4(x),
where S(y - x) is the Dirac function. Thus, ‘(Y
I 4 = J (1 - ~(4) P’(Y I 4 - 42 I 4 dx +
q(y)
. S(Y I 4.
Note that 4(x, x) andf(x) have to be distinguished.Since the influence of mutation onf’ and 4’ can be taken into account easily at any stage,derivations will be given without considering this parameter. Definitions of q$f, #‘, and f’ refer to one locus only and are usedin agreementwith Malecot (1948). a. Monoecism Computation off I. To obtain the probability that two alleles at one locus contained in a seedthat landed at place x are identical by descent,it is necessary to identify the male as well as the female parent of this seed, their coefficient of kinship, and their probability of contributing an offspring (seed)at place X. In caseself-fertilization is allowed, these parents may be the sameindividual, if they are assumedto come from the same infinitesimal area (colony) centered around a specifiedplace. First, let the maleand femaleparent come from disjoint
334
HANS-ROLF
GREGORIUS
areas dv and du centered around places v and u; then the probability that the male and female parents of an offspring at place x come from area dv resp. du and pass on identical genes to this offspring is p(v 1u) dv . s(u 1x) du . +(v, u) = (1 - q(u)) p’(v / u) dv . s(u j x) du . d(z), u). If both parents are the same individual, i.e., u = v and du = dv, they pass on identical genes to an offspring with probability 8(1 +f(u)), and thus, the above probability now is p(u / u) du * s(u / x) du * +(I +f(u))
= q(u) * s(u j x) du * &(I +f(u)).
Summing these terms, we get f’(x)
= 1 (1 - Q(U)) P’(V I 4 4~ I 4 NV> 4 dv du
+ j- ~(4 4~ I 4 HI +fM> da Computation of 4’. The procedure is essentiallythe sameasbefore, except that insteadof two allelesin one seed,we now have to consider two allelesfrom two different seedsthat landed at places x and y, say, and thus have probability g(u 1x) du resp. g(v Iy) dv of originating from a parent (male or female) from areadu centered around place u resp. from area dv centered around place v. The assumptionof equal pollen as well as seed production of all individuals within each colony implies, in caseu = v and du = dv, that all the p(u) du individuals within area du are equally probable asparents. Therefore, if two parents of two given offspring come from the samearea du, they are the sameindividual with probability l/(p(u) du) and passon identical genesto those two offspring with probability (l/2)( 1 + f(u)); on the other hand, the probability of both parents being different individuals and passing on identical genes to each of their offspring is (1 - l/(p(u) du)) *+(u, u). Th us, the probability that two parents of two offspring at placesx resp. y comefrom areadu and passon identical genes to these offspring is [ &-;(I
+fW)+(l
=-- &)
-&)+(uij
[; (1 + f (u)) - %4
*g(uIx)du*g(uIy)du u)] du
I 4 g(u I Y) du
+ C(u, 4 Au I 4 & I Y> du da Adding the obvious result for parents from disjoint areas,we have
4% Y) = I g(u I 4 cdv I Y>4h 4 du dv +s “‘“%pl
[; (1 + f (u)) - c$(u, u)] du
(lb)
CONTINUOUS
335
MIGRATION
Formula (1 b) looks very much like Malecot’s representation; nevertheless, it is a different result because the coefficient of inbreeding f’ in general cannot be calculated from 4(x, x), which is just limy+Z+‘(x,y). This difference is due to the special migration mode of seed plants which implies that offspring generally do not grow up at the place where their parents mate. Furthermore, note that the influence of the rate of self-fertilization explicitly appears in (la), while in (lb) it is hidden behind the g’s. The great importance of this influence is shown as we consider the two extremes for the values of 4, namely, Q(U) = 0 and q(u) = I. The expression q(u) = 0 means self-fertilization is prevented and changes (la) into f’(x) = j p’(v 1 u) s(u 1 x) +(v, u) dv du. (2) Here, q(u) = 1 of courseimpliesf(u) = fandf’(u) = f’, and thus,f’ = t(l +f), which is the well-known result for complete self-fertilization, i.e., crossfertilization prevented. b. Dioecism
By CL’resp. I-L”we denote the population density of the male resp. female population. Clearly, f’ again is given by Eq. (2). Computation of 4(x, y). As bef ore, the probability that two alleles, each contained in a seed that landed at place x resp. y, are derived from parents occupying disjoint areasdu resp. dv and are identical is
Au I 4 du - g(v I Y>dv . d(u, 4. For u = o, the above two allelesmay be derived from the samemaleor female parent and in all other cases, necessarily originate from different parents. Consequently, all the possiblecombinationslead to the following sum of probabilities: $. T(U j x) du * Y(U j y) du
1 .[ P’(U) du - ; (I +fk))
+ (l - T&d;)
d@, ‘11
-f ; * s(u I x) du . s(u 1y) du
z g(u 1 x) g(u [ y) +(u, u) du du + ; * [ ‘(’ . [; (1 +f(u))
- 4(u* u)] du.
