Ideal free distributions when individuals differ in competitive ability: phenotype-limited ideal free models

Anita. Behav., 1986, 34, 1222-1242

Ideal free distributions when individuals differ in competitive ability: phenotype-limited ideal free models G. A. P A R K E R *

& W. J. SUTHERLAND~':~

*Department of Zoology, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K. ~Sehool of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, U.K.

Abstract. A series of prospective models is developed to investigate ideal free distributions in populations

where individuals differ in competitive ability. The models are of three types. In the continuous-input models, there is continuous arrival of food or mates into each habitat patch, and competitors scramble to obtain as large a share as possible. In the interference models, the prey density in a particular patch stays constant but the presence of competitors slows down the rate at which prey are captured. In the kleptoparasitism model, individuals have food or females stolen from them by competitors higher in the dominance hierarchy, and in turn steal items from subordinates. A general result of the continuous-input and interference models is that the population of competitors can be truncated between patches so that the individuals with the highest competitive ability occur in the best patches, or in the patches where competitive differences are greatest. Individuals of lowest competitive ability occur in the poorest patches or where competitive differences are least, and intermediate phenotypes are ranked between these two extremes. Thus the ideal free prediction that all individuals will achieve equal fitness will not apply. However, in continuous-input cases where competitive differences between phenotypes remain constant across patches, this solution is only neutrally stable, and forms only one element of a set of equilibrium distributions. The fact that many empirical studies of continuous-input have found approximately equal mean fitness across patches may relate to this finding. Most interference studies contradict the simple ideal free solution by having different mean intake rates across patches; this may relate to the predicted positive correlation of competitive ability with patch quality. The kleptoparasitism model usually generated continuous cycling of individuals between habitat patches, though some correlation could be found between competitive ability and patch quality. We both have spent some effort on the development and application of the concept of the ideal free distribution (e.g. Parker 1970, 1974, 1978; Sutherland 1983; Krebs et al. 1983). Nevertheless, a basic assumption of the concept, that all individuals have equal competitive ability, is unlikely ever to be met in the real world, and can quite dramatically alter the predictions of the theory. The present paper makes a prospective attempt to include competitive asymmetries into the ideal free theory. A possibility exists that some of the observed fits between data and the predictions of ideal free theory (for equal competitors) arise for the 'wrong' reason. In an ideal free distribution (Fretwell & Lucas 1970; Fretwel11972; see also Parker 1970, 1974) it is assumed that all competitors are equal and each is free to move to the place where its gains are highest. As a consequence, the average gain per individual should stabilize to be equal in all sites. If rewards :~Present address: School of Biological Sciences, University of East Anglia, Norwich NR4 7TJ, U.K.

are unequal, we might expect individuals to move to the better sites until rewards equalize. Table I summarizes the results of the known tests of the ideal free distribution. We have divided the investigations into two categories: continuousinput and interference. With continuous-input studies, there is a continuous arrival of food or mates to each site, and the competitors scramble to obtain as large a share as possible. With interference studies, the prey density in a site is approximately constant but the presence of competitors makes prey harder to find. Conclusions appear different for continuousinput and interferences cases (Table I). Studies involving continuous-input usually find that the number of competitors at a site is proportional to the input rate (rate at which resource units input to the site) as predicted by the ideal free distribution (see below); but each study records considerable and consistent differences in intake between individuals, contrary to ideal free prediction. In studies involving interference, the ideal free distribution

1222

Parker & Sutherland: Phenotypic idealfree models

1223

Table I. Summary of studies relating to the ideal free distribution

Reference

Species

(A) With interference Buxton (1981)

Shelduck

Goss-Custard et al. (1984)

Oystercatcher

Sibly & McCleery (1983)

Herring gull

Monaghan (1980)

Herring gull

Sutherland (1982)

Oystercatcher

Tamisier (1974)

Teal

Thompson (1981)

Shelduck

Thompson (1984)

Lapwing and golden plover Oystercatcher

Zwarts & Drent (1981)

Result Spend less time feeding in areas of dense prey Average intake differs consistently between mussel beds Average intake consistently higher on open tip than elsewhere Average intake about five times greater in better area of rubbish tip Average intake differs between parts of cockle bed Feeding duration independent of prey density Time spent feeding similar between areas with different prey densities Rate of intake greater in fields where prey is most abundant Intake constant in years of different mussel availability

