Evolutionarily stable strategies for localized dispersal in two dimensions

J. theor. Biol. (1982) 94, 579-606

Evolntionarily

Stable Strategies for Localized Dispersal in Two Dimensions HUGH

Environmental (Received

N.

Biology Department, University, Canberra

28 January

COMINS

R.S.B.S., Australian 2600, Australia

National

1981 and in revised form 11 August

1981)

Evolutionarily stable dispersal strategies are calculated for a generalized two-dimensional stepping-stone model in which dispersal is not restricted to nearest neighbour sites. It 1s shown analytically that to an excellent approximation the ESS is independent of the dispersal pattern and is the same as the result of an earlier paper which used the simple but unrealistic island model (the dispersal survival rate of the previous paper must be replaced by an average value). When dispersal to different distances is under separate genetic control it is also possible to make general predictions concerning the fall-off of genetic correlation wlth distance. Two generalizations of the model are considered: random extinction of sites (creating empty sites which are opportunities for colonization), and perennation of existing occupants. These modifications are intended to bring the model closer to two real-world circumtiances: dispersal between discrete colonies of animals living on a patchy resource, and seed and pollen dispersal in evenly distributed plant species. Simulations are required to validate any extensions of the analytic model; I present a relatively fast technique for determining the ESS, which relies in part on the analytic results.

1. Introduction Dispersal is an apparently universal charqcteristic of living things. Indeed no species could long survive without the ability, actively or passively, to kolonize new habitats. It has recently been realized, however, that the colonization of new habitats is not the only advantage of dispersal; a non-zero dispersal rate is selected even if all suitable habitats are assumed to be permanently occupied (Hamilton & May, 1977). This dispersal, which of no benefit to the species, may be understood at the deeper level of gene selection. A gene which fails to compete for habitats where it is not represented must die out in the long term if it faces competition from an alternative allele which occasionally competes successfully for its habitats. 579

0022~5193/82/030579+28$02.00/0

@ 1982 Academic

Press Inc. (London)

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In the model there is an optimal balance between attempted “recolonization” (i.e. annexation of sites already occupied by the species, but dominated by an alternative allele) and the retention of currently occupied sites. A genome coding for this optimal dispersal rate has the unique property of being resistant to any mutation giving a slightly different rate. This rate should eventually be arrived at by selection, and thereafter resists further change; thus it is called an evolutionarily stable strategy, or ESS. Hamilton & May’s results have been extended (Comins, Hamilton & May, 1980; henceforth referred to as CHM) to the case where both colonization and recolonization are important. In the model of CHM sites can become vacant in two ways: all offspring in a particular generation may disperse and not be replaced by immigrants, or the site may be temporarily obliterated by “exogenous extinction”. The latter process, which takes no account of the genetic structure of the site population, is intended to model either the occurrence of natural calamities, or the depletion of resources in a site (in which case the “mortality” curve for sites is assumed to be exponential, or Type II). The generalized model also allows sites to contain more than one individual. This permits the breakdown of the ESS dispersal rate into two parts: the colonization part, which applies even if sites are indefinitely large, and may be derived by considering the reproductive value of individual dispersers, and the recolonization part, which falls off rapidly with increasing site size, and thus appears to be influenced by kin selection (Hamilton, 1975). An alternative explanation of recolonization may in fact be framed in kin selection terms (CHM). Although the results of CHM are suggestive, their quantitative application is made difficult by the mathematically convenient but highly unrealistic dispersal model. In the island model (Wright, 1969) dispersers leaving any site enter a pool; if they survive they are equally likely to return to any site whatever. This mechanism avoids the genetic correlations which build up if a more realistic limited range dispersal mechanism is used. Since recolonization depends on competing with different alleles, it seems that such correlations might well influence the ESS. The purpose of the present paper is to extend CHM’s results as far as possible to include localized dispersal. An immediate limitation, for mathematical tractability, is that it is no longer possible to consider the effects of having small numbers of offspring (which among other things can lead to more sites being empty). The probable effects of having few offspring must be inferred from CHM’s results. It has also not been possible to consider sexual reproduction. This does not appear to be important per se (CHM, 1980, Appendix C); however, some related phenomena warrant further study, for example, the differential dispersal of plant seed and pollen.

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The main results of this paper are obtained for localized dispersal on a regular two-dimensional square grid of sites. It is assumed that the probability distribution of dispersal distances and directions is the same for all dispersers of a given genotype, regardless of the position of their original site (for example, the probability of dispersing two sites to the right might have a fixed probability of O-1). Furthermore the disperser survival rate (calculated relative to the survival rate of non-dispersing siblings) may depend on genotype and on the distance and direction of dispersal, but not on the position of the original site. An additional assumption is required by the mathematics, namely that the dispersal and survival probabilities for each genotype must be “inversion symmetric”. This means, for example, that if the probability of dispersing two steps up and one to the right is 0.1, then the probability of dispersing two steps down and one to the left must also be 0.1. Thus the dispersal pattern (cf. Fig. l(a)) must appear the same when reflected twice; first in

..... ..... ... .. . .;. ;;+I; ;+d; .$ (0)

. J . . .

(bl

. . . . .

(cl

. . . . .