I ;?(;I”
I y> + w;!;;;
’ ‘) ]
336
HANS-ROLF GREGORIUS
Putting all this together, we finally get
+‘4.s1 p’(u)
y(u I 4 y(u IY) + s(u I 4 4u I Y) P”(U)
1
*[;(1+f(u>) - Wu,]du.
(3)
Thus, even for p’ = p”, this result clearly differs from that obtained for mating type a.
2. HOMOGENEOUS AND TRANSLATION-INVARIANT
MIGRATION
The term homogeneousis used to describea situation where the populationdensity is the sameeverywhere in the habitat, i.e., p(x) = p (constant). Translation-invariant migration occurs if distribution of pollen as well as seed takes place independently of the location of the producing plants, i.e., p(x Iy) = p(x + a 1y + a) and s(x 1y) = s(x + a 1y + a), which implies p(x 1y) = p(x-y/O) =:P(x--y) ands(xIy) = s(x-~10) =: s(x-y). It might be difficult to realize a situation like this with finite size of habitat exactly, but it will be a good approximation in case migration distances are small compared to the whole habitat of the population. Naturally, translation-invariance is preserved with T and g. Furthermore, the validity of the relations 4(x, y) =: 4(x - y) = qS(y- x), which also holds for 4 and f(x) = f, f’(x) = f ‘, and q(x) E q is obvious. With these assumptions, the previous results obtained for two mating-types now appearto be: Mating-type (a). f’ = (1 - 4) - j ~‘(4 4(4 du + PU/‘U~ + f>
(44
4'(x) = j g(u)g(v) +(a - v + 4 du du +
(UP>[WW
+ f 1 - #91 * j & - 4 m da
(4)
If self-fertilization is prevented, (4a) changesto
f’ = j- P’(U)4(u) du
(5)
CONTINUOUS
337
MIGRATION
Muting-type (b). f’ = 1 P(U) co4 du
$‘(x) = J-g(u)g(w)4(u - w+ x>du de, + %(l +f) -C(O) . s
4P’P”
[(r(u)
(6)
r(u - x) + p’s(u) s(u -
x)] du.
Comparing Eq. (4) with the correspondingresult of Malecot (1967, Eq. 2.13), it now can be stated that they are even identical, but that specification of the inbreeding coefficient again differs greatly. One of the main questions in population genetics is whether there exists a state in which a population doesnot changewith respectto certain parametersin the course of time. In our case,this so-called stationary or equilibrium state is reached if +’ = + andf’ = f. To arrive at reasonablesolutionsof this systemof equations, we now include the possibility of mutation at a rate c, say, which implies, ascan be easily seenby the initial derivations, that the right-hand sides of Eqs. (la)-(6) must be multiplied by (1 - c)“. We will confine ourselvesto solutions for which j+(z) d x is finite. Thus, the Fourier-transforms of all functions appearing in Eqs. (4a)-(6) will exist and will be characterized by double points, e.g., for a function g given, its Fourier transform will be g(y)
= 1 eizyg(x) dx,
where i = - 11/2and yx is the scalarproduct of vectors x and y. Muting-type
(a). Equating 4 to + andf’ tofin (4) and (4a), we get
#J”(x) = g(x) g(-x)
. $.(X)(1 - c)” + h . g(-x)&c)
* (1 - c)2,
where
The inversion of this Fourier transform resultsin
Ye9 = p$
I dY)12 . (1 - cl2 * s e-zsy . 1 _ 1g(y)12 (1 - c)” dy,
where n = dimension of the habitat, and especially4(O) = h . G, where “‘+ 653/8/3-7
I B(Y)12 s
(1 - cl2
1 - 1g(y)12 * (1 - c)” dy
338
HANS-ROLF
GREGORIUS
and therefore
From that, Jp’(x) 4(x) dx = h . R, where R = &
.