Agreement with ideal free model for identical competitors Possibly contradicts model Contradicts model Contradicts model Contradicts model Contradicts model Possibly supports model Weak support for model Contradicts model Weak support for model

(B) With continuous-input Courtney & Parker (1985)

Tiger blue butterfly

Davies & Halliday (1979)

Toads

Godin & Keenleyside (1984) Cichlid fish Harper (1982)

Mallard

Milinski (1979, 1984)

Stickleback

Parker (1970, 1974)

Dungflies

Thornhill (1980)

Lovebug

Whitham (1980)

Aphids

Density of searching males on different- Yes, but fit not sized bushes fits ideal free better perfect. Individual than random distribution differences not studied Density of searching males in pond Yes, but size and edge consistent with obtaining equal differences in numbers of females SUCCESS Average intake same in two sites Yes, but consistent individual differences Average intake same in two sites Yes, but consistent individual differences Yes, but consistent Average intake same in two sites individual differences Average mating success same in Yes, but bigger different sites males more successful and tend to occur on dropping (Borgia 1979, 1980; Sigurjonsdottir 1980) Larger males at bottom of swarm, smaller Contradicts model males at top. Mating success increases with size and position in swarm Average fitness equal on different sized Yes, but consistent individual differences leaves

seems to provide a p o o r description o f dispersion because the average intake is usually m u c h higher in the best areas. I n some of these investigations, it a p p e a r e d t h a t the discrepancy was due to individuals differing in competitive ability. It seems likely t h a t realistic models of dispersion will have to i n c o r p o r a t e differences between individuals in terms o f competitive ability.

Fretwell & Lucas (1970) i n t r o d u c e d the concept of the ideal despotic distribution, which describes the expected distribution of a territorial species. The first individuals to arrive establish territories from which they exclude competitors. Subsequent arrivals have a choice between a small territory in the unoccupied parts of the better sites, or a larger territory in a p o o r e r site. This differs from o u r

1224

Animal Behaviour, 34, 4

models in which there is no territorial defence and individuals are free to move to any part of the habitat. We therefore consider some of the theoretical possibilities for ideal free systems in which individuals are unequal in competitive ability. The subject of ideal free distribution of unequal competitors is a complex one and there are many possible sorts of solution, depending on the ways in which different phenotypes interact and on the numbers of individuals typically in competition. The present paper must be regarded as a prospective survey only, since we consider only a few out of a very wide range of possible types of interaction between phenotypes. Many other sorts of solution may also be possible. In cases where there is a continuous distribution of competitive abilities the solution can be for the best competitors to occupy the best patches, and for the poor competitors to occupy the worst, with intermediate phenotypes ranged in between. This correlation between patch quality and competitive ability is characterized by the fact that the 'boundary phenotypes' (individuals of highest or lowest competitive ability in a given patch) have payoffs equal to that of the corresponding phenotype in one of the adjacent patch types (i.e. adjacent in terms of patch quality). However, features such as discontinuities in the phenotype distribution, or wide variance in competitive ability, can result in distributions in which some of the best competitors occur in poor patches. In other models, where the best strategy is to minimize the number of competitors that have higher dominance than self, there may be no stable distribution. In yet other cases, there can be sets of possible combinations of phenotypes across patches, all of which are equilibrium distributions. Some discussions of ideal free distributions with unequal competitors have been given previously by Harper (1982), Parker (1982), Ens & Goss-Custard (1984), Milinski (1984), Regelmann (1984), Kacelnik & Krebs (1985) and Sutherland & Parker (1985).

PREVIOUS MODELS Before incorporating differences in competitive ability we describe briefly previous models of the ideal free distribution and their predictions. Ar/ individual's payoff Wi in patch i depends on the

number ni of competitors in the same patch. The ideal free distribution is an evolutionarily stable strategy or ESS (Maynard Smith 1982) such that all individuals achieve equal payoffs in all patches, i.e. Wi(ni) = c (constant)

(1)

for all patch types i, j, k . . . . n. The terms continuous-input and interference are useful not only for empirical studies but also for the purpose of modelling. It is important to distinguish clearly between them as follows: Continuous-input Models Resource items are utilized as soon as they are input into the patch of competitors. For the case of identical competitors, an individual's gain rate is simply: W i ( n i ) = Q i / n i = c,

for all i, j, k . . . . n

(2)

where Qi is the input rate of resource items to the patch i. This sort of model has been of value to the analysis of competitive mate searching (Parker 1978), and in experiments in which competitors are fed from two stations (Milinski 1979; Harper 1982; Godin & Keenleyside t984). It gives the input matching rule: ni = Oi/c