FIG. 1. (a) Example of an inversion symmetricdispersal pattern. Equal fractions of offspring disperse to each site in a diametrically opposite pair (b) Stepping-stone dispersal: each arrow represents one quarter of the dispersers (c) Hexagonal grid stepping-stone dispersal pattern transformed to a square grid: each arrow represents one sixth of the dispersers.

the x-axis and then in the y-axis. (In vector notation this operation results in both components changing sign, i.e. u+ -u. Thus I assume that the dispersal rate is the same to relative positions u and -u, for all vector displacements u). Inversion symmetry implies that there is no net drift of the population in any direction. The model can probably be extended to all cases where there is zero net drift (not all of which are inversion symmetric), however this would require more complicated mathematics. The present assumptions allow a comprehensive range of dispersal patterns (e.g. Fig. l(a)), including the standard stepping-stone model (Fig. l(b)) (Kimura, 1953). The results also apply to dispersal on a hexagonal grid (e.g. Fig. l(c)). This may be seen to be equivalent to a square grid by a linear transformation in which the ith row moves i/2 spaces to the right. Any inversion-symmetric dispersal pattern remains inversion-symmetric in

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the resulting square grid. Note that it is necessary (Weiss & Kimura, 1965; Kimura & Weiss, 1964) to retain a non-zero probability of island-model dispersal. This may be regarded either as an approximate treatment of residual long-range dispersal, or as a surrogate for mutation. In either case it prevents the slow spread of areas of genetic fixation, which is otherwise an unrealistic feature of all localized dispersal models (ibid.). Those results which do not depend on the proportion of island-model dispersal also hold in the limit of purely local dispersal. Exact ESS dispersal rates are calculated for two special cases: the stepping-stone model (Kimura, 1953) of nearest-neighbour dispersal on a square grid, and the equivalent hexagonal grid model. I also derive approximate results which apply to any dispersal pattern.

2. The Model

The model is basically intended to represent an animal population which, because of the patchy availability of some resources, is divided into numerous discrete colonies. Mixing occurs freely between the animals in any one colony. However, dispersal to other nearby colonies involves a degree of risk, depending on the distance dispersed. The colonizing lifestage is assumed to be very numerous (in the case of insects it may be more convenient to regard eggs as the colonizing stage, even though their dispersal depends on adult movement), but in each generation there is a genetic bottleneck, in which the gene pool of a site resides in only a few individuals. The width of this bottleneck determines the degree to which future offspring in a site are related. An extension of the model can be applied to the case of an evenly distributed population of plants. In this case the site is the area occupied by a single mature individual, and the bottleneck population is one. In the absence of exogenous extinction, I assume that each individual lives an average of Y “time steps” (one “time step” being the time to maturity) with a type II mortality curve. Thus on each time step there is a l/ Y probability that the new occupant is determined by competition between propagules (seeds and pollen cannot be distinguished in the present asexual model), and a 1 - l/Y probability that the previous occupant remains. This modification to allow survival of the previous occupant has no effect on the island-model ESS with one mature individual per site (Hamilton & May, 1977); however, in other cases it affects the inbreeding coefficient and the correlations between sites, and must be included explicitly in this part of the analysis.

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The actual model is further idealized. It is assumed that the possible sites form an infinite two-dimensional square lattice, and that dispersal occurs in an inversion symmetric pattern which is the same for each site. The sequence of events in each synchronous generation is shown in Fig. 2. Note that there are two life stages involved: the mature (adult) organisms Occupied

kadults

FIG.

2. Sequence

of events

I

!

Empty

0 adults

in each time step of the model.

which occupy sites, and their offspring, which disperse and compete for sites. Following dispersal the offspring of a given generation compete fairly among themselves and are reduced by predation and mortality to give the k adult organisms which the site can support (this number is assumed constant; if it were allowed to vary then the ESS would be that applicable to some intermediate value, which could only be obtained by simulation in the localized dispersal model). At any time during competition, the site can be obliterated by exogeneous forces; this happens with probability X and is assumed to occur completely independently in the various sites. Finally, the adults in each site produce a very large number of offspring, which remain at the home site or disperse to various destinations with genetically determined probabilities (in the case of insects the adults may leave the site to distribute their eggs in a genetically determined pattern). The factors influencing the dispersal destination are genetical, developmental and environmental. Consider the example of plant seeds. In its most general form the model supposes that plants of a particular genotype produce a range of seeds with different dispersal tendencies (e.g. winged or not winged, or having different weights or strengths of attachment to the capsule). The actual destination of each seed, and whether it survives

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dispersal, depends on environmental factors, such as the wind direction at the time of its release. In the model the range of different dispersal tendencies is represented by a finite number of “disperser types” (there are also non-dispersing “sedentary” offspring). Although some general results can be proved regardless of the number of disperser types, exact calculations become much more complicated as the number of types increases, so there is some incentive to keep this small. Due to the operation of environmental factors each disperser type has a probability distribution of dispersal destinations (all the distributions must be inversion symmetric; see above). Since the number of dispersers is assumed to be very large, the distributions give the proportion of dispersers of a given type which disperse from each site to a given relative position (represented by Via, where i specifies one of the possible relative destinations and a is the disperser type). The assumptions regarding dispersal mortality are also very general; the survival rate Pia may depend on both the relative destination i and the disperser type a (as in Hamilton & May, 1978 this is calculated relative to the survival rate of sedentary offspring). Note that there is an observational problem here, since the destination of a non-surviving disperser may not be apparent. However the model results only depend on the product Piauia (the fraction of type u reaching relative position i), SOany convenient definition of Via and pia can be used, provided the observable quantities PiaUia are correct, and the via’s sum to one for each disperser type. Inversion symmetry requires that Piouia = pinvia for any diammetrically opposite relative positions i and j. The model developed so far assumes that every adult organism produces the same total number of offspring, whatever the fractions of the various disperser types; that is, all disperser types (and sedentary offspring) involve the same maternal effort. Since survival probabilities are specified separately for each disperser type, this assumption is not an essential mathematical limitation. Differing maternal efforts may be accounted for by reducing the survival rates for the more difficult to produce disperser types. The “survival rate” pia is then reinterpreted to mean the number of disperser type a which can be delivered to relative location i using the same effort as is required to produce one sedentary offspring. As noted previously, a certain amount of island-model dispersal must be included in the model to prevent local genetic fixation. This is incorporated in the above scheme by requiring a certain proportion of at least one disperser type to undergo island-model dispersal, denoted by ecoe. Also the model requires that there be completely non-dispersing (sedentary) offspring, however a proportion of any disperser type (denoted uOp) may