s
I dY)12 (1 - cJ2
li;‘(-JJ)
I - j g(y)12 (1 - c)”
dy,
Furthermore,
f=
(1 -c12.4
+
2 - (1 - c)” . q from which, considering rearrangements:
2(1 - ‘j2 ’ (’ - 9) , $@) #@) du , I’ 2 - (1 - c)” * q
the previous results,
f
can be computed after some
-1 f=[
(74
((1 - q)(R/(~ + & + 4) . (1 - 4" - ' I
From this, again A=
(2-qq(1
-c)‘)(p+G;-(1
-q).(1-c)2R
*
Thus, finally
d(x> = .i e-ixu. [(I ~Y)I” . (1 - ~)~)/(l- Ia(r
. (1 - cJ2)14
(27~)~ [(2 - q(l - c)~)(~ + G) - (1 - q) . (1 - c)~ R] . For the case of pure cross-fertilization, imply
f=($$k
(7)
i.e., q = 0, the preceding formulae
__ - 1)-l
(84
and
w =
s e-izu . [(I gW12 (1 - c)“)/U - I &TY)/~ (1 (27+ [2(G
+ p)- R(1- c)“]
Equation (7a) shows that for pure self-fertilization, to be.
4”)l 4
(8)
(q = 1) f = 1, as it has
Muting-type (6). Apart from a few steps, all essential derivations again follow the above pattern and altogether are straightforward. The results can be stated as
f=(
2(G1 + ~&CL”) _ 1)-l R,(l - c)”
(9a)
CONTINUOUS
339
MIGRATION
and
G>
= -
s
e-&” . I-1” I +(Y)l” * (1 - CT + P’ I s’(YV (1 - 4” 1 - I ‘f(Y)12 (1 - cJ2 (2~)” [2(G, + 4p’p”) - I?,(1 - c)“]
I RI :=&-
(jr Y
(9)
w I +(Y)l” * (1 - de + CL’I s’(Y)12(1 - q dY and , 1 - I dY)12 (1 - 4” P” I f(Y>l” . (1 - cj2 + CL’I f(Y)12 (1 - 4” dy
I PC-Y)
1-
I a(Y
(1 - CY
Note that the properties of the Fourier transform imply i-‘(x) = #(LX) . s’(x) and because of that, g(x) = $(l + l;(x)) . s’(x), which makes it possible to give a representation of the “steady state” equations just by using the Fourier transforms of the initial probability densities p and s. Furthermore, the existence of solutions (7a)-(9) of course requires the finiteness of the integral of the functions
I ec(YN2(1 - 4” 1-
/g(y)12 (1 - c)2
resp.
CL” I L’(y)? (1 - c)” + CL’ I i(
(1 - C)2.