(3)

that the number of competitors in a given patch should match or track the input rate of resource items (Parker 1978), and there is evidence that this can occur in natural populations. In certain instances (e.g. the tiger blue butterfly, Tarucus theophrastus, see Courtney & Parker 1985) resource items (females) are sometimes 'missed' by the competing males, and disproportionately so as the number of competitors increases. The consequence is that fitness is no longer simply input rate divided by competitor density (equation 2) which complicates the predicted distribution. Interference Models The notion here is generally that a number of competitors (usually predators) search in patches for resource items (usually prey). The patches differ in the density of resource items, and it is usually assumed that (at least in the short term) the predators exert a negligible effect on prey densities. If competitors have no effect on each other's intake,

1225

Parker & Sutherland." Phenotypic ideal f r e e models

then an ESS consists of all the predators exploiting the patch with the highest prey density. However, 'interference' (i.e. a slowing down in an individual's rate of prey capture due to interactions with other predators) is a widespread phenomenon (Hassell & Varley 1969; Goss-Custard 1980) and can cause predators to move to patches of lower quality. Typically, interference is modelled by incorporating a constant, m (usually 0 < m ~
aini -m

which is similar to the approach of Hassell & Varley (1969), and becomes formally equivalent to equation (2) if m = 1, as is the case for continuousinput (Sutherland & Parker 1985). F o r an ideal free solution with identical competitors, we need I'Ve(hi) : Qini -m = c (constant)

(4)

so that the number of competitors in patch i is ni = (Qi/c) l/m

(5)

(Sutherland 1983).

IDEAL FREE D I S T R I B U T I O N S WITH UNEQUAL COMPETITORS For a distribution to be stable, it is clear that no individual competitor must be able to profit by moving to another patch. In game theoretical terms, a stable distribution will be a Nash equilibrium in the sense that no player can achieve a higher payoff by deviating its strategy unilaterally from the equilibrium combination of strategies (one for each player). A strategy is thus a choice of patch. In both types of model (continuous-input and interference), individual payoffs decline with the numbers of competitors exploiting a patch. Fitnesses of strategies are therefore frequency-dependent. If we observe a mixed ESS (i.e. individuals of the same phenotype play more than one strategy and thus occur in more that one patch) it is clear that the payoffs to the phenotype in each patch must be equal. An ideal free distribution for a population consisting of a set of phenotypes differing in competitive ability will have the following characteristics: (1) payoffs to the various phenotypes will be different; (2) payoffs to individuals of a given phenotype will be identical, whether they occur in more than one patch type or not; (3) no individual will be able to increase its payoff by

shifting to another patch; and (4) the ESS will consist of a set of 'phenotype-limited' strategies (Parker 1982), each prescribing where a particular phenotype should be found. Continuous-input Models with Two Patch Types and Two Phenotypes We begin by considering the continuous-input model because of its analytical tractability. For further simplicity, we examine the simple case where there are two patch types, i and j, and two phenotypes, A and B. Competitive weights

To make progress, we need some assumptions about how the different phenotypes interact in competition. Following Parker (1982) we shall assume that each individual has a 'competitive weight', K, and that its share of the gains in a patch is equal to its competitive weight in the patch, divided by the total competitive weight for that patch. Phenotype A has competitive weight KAi in patch i and KAj in j; B has correspondingly gBi in i and Ksj in j. We seek an ideal free distribution for the frequency PAl of A phenotypes in i, and Phi in j ( P A l = 1 - P A j ) , and similarly PBi and PBj for B. Theoretical conclusions

In Appendix I, we derive the following conclusions analytically. There are two cases to consider, depending upon whether the relative competitive weights differ between patches. For example, ifA is always twice as good as B then apply case 1. I r A is twice as good as B in one site, but (say) three times as good in another, then case 2 is required. Case 1 is as follows. Khi

KBi

KAj

KBj

Thus, competitive weights of A and B are either constant, or change in the same proportion between i and j. There is a set of possible distributions of phenotypes, all of which are equilibrium distributions. The proportion of a given phenotype in a given patch can vary (maximally) between 0 and 1; see Fig. la. The relative fitnesses of the two phenotypes are equal to their relative competitive weights, whatever the distribution. For the case where the competitive weight of each phenotype is unaffected by the patch type, the ratio of patch quality and total competitive weight in the patch