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also fail to disperse (i.e. their dispersal distance may be so short that it falls within the original site), It was shown by CHM that the condition for a non-polymorphic dispersal ESS could be obtained by considering the interaction of two phenotypes; a wild-type with dispersal probability u, and a mutant form having a slightly different dispersal probability u + E. In general the average change in mutant frequency from one generation to the next is zero if E = 0, but has a term linear in E. If u is to be an ESS the linear term must also be zero, otherwise either LI + E or tr -E would be a superior strategy. This is only a necessary condition for the ESS: however, in CHM’s model it is also sufficient, since it specifies a unique dispersal probability, which must be the ESS if one exists. In the notation of the present model, CHM has only one disperser type (uol = 0, uaol = 1). The two alternative phenotypes differ in the relative proportions of dispersers and sedentary offspring. In the obvious generalization, I now suppose that mutations can slightly alter the proportions of any of the disperser types. The set of ESS conditions is obtained by considering the interaction of a wild-type phenotype with a series of mutants, each having a slightly increased frequency of one of the disperser types at the expense of the sedentary type. These equations are again sufficient to specify a unique ESS. 3. The ESS conditions Suppose that the ESS (or “wild-type”) genotype is being challenged by a mutant which has a slightly increased proportion of disperser type b, and correspondingly fewer sedentary offspring. Given the spatial distribution of mutant frequencies in the current generation, it is required to calculate

TABLE

1

Parameters and uariables in the model k

Y P 4. via

Pi.

mi fii K

number of adults per site proportion of mutants in a site exogenous extinction rate longevity of adults average mutant gene frequency proportion of offspring which are of a given disperser type proportion of disperser type a dispersing to a given relative destination i probability of surviving dispersal proportion of competitors from relative location j covariance, divided by p(1 -p) to remove dependence on mutant gene frequency recolonization value of relative location i

586

H.

the expected value of the to see whether the mutant Following competition ticular site are (see Table

N.

COMINS

overall mutant frequency in the next generation, frequency increases or decreases on average. and possible extinction the frequencies in a par1): Mutants (M): Wildtype

xn (1)

x(1- 77)

(W):

where x is a random variable, taking the value zero with probability X and one with probability 1 -X, and is uncorrelated between sites. If x is non-zero then n = i/k, where i is the number of mutant adults, and k is the total number of adults. After dispersal the number of competitors of each type in this site becomes

AM:[ 40+

W:

& + C POClUOCZ4Cl n

c thxdkw~a [ a

[4O+IPOaUOA.]

+CP:aUma4n(l a

+PObUObE

+PmbUcob&

I

1 (1

XT

+C

C Pi&i&n i [ a

+ PibUib&

1

XiTi

-x)p

(2)

X(1-7))+~~PinUia~aXi(l-~i)

-X)(1

-p).

In this equation +. and 4, are the wild-type proportions of sedentary offspring and disperser type a respectively. In the mutant 4. becomes 4. - E, while & changes to C$b+ E. The sums are over all disperser types a, and all relative displacement indices i, in the notation of the previous section (e.g. for the square grid stepping-stone model (Fig. lb), i = 1 for vector displacement (+l, 0), 2 for (0, +l), 3 for (--1,0) and 4 for (0, -1)). Additional terms are included for not dispersing (ponuoa), and island-model dispersal (pooauoon). The disperser type and destination frequencies must satisfy the conservation equations (3)

The frequency of mutant competitors 7” is calculated by dividing the first expression in (2) by the sum of the two expressions. Assuming E is small we have $‘+A/C+eB(C-A)/C2

(4)

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where

B

=X’V7CPOb~Ob

-

ll+C I

pibvibxiqi

+prnbvcob(l

-X)P

co= 40 f cn Poa~odPa. Since fair competition does not change the mean gene frequency, equation (4) can be averaged over all sites to give the’average mutant frequency p’ for the next generation. In the general case with X > 0 all possible patterns of extinctions must be taken into account. Although this is possible for simple dispersal patterns, the calculation of genetic correlations (see appendix) is always intractable in this case. Thus the general form of equation (4) is only useful in simulation (discussed later), and I assume X = 0 (no exogenous extinction) for the remaining analytic calculations. With X = 0 we have x = 1, and the variable C in (4) is the same for all sites. Thus (4) can be averaged to give the new mutant frequency p’=($‘)=p+e(B(C-A))/C*

(5)

where ( ) denotes the average over all sites. The mutant increases if E(B (C -A)) > 0. Therefore, as discussed above, a necessary condition for an ESS is that either & = 0 or (B(C -A)) = 0 (the other extreme, &, = 1, is not encountered since there are always some sedentary offspring in the ESS). This condition may be written informatively as Ab = Ao, where Ab and A0 are “values” of disperser type b and sedentary offspring, whose somewhat involved definitions appear below. Since a similar condition holds for every disperser type, the full ESS condition is A0 = A1 = A2 = . . . = AN. That is, the value of sedentary offspring and of every disperser type must be the same. This is an intuitively reasonable ESS condition, since the different types of offspring are assumed interconvertible. If the Abs differed, the postulated ESS could be improved by reducing the frequency of the disperser type with the smallest value. The value Ab of disperser type b can be written as a sum of site values, which are realized for every offspring that survives to reach the site at a

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given relative position: Ab

= POboOb

VO + 1 PibVib I

vi

+ PmbuoDb

(6)

where V. and Vi are the values of the home site and that at relative displacement i (note that V, = 1 for island-model dispersal). Since there is no colonization in the current fully occupied model (with X = 0), I call Vi the “recolonization value”. Thus Ab is a weighted sum of recolonization values over the destinations of disperser type 6. The recolonization value of a site is determined by the genetic dissimilarity between its inhabitants and those at the home site, and takes a maximum value of one for island-model dispersal, and a minimum value V. for not dispersing. The recolonization values Vi are themselves defined by a weighted sum, although less complicated approximate expressions will be discussed later. Let mj be the proportion of competitors in a site (i.e. offspring prior to competition) which are immigrants from the site at relative location j, moo be the proportion which are island-model immigrants, and m. be the proportion of sedentary competitors. Then we have m0= (

+o+CPO~VO~~~ a

C )/

(7)

where C is defined in equation (4). The Vi are then defined by Vi=mo(l-Foi)+Cmi(l-Fjj)+m,

(8)

where the “covariance index” Fi/ is a measure of the genetic similarity between pairs of individuals chosen randomly from sites i and j, and lies in the range zero to one (covariance index equals covariance divided by ~(1 -p), and Fii = Fji). A familiar example is the “inbreeding coefficient” Fm (Wright, 1969). The equilibrium covariance indices are calculated in the appendix, and are functions of the fractions of competitors from each source (mo, ml, . . .), the site size k, and the longevity Y (perennation does not affect the ESS condition and was therefore ignored above). Although the expressions are complicated, they satisfy certain recurrence relations which allow the

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combinations Vi to be written fairly simply (see next section). In addition the Vi themselves satisfy a summation condition. In order to use this we set equation (6) equal to A0 (this is the ESS condition, note also that A0 = Vo), multiply by &, and sum over b: VlJ/C=moVo+CmiVi+m,.