1 - I dY>l” (1 - 4”
Disregarding the type of the densities p and s, the steady state values of +, as given in (7)-(g), have the remarkable property that, because 1e-imy / = 1, necessarily 4(x) < 4(O), which by no means is a trivial statement if we consider, for instance, a situation in which all pollen coming to fertilization at a given plant originate from outside a certain circle around this plant and all the seed it produces again goes down outside a certain circle. Furthermore, the possibility of a population of seed plants attaining a state in which all individuals are equally related, i.e., in which the coefficient of kinship does not depend on distances, is to be excluded; a fact that can be deduced directly from formulae (4) resp. (6) by putting +(x) = 4 == +’ z b’(x), which leads to a contradiction. Analytical representations of the above equilibrium values for specific pollen and seed distributions require immense mathematical expense, even if most simple assumptions, such as one-dimensional habitat and uniform distributions of gametes, are chosen. On the other hand, formulae (7a)-(9) in many cases may be applied to obtain reasonable results by means of numerical integration. As an example for gamete dispersal in the two-dimensional plane, we choose an isotropic (radially-symmetric) negative-exponential distribution, which is widely accepted as realistic for most seed plant species (see for instance Bateman, 1947), i.e., for monoecious individuals
p’(x) = g
* exP(--ol,
II x II)>
resp.
s(x) = g
. exd--ol,
I! x IIL
340
HANS-ROLF
GREGORIUS
where 0 < 01~, 0~~, and Ij x /j is the Euclidean length of the two-dimensional vector x. In this case, the Fourier transform is known to be equal to the Hankel transform of order zero and thus, given by &‘(a2 + I/ x l12)-a/2(see, e.g., Sneddon, 1972, pp. 79, 315). From this g(x) = $(x) . (1 + 4 + (1 - 4) B’(4) = gag3(01: + I/ x 11y’2 * [I + Q + (1 - Q) %3(%2 +
II32ll2)-3’2l1
which enters into the calculations of G, R, and the integral part of c$,asgiven in (7). The complete representationis given in the Appendix. Finally, it should be stated that at least with radially symmetric probability densitiesp’ and s, the coefficient of inbreeding asgiven in (7a) may be regardedas an increasingfunction of the rate of self-fertilization q and a, decreasingfunction of the population density CL.The latter, of course, also applies to nonradiallysymmetric distributions.
3.
SOME
SPECIAL
CASES
The following specificationswere chosenwith the intention of giving an idea of how the singlecomponentsof the model affect genetic relationship. (1) Finite Population Size N = sp(x) dx. Each plant of a given generation results from a seed, randomly drawn from the whole seed production of the preceding generation; the plants are distributed over the area according to a population density given by CL. In terms of the present model this situation can be describedby assumingthe dispersalof seedto be at random, i.e., s(x / y) = p
=: c(x).
Thus, g(x I y) = $[J(l - q(u)) P’(X I 4 44 and therefore, from formulae (la) and (lb):
du + (1 + q(x)) 441 =: f(x),
f’ = + . j- (1 - q(u)) p’(v I u) 4~) dv du + Hl + f) * j- qO4 44 du = 4 * (1 - q) + &Cl +f)(r, where Q = J’q(u) c(u) d u is the averagerate of self-fertilization in the population, and 4’ = 4 + ((1/2W +A
- 4 . M,
with
jti’ = s (E(42/~(4) dx-
CONTINUOUS
341
MIGRATION
The influence of varying rates of self-fertilization clearly if, additionally, we assume cross-fertilization $‘(x 1y) = ~(x)/(N - 1). In this case, M = (1 /N)(l + where V = s(q(x) - q)s . C(X) dx is the variance of the (obviously I’ < a(1 - @).
can be displayed more to be at random, i.e., (1/4)(N/(N - 1))2 . V), rates of self-fertilization
(2) Homogeneous and Translation-invariant Migration; Cross Fertilization at Random. As has been stated before, this case requires a hypothetically infinite population size (resp. size of habitat), which may be regarded as the limit of increasing, finite size S, of habitat H. If we assume S, large enough, so that (4a) and (4) hold approximately, then p’(x) has to be replaced by S;;” and g(x) by a[(1 - 4) S;’ + (1 + a) s(x)]. Letting S, approach infinity and assuming the existence of lim++, S;;’ . s,, C(X) dx =: 4 (which then is the average coefficient of kinship), consequently, (4a) and (4) change to f’ = (1 - 4) - Q, + 4 . U/2)(1 +a
resp.
C’(x) = B . [; (1 - q2)+ + (1 - q)2] + (9)’
- / s(u) s(v) +(u - v + x) du dv
+ TT
[; (1 + f) - 4(O)] -1 s(u - x) s(u) du.