Animal Behaviour, 34, 4

1226

a R=4

p~

R=I

o.~

0'5

between patches at the extremes of each set of equilibrium distributions (Fig. 1). However, where there are large numbers of individuals in patches, the chances of generating such extremes are very small. Calculation of the distribution of mean fitness for patch types is a complex problem that is well beyond the scope of the present paper. However, suppose that individuals distribute themselves randomly with respect to phenotype, but 'ideal free' with respect to resource quality. Then for large numbers of competitors in patches, the phenotype frequencies will be approximately equal in each patch, though the total number of competitors will be proportional to patch quality. Thus R = input ~ rate to i _ PAl _ Pal input rate to j PAj PBj It is easy to show that this random distribution always forms part of the equilibrium set, by substituting for R in equations (A2) and (A3). Such a distribution will have equal mean fitness in each patch, yet closer examination will reveal that individual payoffs are always proportional to competitive weights. Even when numbers in patches are relatively small, mean fitness across patches may appear approximately equal as a result of averaging data from several patches of each type. Case 2 is as follows.

0,5

\

\ \\

ESS i

ESSii

o.5

Ess iii

Figure 1.(a) Case 1: constant ratios of competitive weights. The lines are sets of equilibrium distributions of PAl, Pro, the frequencies of A in i, and B in i, at different ratios of R = Qi/Qj and for the case where ei/fli = 1 (see Appendix I). (b) Case 2: changing ratios of competitive weights. The lines are Phi or Phi plotted against PA, the frequency of the A phenotype in the population. K= 5, L=I,R=I.

must remain constant for all patches, though any distribution of phenotypes that satisfies this rule will be at equilibrium. Although the fitnesses of the two phenotypes are proportional to their competitive weights, it is interesting to consider what will be the mean fitness of individuals in a given patch. In empirical studies, this is often measured and compared across patches; equality across patches is generally claimed as evidence in support of the ideal free distribution. The greatest difference in mean fitness will occur

KAi

KBi

KAj XBj Thus, competitive weights for different phenotypes do not alter in the same proportion when individuals are switched between patches. For example, A might do twice as well as B in patch i, but only equal B in patch type j. This is the case studied explicitly by Parker (1982) in relation to alternative mating strategies: a summary of the conclusions in the context of ideal free theory is given here for the sake of completeness (see Appendix I). There is always a unique ESS distribution at a given set of values for the parameters, and not a continuum of possible distributions as in case 1. At the ESS, at least one of the probabilities PAi, PB~, PAj, Puj must be zero. It is convenient to define

Kai

KAj

L

Thus A does K times as well as B in i, but only L times as well in j, where K > L . The phenotype

Parker & Sutherland." Phenotypic idealfree models distribution is now predictable exactly and, depending on the conditions, will fall into one of three possibilities (see Parker 1982): (1) A occurs in i only, B occurs in both i and j; (2) B occurs inj only, A occurs in both i and j; or (3) A occurs in i only, B occurs in j only. Note that the other three possibilities cannot occur; if A is restricted to one patch type, it is the patch (i) in which it does best against B, and if B occurs in one patch type, it is where (j) it does best against A. The ESS is therefore always 'commonsense', but note that i is not defined as a better patch than j in the sense of Qi > Qj. It is simply the patch in which A does best against B. Appendix I contains details of the three types of ESS converted from the notation of Parker (1982). A graphical illustration is given in Fig. lb. The ratio of fitnesses of the two phenotypes is K in ESS 1, L in ESS 2 and Qi(1 --PA)/QjPAin ESS 3. Unlike case 1, the relative fitness of the phenotypes is not constant but now depends on the patch qualities Qi, Qj, and on the population frequencies PA, PB, of the two phenotypes.

i

j

1227 k

;