(9)

It is shown in the appendix that C miVi has a simple form, and that equation (9) can be reduced to a condition on the site size k: k = 1 + [co/(mo - co)][ Y* - 2 YAz(O){mo/co+

2 Y - 2}]

(10)

where co is defined in equation 4 and AZ(O) is defined in equation (A32). This expression can be further simplified, since to an excellent approximation AZ(O) = $(2 - Y-’ + moY-‘)-‘. The final results when either Y = 1 or k = 1 (the cases of interest for animal colonies and plant populations respectively) are: k + 1 + (mo - CO)-‘(1 + mo)-‘(moco

- mo + co)

m0 = co(l + mo).

To appreciate the remarkable co=pl#

(114

(lib)

nature of these results, we observe that -V)

m0=P0~~-v~l[~0o(l-u)+~ul

(12)

where PO is the average survival rate for offspring which do not leave the home site po=

4O+Cpoa~o. 4 a]/[ 1 a

4,+;

VodPa]

(13)

and p is the average dispersal survival rate

and u is the total ESS dispersal rate. fJ = C C ~ia4n i a

+ C ~cOa4a. a

(15)

When co and m. are substituted from equations (12), equations (11) link the total ESS dispersal rate 21to the site size k and the average dispersal and non-dispersal survival rates p and PO. But in the case where PO= 1 (true

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in particular if only sedentary off spring remain in the home site), equations (11) are exactly the same expressions as are obtained for the island model (i.e. setting X = 0 in CHM’s results). Therefore if PO = 1, then any pattern of local dispersal can be replaced by simple island-model dispersal, and the total ESS dispersal rate will be unchanged, provided the island-model dispersal survival is equal to the average dispersal survival p in the original case. The result applies whenever all sites are occupied (i.e. X = 0), regardless of the number of disperser types. In general this result does not allow immediate calculation of the ESS total dispersal rate, unless there is only one disperser type or the average survival rates for all types happen to be equal. Otherwise the average dispersal survival depends on the ESS dispersal pattern. However equation (11) could be used to check if an observed dispersal pattern satisfies the ESS condition, or to provide limits on possible ESS dispersal rates if all survival rates lie in a range p1 < p < ~2. Figure 3 illustrates the results for

FIG. 3. Comparison of ESS total dispersal rate for localized dispersal and island-model dispersal: (solid curve) island-model with survival p = 0.7, (squares) square-grid stepping-stone model with pl= 0.7, pm = 0.7 and one disperser type, (stars) hexagonal-grid stepping-stone model with p1= 0.7, pm = 0.7 and one disperser type.

two simple cases (with only one disperser type): stepping-stone (nearest neighbour) dispersal on a square grid (Kimura, 1953) with equal survival rates for short-range and island-model dispersal, and a similar case for a hexagonal grid. The square-grid stepping-stone model represents a worst case for the approximation (ll), however the results are still extremely close to the island-model values.

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4. Patterns of loai

591

diapersal

Equation (6) states that the value Ab must be the same for each disperser type which plays a part in the ESS. This condition was interpreted as requiring that each disperser type must have the same “average recolonization value”, where the recolonization value of a site at relative position i is Vi (home site value is Vo, island-model value is 1) for each offspring arriving there. It is shown in the appendix (A36) that to a quite good approximation fori#O

Vi+l-FtJiy

(16)

where Foi is the average covariance index [covariance divided by ~(1 -p)] between the ith relative location and the home site. I now wish to consider the case where offspring of a particular disperser type always disperse to a roughly similar distance. I assume that all islandmodel dispersal is performed by a separate disperser type, and that nonsedentary types always disperse (oOb = 0 for all b). Then the ESS conditions become pm= vo T pibuib

where pa is the island-model (18) we can write

K

=

VO

07) (18)

survival rate. Using equations (16), (17) and (pit1

-FOi))b

+POS

(19)

where the expectation is over attempted destinations for type b. In particular, if the survival rate for type b offspring has little variation (due to their narrow band of dispersal ranges) then l-

(FOi)b

+ pco/(pi)b-

(20)

That is, one minus the covariance index at the dispersal range of type b is inversely proportional to the survival rate for type b offspring. The intuitive basis of the ESS can now be comprehended. If there is a range where sites may be found which are little correlated with the home site (i.e. have a high recolonization value) then the disperser type with a corresponding range is increased in frequency. However this dispersal tends to increase the correlation with those sites. Eventually their value is lowered to the point where no further dispersal is worthwhile, due to the survival penalty associated with this particular disperser type. At this point the covariance satisfies equation (20).