It can be shown easily that $’ = 4, which implies 6 = 0, in case the population started with an average genetic relationship equal to zero. Thus, the most apparent difference in our case compared to nonrandom cross-fertilization is that the coefficient of inbreeding does not depend on the coefficients of kinship; whereas conversely, there is a dependence. Because this case allows a comparatively simple analytical solution for the equilibrium value of 4(O), if again a radially symmetric two-dimensional negativeexponential distribution is applied (see Appendix, Eq. (i)), we shall present some numerical examples that refer to monoecious populations. An important property of this specification is that the equilibrium value of 4(O) just can be increased if nonrandom cross-fertilization is added (i.e., (Ye > 0; see Appendix, inequality (ii)). Our intention is to investigate the influence of the range of seed dispersal on 4(O). To give the term “range” a precise meaning, it is defined to be the radius D of a circle CD for which the probability SC, s(x) dx (i.e., the probability that a seed, which landed at a given place, is produced by a plant located at a distance not exceeding size D) has a specific value. In our case lc, s(x) dx = (or,2/2rr) * 1 exp(--(Ys // x 11)dx = cc,2 . iD t * exp(-a$) CD = 1 - (1 + CQD). exp(--or,D),
dt
342
HANS-ROLF
GREGORIUS
which adopts the specific value 0.96, if D is chosen as equal to 5101,. Keeping the probability 0.96 fixed, 01, can be calculated for arbitrary D according to CY,= 5/D and thus, C(O) may b e regarded as a function of D. The following graphic representations are based on these preassumptions and on $(O) = G/((P + GN2 - dl - 4”) ((f rom (7)), where G is given by Eq. (i) in the Appendix. The order of magnitude chosen for the parameters in question may be applied to several conifer species up to a certain extent.
07 p= G B
003[m-21,c~10~5
05
02 01 10
0
20
FICURG 1.0
70
50 D i-m1
100
1
t-
Ic
D Cm1
FIG. 2. The equilibrium value of the coefficient of kinship 4(O) of two very closely D of the seed dispersal. 9 = rate of self situated plants as a function of the “range” fertilization, c = mutation rate, and p = population density [Ind./m2].
CONTINUOUS
343
MIGRATION
Figure 1 demonstrates that severe differences in $(O), according to different rates of self-fertilization, may be expected for comparatively low ranges of seed dispersal only (D < 15 m), while the+(O) va 1ues for different population densities grow more and more alike with decreasing size of D, after having passeda certain range of greater diversity (5 m < D < 30 m), as shown in Fig. 2. In all cases,the quick decreaseif+(O) is striking. (3) Homogeneous and translation-invariant migration; random dispersal of seed. If we again apply the method of regarding the results as limiting values of increasing finite size of habitat, two ways of proceeding can be chosen: First, with help of specification (1) by taking q(u) = q, C(U)= S;;l, and p’(~ / U) = p’(~ - u); and second, with help of specification (2), by replacing s(x) with S;;‘. Both procedureslead to the sameresult, namely, $‘(x)
= (1 -
q
.$ = (1 -
42 * 4 = 4’ = $‘,
f’ = (1 - c)2(1- q) YJ + (1 - cyq. $(l +f). Thus, the equilibrium valuesare
+c4 = $7
f = (1 - 4) +4 + (q/2) 1 - (d2) ’
d(x) = 0,
f = 2 2 (y”2c;2 q ,
in case c = 0
and in case c # 0.
Specifications (2) and (3) show that random cross-fertilization as well as random seeddispersalproduce the sameamount of inbreeding, which depends solely on the rate of mutation and self-fertilization.