I I

I

n

a

i

i

:

j

k

i I in /

i I I

I t i

I I I I

I I I I I *

,,

W

',

',

I I

,, I I I

I ~',"-

Continuous-input Models with Continuous Phenotypes and Several Patch Types A graphical solution has been proposed for the evolution of alternative mating strategies in a population with a continuous distribution of competitive abilities (Parker 1984). This can be readily extended to ideal free searching with several patch types, i, j . . . . n (see Fig. 2). Figure 2a shows the frequency distribution in the population of some characteristic (such as size, S) which is positively correlated with competitive ability. We plot P(S) rather than P(K), since S is fixed whereas competitive weight, K, can vary depending on the patch type an animal is in. However, we shall assume that K is always positively correlated with S i n all patch types, i,j . . . . n. At the ESS, the phenotype distribution P(S) can be truncated between the patch types as shown in Fig. 2a, provided that the conditions outlined in Fig. 2b are fulfilled. First, rank the patch types i, j . . . . n in order of increasing effect on competitive ability. Thus for two given phenotypes, $1, $2, where $2 > $1, the ratio

K,(S2)

Kj(S2)

Ki(S,) < ~

Ko(S~) < . . . . Kn(S1)

(6)

Figure 2.(a) A hypothetical frequency distribution, P(S), where Sis a characteristic that correlates with competitive ability. At the ESS, the population is truncated so that the individuals with lowest S occur in patch i and those with highest S occur in patch n, see text. (b) Fitness W in relation to S for the ESS distribution. The dots represent the boundary phenotypes. The dotted line shows the fitness that a given individual of size S would achieve by switching to patch j.

Figure 2b shows the fitness payoff, W, achieved by each phenotype. F o r this type of solution to be an ESS, we require that two rules are satisfied, as follows.

Boundary phenotype rule In a large population with a continuous phenotype distribution, the payoff to the highest-ranking phenotype in a given patch must equal the payoff of the lowest-ranking phenotype in the next patch higher in the rank of increasing effect on relative competitive abilities. Thus W(Simax) = w(ajrnin), w(ajmax) = W(Skmin), etc. (7)

1228

Animal Behaviour, 34, 4

Note that Simax=Sjmin,_Sjmax=Skmin, and the boundary phenotype rule derives simply from the fact that payoffs to a given phenotype must be equal for each component of a mixed strategy. The boundary phenotypes (represented by dots in Fig. 2b) can therefore by envisaged as playing a mixed strategy between the two adjacent patch types in the rank order. The boundary phenotype rule is important analytically since it permits calculation of the points at which the population is divided into the different patches. Densities of competitors will adjust in each patch until the boundary phenotypes have equal payoffs.

Extrapolation rule Assuming (1) that the relationship K~(S) between competitive weight and size in a given patch is continuous and monotonic, and (2) that a single individual switching to an alternative patch has a negligible effect on the total competitive weight in the patch (as would be the case with many individuals per patch), we can determine whether the distribution is an ESS by extrapolation of Wj(S) as shown by the broken line in Fig. 2b. In order to be an ESS, the extrapolated (broken) lines for each patch type must all fall below the realized (solid) line for W(S). This rule is equivalent to the ESS requirement that no individual must be able to profit by switching to another patch. By way of example, assume that ~ ( S ) is linear for all patch types, j, as in Fig. 2b. Then Wi(S) must be linear if each individual's share of the gains in a patch are equal to its competitive weight over the total competitive weight for the patch. Provided that a switch of any single individual into a patch cannot significantly alter the total competitive weight for the patch, the broken line shows the payoff that any individual of size S would achieve by deviating from the ESS and switching to patch j. For all such cases, the payoffmust be lower (broken line) than would be achieved by following the ESS (solid line). In conjunction, the two rules lead to the following conclusions about the ESS: (1) fitness will correlate positively with competitive ability; and (2) individuals with highest competitive ability will occupy the patch in which competitive effects are greatest (see equation 6), and vice versa. The ESS is essentially a 'commonsense one'. It is possible to create a distribution which obeys the boundary phenotype rule but which is 'paradoxical' in the

i

j

n

m

J

$ Figure 3. Fitness W in relation to S where there are

constant ratios of competitive weights. The distribution is only neutrally stable since an individual does not lose by switching to another patch. sense that the highest-ranking phenotypes occupy the patch in which competitive effects are least, and vice versa. However, such a distribution would violate the extrapolation rule and cannot therefore be an ESS. The present model has several parallels with the simpler model of the previous section. In particular, Appendix II shows that if condition (6) does not apply so that Ki(S2) K~(SI)