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Note that dispersal to one range also builds up correlations to other ranges, since the offspring of the original dispersers may themselves disperse. This means that it is very difficult to get higher correlations between distant sites than between closer ones (I exclude certain perverse cases which rely on the discrete nature of the model grid, for example dispersal to even distances only). Thus if offspring dispersing to a short range have a lower survival rate than those with a longer range then the shorter range disperser type will not form part of the ESS (in particular all survival rates for localized dispersal must exceed the island model survival rate p,). In such cases we have 1 -(F~i)b
(22)

= pm/pa

where po = 1+ (k - l))‘(l

+ ma)-’

(23)

and ma is given by equation (7) (this form of the result is valid for the two cases I consider interesting, namely Y = 1 or Y 2 1, k = 1). Thus compared with the sequence of equation (20) the inbreeding coefficient is unexpectedly large and the central site is particularly unattractive, especially if k is small (i.e. if kin selection is important). As mentioned previously, island-model dispersal can be included either as an approximate treatment of long range dispersal, or as a surrogate for mutation. In the latter case pm = 1. However the mutation rate is small and fixed, so equation (17) no longer holds. Thus the pm in equations (19) and (22) must be replaced by an unknown coefficient of proportionality. Although equations (20) and (22) describe the ESS covariance pattern, they contain insufficient information to calculate the relative frequencies of the various disperser types. The full solution procedure is arcane. If we define 8 = F&Z(O) (see equation A16) then (17) and (18) may be written I-gwo=p, (l-~Ob~Ob)(l-~W0)=C~ib~ib(1-5Uli)

(24)

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where WC, and Wi are combinations of the Z(1) correlation integrals (see Equation A30). Each Wi has an integral representation in terms of the proportions mo, ml,. . . of competitors of a given disperser type. If & is arbitrarily fixed (8 > 0) then the number of free variables equals the number of equations in equations (24) (there is one less of each if island-model dispersal represents mutation) so they can be solved simultaneously for the ESS, This not particularly easy process is further complicated by the possibility that some 4b are equal to zero. In, this case we lose one free ,

I

LO-’

(a)

Site size I

-I

“O-

L ;

(b),-----I

----

----

-------

‘-03

:

j

‘r

3 B

ii i

O.=-

h

\

-x-w, 50

,T 0.7 loo

Site size

FIG. 4. Dispersal ESS as a function of site size for the simplest model with more than one disperser type (via. pure stepping stone dispersers and pure island-model dispersers). A square-grid stepping-stone model is used, with p1 = 0.8, poD= 0.7 and separate genetic control of localized and i&and-model diipersal: (a) local dipersal rate @lid) and island-model diapersal rate (dashed), (b) total dispersal rate (symbols), rverage survival rate (dashed) and the island-model ESS dbperaal rate which would correspond to the average survival rate (solid). The coincidence of the symbols with the solid curve illustrates equation (12).

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variable and one equation for each such &, (thus we still have point solutions). Finally equation (A16) is used to determine the site size for which this particular solution is an ESS. Figure 4 shows the results of such a process for a square grid steppingstone model with unequal local and island-model survivals. As well as confirming the predictions of the last section the results show that the range of dispersal tends to increase as k decreases and inbreeding increases. This is in accord with the idea that kin selection leads to more altruistic behaviour. An alternative explanation is that if there are fewer individuals to recolonize per unit area, then dispersal must go further afield. 5. Non-zero

exogenous

extinction

Although the exogenous extinction case cannot be handled analytically, it is not necessary to resort to straightforward multiple allele simulations as used by Roff (1975). Instead the case of two dispersal-neutral alleles is simulated, and this generates a representative sequence of circumstances in which two almost neutral alleles (see section 3) might compete. The second term of equation (4) is calculated to determine to what extent modified dispersal would have been advantageous in this circumstance, and the average of such terms (cf. equation 5) is multiplied by a small constant and added to the assumed ESS dispersal rate, in order to generate the correct ESS by a simulated “learning” process. This technique is easily generalized to more than one disperser type without requiring any additional alleles to be simulated. The simulations so far performed are of the stepping-stone (nearest neighbour dispersal) type on square and hexagonal grids (only one disperser type is considered). The arena is a 10 X 10 grid which is assumed to be part of an infinite (unsimulated) grid in respect of island-model dispersal. Since the simulation is of neutral alleles the genetic composition of island-model immigrants is invariant (this is of practical importance since it prevents fixation of either allele, an event which would interrupt the sequence of inter-allele comparisons). In order to avoid edge effects I use periodic FIG. 5. Dispersal ESSes obtained by simulation of the square-grid stepping-stone model (squares), and hexagonal-grid stepping-stone model (stars): (a) validation runs with survivals p1= po3= 0.8 (upper) and p1= po3 = 0.2 (lower), 10 per cent island-model dispersal and no extinction (compared with analytic results of equation (12)), (b) validation runs with 0.8 and 0.2 survivals, 95 per cent island-model dispersal and 50 per cent extinction (compared with analytic result of CHM for the island model), (c) runs with 0.8 and 0.2 survivals, 10 per cent island-model dispersal and 50 per cent extinction. The comparison curves are CHM’s analytic results for the island model. Note all results are for a single dispersal mode.

ESSES

FOR

LOCALIZED

DISPERSAL

Site size

I 0-l

2 6e 2 g 05‘S = F

q(b) F--m A\ hn- -+I----

I

!*

0

.

----f-J-------------m.

0-,0

I 50 Site size

Ji----l 0

Site

50 we

FIG. 5

100

596

H.

N.

COMINS

boundary conditions on both pairs of edges (i.e. in jumping off one edge one reappears at the opposite one). The hexagonal grid is similar to the square one except that every second row is shifted half a space to the right. In each step it is first decided randomly whether or not a site will go extinct, in which case further calculations are skipped. Dispersal is then simulated (this is deterministic because of the assumed large number of offspring). The number of mutant adults is found from the frequency of mutant competitors by computing binomially distributed pseudo-random numbers. Figure 5a shows a comparison of square and hexagonal grid simulations with X = 0 with the corresponding analytical result. The simulation results are very close to the analytical predictions. Figure 5b further validates the simulation technique by comparing the analytical island-model results of CHM with simulations for X > 0 and very little local dispersal. Finally, Fig. 5c shows a comparison of simulation results for X > 0 and localized dispersal, with the corresponding analytical island-model results. The islandmodel is evidently still a good approximation, even when there are vacant sites. This may be understood from the following argument. Dispersal to a neighbouring site is inferior to dispersal to a randomly selected site since the nearby home site is known not to be empty, so the destination has a greater number of competitors than average. However for moderate extinction rates this consideration only affects about $ of the immigrants to a site (2 for a hexagonal grid) so that the island-model assumption of random dispersal has only a minor effect on the calculated reproductive value for dispersers. This is true for general dispersal patterns, provided that the range of dispersal is much greater than the correlation range of extinction (in the present model extinction is only correlated within a site). 6. Conclusion The results of earlier calculations of ESS dispersal (CHM, Hamilton & May, 1978) suggest that in addition to the benefit of colonizing empty sites, dispersal confers an additional advantage in enabling “recolonization” of sites where a competing allele predominates. Thus the ESS dispersal rate is non-zero even if all sites in the model environment are continuously occupied. In this paper I introduce an additional sophistication by allowing dispersal to be localized; that is, it takes place between nearby sites in a twodimensional square or hexagonal grid, rather than between any two sites with equal likelihood (as previously assumed). It might be expected that the gene frequency correlations introduced by localized dispersal would tend to reduce the chance of competing with alternative alleles at dispersal