APPENDIX
The explicit representation of g(y), when using two-dimensional radially symmetric negative-exponential probability laws for seed and pollen distributions, has been stated before. Applying this result to
I aY)12
1-
1g(y)12
(1 - 4” (1 -
c)2
:=
W),
with
r = 11 y //,
344
HANS-ROLF
GREGORIUS
we get (l/4) %y@,2 + Yy 1 - (l/4) (Y:(oL: +
T(r)=
G = &
* [ 1 + q + (1 - q) (Y93(ay92+ y2)--31212 . [l + q + (1 - q) 019s(0192 +y2)-3/2](l R+j”
dr,
- j-@ rT(r)
(1
-
c)" -
y2)--3
42
9
YOL~~(~~~ + Y~)-~/~ . T(Y) dr,
0
sr yT(*)
d(x) =
. JoPo- II x II) dy
27~[(2- q(l - c)“)& + G) - (1 - q)(l - c)” R] ’
where Jo(.**) is the Besselfunction of order 0, defined as Jo(t) =
f
(--1)‘;~~~~2)t)2’
= $
. 6’
cos(t . cos r) dy
?&=O
For computation of G, R, and 4 by means of numerical integration, it is convenient to get an impressionof how fast the respective integrals converge to zero if the lower limit of integration approachesinfinity. This can be taken from the fact that LY~~(cQ,~ + y2)e3i2 < 1, 1Jo(t)/ < 1, and T(y) ’
a,6(01,2+ y2p-3 (1 - c)2 1 - a,8(Ly,6+ y2)-3 (1 - c)”
(1 - c)2 =
(1
+
(Y/c@)3
-
(17
*
Thus, 2
Ia
m y . T(y)
dy
G
=
s.y
(1+
;r;ayf8)J!(1
_
(1 - c)2 a,2 .
m s1+(ata,P
2
+ &
[ (1 -
dy 2
1 P-(1-c)”
dt = 2(laB 4”
1
*[ -+F-6 * In
c)2
((I - c)2 - 1 - (a/01s)2)2 q + (1 - cj2 (1 + (&“) + (1 + (4%)2)2 I
. arctan
[
2 + q+?d2 + (1(1 - cy 3112
4” II ’
where we used the substitution t = 1 + (r/01J2. In casecross-fertilization is assumedto take place at random, ashasbeen done in specification (2) of Section 3, this is equivalent to o/== 0. Employing the samereasoningasbefore and putting b = a(1 + q)2( 1 - c)~, we are led to .arctan [+-$I
-&.ln
[ ,a(iilJl]
-&. (9
CONTINUOUS
345
MIGRATION
Now, since R = 0, we get from Eq. (7), $(O) = G/((p + G)(2 - q(l - c)~)), where G has to be substituted by the expression (i). Furthermore, if we denote the right-hand side of(i) by B, in case 6 = i(l + f#(l
- c)”
and by B, in case 6 = (1 - c)~, it can be easily shown that in general (i.e., for arbitrary (YJ, the following inequality for G holds: B, < G < B, . From this and from the fact that R < G, it can be deduced that B2
(p + B,)(2:
q(l - c)~) ’ ‘(O) ’
p(2 - q( 1 - c)~) + 442 - (1 - c)“) ’ (ii)
A. J. 1947. Contamination in seed crops. III. Relation with isolation distance, Heredity 1, 303-336. FINNEY, D. J. 1952. The equilibrium of a self-incompatible polymorphic species, Genetica 26, 33-64. MALBCOT, G. 1948. “Les mathematiques de l’h.%CditC,” vol. 1, Masson et Cie, Paris. MALBCOT, G. 1950. Quelques schemas probabilistes sur la variabilitt des populations naturelles, Ann. Univ. Lyon, Sci. Sec. A 13, 37-60. MALBCOT, G. 1967. Identical loci and relationship, Proc. 5th Berkeley Symp. Math. Stat. Prob. 4, 317-332. MARUYAMA, T. 1972. Rate of decrease of genetic variability in a two-dimensional continuous population of finite size, Genetics 70, 639-651. SNEDDON, IAN N. 1972. “The use of integral transforms,” McGraw-Hill, New York. WRIGHT, S. 1943. Isolation by distance, Genetics 23, 114-138. WRIGHT, S. 1946. Isolation by distance under diverse systems of mating, Genetics 31, 39-59. BATEMAN,