Kj(S2) Kn(S2) Kj(SI) . . . . K,(SI)

then an ESS in which the phenotype distribution is truncated between patches has only neutral stability, since the extrapolated lines fall on the same line as W(S); see Fig. 3. Hence a switch to an alternative patch yields the same payoff as staying. As before, many possible distributions of phenotypes can be stable. An example of how the ESS distribution can be calculated for cases obeying condition (6) is given in Appendix III. It derives the distribution of phenotypes when there are three patch types, i, j and n, for linear K(S), and for the case where the phenotype distribution decreases linearly from 1 to 0. Some results are shown in Fig. 4. Curves 1 and 2 (continuous lines) show ESSs obeying both of the rules listed above. Case 3 (broken line) is a distribution of the paradoxical type (the most dominant individuals in the patch where competitive effects are least, and vice versa). It obeys the boundary phenotype rule, but not the extrapolation rule, and hence cannot be an ESS.

Parker & Sutherland." Phenotypic ideal free models

/i

/

"I

I

i

Jj

/

/

1

1229

2

Figure 4. Curves I and 2 are ESSs for a three-patch case using the explicit functions for the continuous-input model analysed in Appendix III; the distributions have poorest competitors in the patch i in which the competitive differences are least, and the best competitors in patch n for which competitive differences are highest. All patches have equal quality; Qi = Oj = Qn = I. In curve 1, competitive differences are relatively high (I= 1, J=2, N=3) and in curve 2 they are relatively low (1=0-1, J=0-2, N=0.3); reducing the competitive differences reduces the differences in fitness, as would be expected. In curve 3 competitive differences are reversed so that I = 3, J=2, N= 1, but the population is truncated in the same way as in curves 1 and 2 (with the poorest competitors in patch i). This (unlike 1 and 2) is clearly not an ESS distribution since the extrapolation rule is violated. In all three curves, b = 1.

patch with the most d o m i n a n t individuals, and vice versa. Patches in which individuals get highest payoffs at the ESS are not strictly the 'best' patches in terms of resource item input, they are patches in which phenotypic differences exert the biggest effect on payoffs. However, the most plausible biological interpretation is that such patches are also the best patches in terms of resource item input. A high resource item input will demand a high total competitive weight within that patch, which may mean that the competitive effects between individuals are also high. If there is a positive correlation between the total competitive weight in a patch and an increased effect of competition (e.g. Begon 1984), then we should expect to see the best competitors in the best patches, and vice versa. This effect occurs explicitly as a result of the next model for r a n d o m search.

The Interference Model with Continuous Phenotypes

The degree of interference experienced is here assumed explicitly to increase with the density of competitors in a patch, so that payoffs decline as the interference constant, m, is increased in equation (4). Following Sutherland & Parker (1985), we assume that the degree of interference is proportional to g _ mean competitive weight in patch i K(S) self's competitive weight in patch i so that, from equation (4)

Biological interpretation Since the predictions are rather complex, a brief review of the most plausible biological interpretations seems advisable. The model is for cases where there are large numbers of competitors in each patch, and where the range of competitive abilities is not extreme. The prediction is that the most d o m i n a n t phenotypes should aggregate in the patch in which the effects of dominance will be most manifest. The next most d o m i n a n t competitors should aggregate in the patch where dominance effects are next highest, and so on, so that the least dominant individuals occur in the patch in which there is least effect of competition. The phenotype distribution is thus truncated between patches, fitness will correlate with competitive ability, and the gradient of the relationship between fitness and competitive ability will be highest in the

Wi(S, hi) = Qini-"(~i/~:i(s~)

(8)

where m is a positive constant that scales the magnitude of interference. Thus an individual of average competitive ability for a patch sustains interference m, whereas an individual of aboveaverage competitive ability suffers less interference, and vice versa. Qualitatively therefore, model (8) has the appropriate biological realism. An ESS distribution for this model must also obey the boundary phenotype rule and the extrapolation rule. It is again easy to calculate the ideal free distribution of phenotypes if K(S) and P(S) are stated explicitly. A n example is given in Appendix IV and Fig. 5, which shows a numerical example when P(S) decreases linearly from 1 to 0 (as in Fig. 4). We calculated the ESS distribution using the boundary phenotype rule, assuming that the least

Animal Behaviour, 34, 4

1230

//

i

//'/

petitive phenotypes in the worst patch (and vice versa) violates the extrapolation rule and is unstable. The boundary phenotype $2 is very insensitive to changes in the interference constant m over a wide range, unless the total number of competitors (N) is very low (Fig. 6). If N is low, the approximation that phenotypes form a continuous distribution becomes unreliable. In contrast, boundary $1, is much more sensitive to changes in m. However, if m ~ 0 , there is no interference and all competitors should colonize the best patch.