ESSES

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DISPERSAL

597

destinations, and that the ensuing lack of recolonization benefits would result in the ESS dispersal rate being lower. However it might equally be argued that the dispersal rate should increase, in order to seek recolonization further afield. In fact it is shown here that these two effects cancel almost exactly, no matter what pattern of local dispersal is used. Thus the ESS total dispersal rate is essentially unaffected by the “island model” approximation of an infinite dispersal range. This result also holds if there is more than one disperser type (e.g. both heavy and light seeds). The result is shown in general only for the case with no vacant sites. However, using a simulation technique, it is also shown to hold approximately in the more general case where some sites are unoccupied, at least when dispersal is to nearest neighbours in a square or hexagonal grid. In fact, this approximation is expected to apply to all dispersal patterns if the site occupation rate is fairly high. An additional condition (satisfied in the current model) is that the range of dispersal is much greater than the correlation range of the events which create vacant sites. If the distance of dispersal is under genetic control then the localized dispersal model makes specific predictions about the fall-off of genetic covariance with distance which is induced by ESS dispersal. For a fully occupied environment the ESS requirement is that the average “recolonization value” must be the same for each separately controllable dispersal mode, where the recolonization value for the site at the Ah relative location is V, for each offspring arriving there (see equation 6). It is found that Vi is approximately equal to the covariance index between the ith relative location and the home site (covariance divided by ~(1 -p), where p is the average mutant gene frequency). If the ranges and survival rates (which may include parental effort per offspring) for a particular disperser type are relatively constant, then the ESS condition implies that one minus the covariance index at the given range is inversely proportional to the corresponding survival rate (in some cases the constant of proportionality is also predicted). Although this result describes the covariance induced by ESS dispersal, it is not sufficient information to ‘calculate the ESS dispersal distribution; a rather involved procedure is required, along the lines established by Weiss & Kimura (1965). I suggest two applications for the localized dispersal model: dispersal between discrete colonies of animals living on a patchy resource, and seed and pollen distributions in continuous populations of plants. In the first case the above results predict an island-model type relationship between total dispersal and the average dispersal survival rate and thus provide an interpretation of the dispersal survival rate used in CHM. The plant case is more difficult for several reasons: seed and pollen have markedly different

598

H. N. COMINS

dispersal distributions, perennial plants such as trees may retain their sites during the time when propagules are attempting to colonize, and the definition of non-dispersal is not as straightforward as in the discrete colony case. The distinction between seeds and pollen has intriguing implications, since seeds are useful for both colonization and recolonization, whereas pollen permits long distance recolonization but relies on seed dispersal for colonization. The analytical model should predict pollen distributions in species with low colonizing ability (seeds being regarded as effectively non-dispersing). The more general seed and pollen case, where colonization is also important, is an interesting topic for future simulation studies. It is shown in the analytical treatment that the absolute level of site retention has no effect on the dispersal ESS. However, it is possible that there is a physiological tradeoff between site retention and reproductive rate. In this case it can be shown (for one adult per site and no empty sites, see Appendix) that the balance between site retention and reproduction depends on the intensity of competition between propagules (cf. equation A48). The fact that “sites” are contiguous may mean that there is no class of propagules which can convincingly be regarded as non-dispersing. The analytic treatment leading to the correspondence with the island model requires that there are sedentary offspring. Otherwise solving the ESS conditions is much more complex. It might be thought that adaptations leading to increased site retention are in some way equivalent to producing sedentary offspring. If there are no sedentary offspring but longevity is interconvertible with reproduction, then a new ESS condition is obtained (A49), involving the “marginal conversion efficiency” into site retention. If the conversion efficiency is known from physiology then this condition may be used to predict the ESS proportion of abortive dispersal (returning to original site), and replaces the island model ESS condition. I would like to thank Drs W. D. Hamilton, R. M. May and I. R. Noble for helpful discussions. I would also like to thank the referees and P. M. Cochrane for helping to clarify the text.

REFERENCES ABRAMOWITZ, M. 8~ STEGUN, I. A. (1964). Handbook ofMathematicalFunctions, Washington D.C.: National Bureau of Standards Applied Mathematics Series 55. COMINS, H. N., HAMILTON, W. D. & MAY, R. M. (1980). J. theor. Biol. 82,205. HAMILTON, W. D. (1975). In: (R. Fox, ed.), Biosocial Anthropology. ASA Studies, London: Malaby Press.

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599

HAMILTON, W. D. & MAY, R. M. (1977). Nature 269,578. KIMURA, M. (1953). Annual Report of the National Institute of Genetics, Japan 3,63. KIMIJRA, M. & WEISS, G. H. (1964). Genetics 49,561. MALECOT, G. (1969). The Mathematics of Heredity. San Francisco: W. H. Freeman and Co. ROFF, D. A. (1975). Gecofogia, Bed. 19,217. WEISS, G. H. & KIMURA, M. (1965). J. appf. Prob. 2,129. WRIGHT, S. (1969). Evolution and the GeneticsofPopulations. Chicago: University of Chicago Press.

APPENDIX

Al.

The Square-grid

Stepping-stone

Model with no Extinction

The equilibrium covariance indices required for equation (8) may be calculated using the method of Weiss & Kimura (1965) (see also Malecot, 1969), except that it is possible to avoid an approximation which limits their results to small dispersal rates. All of the above quantities have terms linear in E, the difference in dispersal probabilities between the two genotypes. However these give only second order terms in equation (5) and disappear in the limit E + 0. We can therefore set E = 0 and consider the classical case of two functionally equivalent alleles. Setting X = 0 and E = 0 in equation (2) we obtain the fraction of mutant competitors at a site previously having mutant gene frequency n : TN =

mOq

+ ml(~l)NCl+

%~p

(Al)

where mo, ml, and ntoo are defined in equation (7). As an aid to computation I introduce the concept of a “neighbour class”, which is a set of relative locations all of which have the same value of PiaUia for all disperser types. Because of the assumption of inversion symmetry diammetrically opposed locations always belong to the same neighbour class. In this appendix the indices i and j run over neighbour classes, not sites. The stepping-stone model has only one neighbour class “NC]“. A total of k survivors is chosen from the competitors to become the new generation of adults. The resulting gene frequency is ?J’= Tf+[(r/“)

642)

where l(n”) is a discrete random variate with mean zero and variance

~!%“N8 = rl”U - 77”)lk

(A3)

and ( >arepresents the average over the probability distribution of random fluctuations in a particular site. Note that these variations are clearly not correlated with the corresponding variations in any other site. Equation (A3) is a standard result for the binomial distribution.

600

H.

N.

COMINS

Let CY= 77-p, cz’ = q’ -p and (Y” = q” -p be the deviations of the various gene frequencies from average. Then Weiss & Kimura rewrite equation (Al) as a”(0) = Lx(O) (A4) where 2 is a shift operator acting on the spatial position at which quantities are to be evaluated. It is defined as follows for the square grid: iC=m,+(mi/4)[S:

+s; +s:

+s,]

645)

where S:, S; are unit shift operators in positive and negative x directions, and Sz, S; are unit shift operators in the positive and negative y directions (for example, S,(Y(~, 4) = a(2,4), Sc(u(3,4) = a(3,5), (S: +S;) a(3,4) = a(4,4)+~~(2,4)). Let c(I) be the covariance in gene frequency between one site and another site a vector distance I away and let r(l) = c(I)/c(O) be the correlation coefficient. Then Weiss & Kimura show that at equilibrium (1 - i*)r(l)

= 0,

for If 0

(1 -JC2)c(0) = (7jYl-

646)

$‘)/k)

(A7)

where ( ) denotes an average over all sites. Using 77”= cr”+p, equation (A4) and Weiss & Kimura’s becomes (1 YC2)c(0) =p(l -p)/k -L^*C(O)/k. Rearranging,

appendix,

(A7) (A8)

and using r(l) = c(I)/c(O), we get

c(0) = ’

PO -PI

(A9)

k -(k - 1)Z2r(0)’

Note that although r(0) = 1, it is not possible to substitute this value in equation (A9), because of the preceding operator. Weiss & Kimura solve (A6) by constructing the following set of integrals: *=cos (l,e,) cos (&) 1 -H*(e) where H(8) = m. + (mJ2)

d8i d&

(AlO)

(cos 8i + cos e2). It can be verified that

(l-i*)Z(l)=O, (1 -i2)Z(0)

forl#O = 1

(All)

Thus a solution of (A6) is given by r(U = Z(O/Z(O)

(A12)

ESSES

FOR

LOCALIZED

601

DISPERSAL

and (A9) becomes PC1 -P)

(A131

‘(‘) = 1+ (k - l)[Z(O)]-” They further show that 1 Z(l)

=;

[CLUl,

12)

+ (-F+%Mh,

I2)I

(A141

where a = 2(1+ m,/m*) b = 2[(2 - m,)/m1-

l]

Ii, I2 are the x and y components of I and Q,(l, I’) = lam e?&)&(t)

dt

(Al3

where I,(x) is a Bessel function of imaginary argument. In order to evaluate equation (8) we need the following quantities taking symmetries into account): Foe = c(O)/[p(l Fo1=

horn

-p)] = (1 + (k - l&Z(O)]-‘}-’

(Ala

(A171

1)

FII =Fc~,[l+2r(l,

(after

l)+r(O,

2)]/4.

0418)

The necessary integrals from equation (A15) can be evaluated using Weiss & Kimura’s analytical and recurrence relation techniques, giving:

Q,(O, 1) = -;+; Q,(l,

l)=;($l)(i)

Q,(O,2)=2aQ,(O,

Qa(O,0)

(Al%

K@+(i) l)-2Q,(l,

l)-Q,(O,O)

where K(x) and E(x) are complete elliptic integrals of the first and second kind respectively (note that these are sometimes defined with arguments which are the square of those used here (e.g. Abramowitz & Stegun, 1964)).

602

H.

A2. General Localized

N.

Dispersal

COMINS

Model with no Exogenous

Extinction

The general model incorporates a probability 1 -V that the previous adults retain the site (where v = l/ Y). In this event the new gene frequency n’ equals the old frequency 7. Alternatively, if the site is opened for competition then the fraction of mutant competitors is (setting X = 0 and e = 0 in equation 2): 77” = m07j + C Thus the shift operator

mi(T)NCi

(A201

+ m4.

L is redefined as i

= m0+C

t.421)

mi(~(y)hci

where j(y) is a shift operator which adds y to the co-ordinate vector, and y runs over the co-ordinates of a site in the appropriate neighbour class (i). Defining (Y = n -p, a’= v’-p, (Y”= +p as above we have a”(0) = Jca(0).

(A22)

We now calculate the new expected correlation between the central site and the neighbour at relative position I. Taking into account the various possibilities regarding site retention (assumed uncorrelated between sites) this is (for I # 0): (cz’(O)a’(I))

= (1- V)2((Y(o)(Y(1))+2Y(l

- v)(a(l)i(Y(O))

(x423)

+ Y2(cY(l)i2a(0)) where I use the inversion symmetry of the terms (see Weiss & Kimura’s (d(O)a’(O)) Assuming to (A6):

equilibrium,

of the shift operator appendix). Similarly

= (1- v)(a(O)cy(O))+

V((Y(0)i2~(O)).

equation (A23) gives the following

{1-[(1-v)+v~]2}r(l)=0,

to rearrange

(~24)

result, analogous

forl#O.

Equation (A24) gives a result identical to (A7), and therefore valid. The solution of (A7) and (A25) is analogous to the previous (A12) and (A13) where Z(1) = (27r)-* j lo2mcos (1.9) de/[1 -H2(8)]

two

W-J (A9) is also case, giving

KW

ESSES

FOR

LOCALIZED

603

DISPERSAL

where (~27) and y runs over co-ordinates in the appropriate class. The proof of this result requires that dispersal be inversion-symmetric. Let us now calculate the neighbour-class correlation factors Fij which are required for the ESS. Equation (A16) still holds for FoO. The remaining Fij are given by Fij = (Foo/Z(O))Zij, where Zj

= U(Yi

and yi and yj run over neighbour

6428)

+ Yj)hCihCj

classes i and j respectively. In particular

Zoi = (2~)~’ j lo*% (COS ( y mO))NCi de/[1 -H*(O)]. Generally we require particular Hence I define

combinations

(~29)

of the fij (cf. equation

Wi = f?ZoZCli+ C WtjZija i

8).

(A30)

That is, m

= (I(Y))NCiv

where I(Y) = cm-*

i 1 J-02T

cos

(y.8) de/[1 -P(e)]

(A31)

and the shift operator acts on y. Note however that the integral (A26) may be split in two by factorizing the denominator; that is Z(1) = AI(I) +A&), where A,(l)=(8n’)-‘1

~02~c0s~.edei[l-~(e)~ (AW

~,(1)=(82)4[

~02wc0s~.edei[l+zqe)~.

It can be shown that these integrals satisfy the recurrence relations (l-l)[A~(l)+Al(-l)]=S,lv

(A33) (;-l+i)

[A2(1)+Az(-Vj=S,/v.

604

H.

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COMINS

where Si = 1 if I = 0 and is zero otherwise. Applying these relations to (A31) we find wi =Zoi -(2/v)(A2(y)hci.

(A34)

However AZ(l) is always much smaller than Al(I) for I# 0, particularly if dispersal rates to individual sites are small. Also AZ(O) is equal to 3[2 - Y + vmo]-’ to first order. Therefore we have the approximate relationships Wi+ZO,

for i#O

w,+z(0)-[v(2-Y+vmo)]-1 or in terms of the recolonization

(A35)

values Vi :

l-Vi+FOi,

fori#O

1-vfJ~&J-[F,,/Z(0)][v(2-V+vmo)]-’.

(A36)

Finally we calculate the combination of Wis required for equation i.e. C miWi. Using (A30), (A33) and (A34) this reduces to (1-mo)Z(0)+(2/v2)A~(0)[2-2v+vmo]-v-2.

(9),

(A37)

Substituting this in (9) and using (A34) for W. we arrive by a series of manipulations at Z(O)(Fii

- l)(mo/co-

1) = v -2-2v-1A~(0)[mo/co+2v-'-2].

(A38) Substituting

(A16) for Foe we have equation (lo), since Y = Y-‘.

A3. ESS Condition on Longevity Suppose that the number of type b dispersers can be increased at the expense of a reduction in longevity Y, and that X = 0 and k = 1. As before let Y = Y-l, and let r be the total number of offspring per adult. Since k = 1 a site is either owned by a wild-type adult or by a mutant, for which Y + Y + Sv, rq5b+ r&, + Sr (r& remains the same for a # b). If the site is owned by a wildtype, then the probability of a mutant takeover is v(Ar + GSr)/(Cr

+ GSr)

(AW

where A and C are defined in equation (4) (with x = 1) and G

=

‘IIPObvOb

+c

Pib&b(v)NCi

+hbUmb@

(A40)

ESSES

FOR

LOCALIZED

605

DISPERSAL

Assuming that Sr is small and averaging over all wildtype sites we have an average takeover probability of

4WCh+ Wr)(G(C -Ah/C*1

(A411

where ( ),, denotes the expectation over wildtype sites only. Similarly average probability of a mutant site being taken over by wildtype is

b + Sv)[((C-A)IC)I - Wd(G(C -Ah/C*1 where ( )1 denotes the expectation over mutant sites. For an equilibrium of gene frequency it is required (A41) equals p times (A42). But (x) = (1 -~)(x)~+p(x)l variable x. Therefore to first order we have (Sr/r)(G(C

the

(~42)

that (1 -p) times for any random

-AN/C* = (SvlvMl -WC)d

(A43)

It can be shown that (WC --4/C) where Ab is defined in equation differential limit)

=A1 -PI&

(6). Thus the ESS condition

d(ln V) -=-. d(ln r)

Ab 1-P C l-(A/Ch’

(A441 is (taking the

@45)

Using simple statistics one can show that if the covariance between the central site and a neighbour is ~(1 -p)F, and the central site is known to be occupied by a mutant then the expected gene frequency at the neighbouring site is p + (1 - P)F (thus if there is no correlation the expectation is p, whereas with perfect correlation (F = 1) the gene frequency is always equal to that at the central site, giving an expected value of one). Using this result to evaluate (Ah we have l-(AICh=(l-p)

[l-mo-$miJ’ot] (A46)

= (1 -p)Vo. Using Y = Y-‘, (A43 becomes (using equation 6): Ab = [-C

d(ln Y)/d(ln r)]&.

(A47)

If there are sedentary offspring then A0 can be substituted for A* in this equation, giving the ESS condition -d(ln

Y)/d(ln

r) = -l/C.

(A481

606

H.

N.

COMINS

If there are no sedentary offspring then multiplying (A47) by & and summing (cf. equation 6) gives (9) with l/C replaced by -d(ln Y)/d(ln r). Following through the same analysis as previously and setting k = 1, it is found that (11 b) becomes -d(ln

Y)/d(ln r) + 1 + mo.

(A@)