SOME SIMULATIONS WITH DISCONTINUOUS PHENOTYPES

/

S Figure 5. ESS for a three-patch case using the explicit functions for the interference model analysed in Appendix IV. The poorest competitors are in the poorest patch (i) and the best are in the best patch (n). Qi= 1, Oj = 2, Qn=3, m=0"l, N= 100, b=l.

competitive phenotypes would be in the poorest patch, and vice versa. Extrapolated lines for W(S) are included to show that the distribution thus obtained is stable; individuals switching to either of the other two patch types fare badly. A distribution that obeys the boundary phenotype rule, but has the most c o r n -

We ran a series of computer simulations of the two types of model (continuous-inputand interference) in which there were 100 individuals of each of 10 phenotypes, and three patch types. As is the usual practice with this sort of simulation, strategies (here a choice of patch type for a given phenotypic state) in a given round were allowed to replicate in proportion to their payoffs. Thus although the population at each round always contains 100 individuals of each of the 10 phenotypes $1, $2 . . . . Sl0, the proportions of each phenotype in patch types i, j, n were set equal to their relative payoffs in that patch in the previous round. A round could be envisaged as a round of reproduction in an evolutionary process, but is probably best considered as a round of sampling and dispersal, with individuals most likely to leave patches in

a

b N=IOOO I

N=IOO

N=IO

S1 /

// f o=,o

// //' S

~----

/

/

i / ,' I

/

,I, / I

0.5 m

I

015 m

Figure 6.(a) Changes in boundary $1 in relation to the interference constant m at different population sizes, N. (b) Comparable changes in boundary $2. Parameters other than m are the same as in Fig. 5.

Parker & Sutherland: Phenotypic idealfree models

1231

Table |I. Continuous-input model simulation with 10 phenotypes Simulation Patch

$1

$2

$3

$4

$5

$6

$7

$8

$9

Sl0

100

100

100

100

51"35

0

0

0

0

0

i

0.102 0.136 0"169 0.203 0.237 0-271 0.305 0'339 0.373 0-407

J

0'079 0.119 0.158 0.198 0.237 0.277 0.316 0'356 0.395 0-435

n

0.068 0.110 0-151 0.192 0.233 0.274 0.315 0'356 0.397 0-438

i

0.104 0.138 0.173 0.207 0.242 0'276 0.311 0-346 0.380 0.415

J

0.086 0'130 0.173 0.216 0'259 0.303 0.346 0.389 0-432 0.475 0 0 0 0 0 0 72.07 100 100 100 0.075 0.120 0.165 0.210 0.255 0'300 0.346 0.391 0.436 0.481

0

A

0 100 0

B

n

100

0 100 0

100

0

0

0

0

92.45 7.55

100

0 100

100

48.65 0

100 0

0

0

100

100

100

100

100 0 0 27"93

100

45.26 54.74

0 100

0

0

0

100

0

100

0 100 0 0

5.35

i

0'007 0.010 0.012 0.015 0.017 0.019 0.022 0.024 0'027 2.43

J

0'005 0"007 0.009 0.012 0.015 0.017 0.019 0.022 0.024 2'43

n

0.004 0.006 0.009 0.011 0.014 0.016 0'019 0.021 0'023 2.43

0

C

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

32'93 61.72

Values in bold type are the numbers of each phenotype ($1, $2, $3 . . . . $10) in each patch i, j, n. The population consists of 100 individuals of each phenotype. Values in ordinary type are the fitnesses achieved by an individual in a given place. In simulation A, Qi = 199, Qj=200, Qn=201 and Ki(S)=I+iS,/(](S)= 1 +iS, Kn(S)=I+nS, where i=0.5, j = 1.0, n= 1.5, and $1 = 1, $2=2 . . . . Sl0 = 10. In B, Qi = 100, Qj =200, Qn=300; other values are as in A. In C, values are the same as B except that one phenotype is overwhelmingly dominant (St0 = 1000). which they experience low payoffs and least likely to leave patches where they experience high payoffs. Either way, the model corresponds qualitatively to a system in which strategies that are more successful are either repeated more often (learning and dispersal) or proliferate at an enhanced rate, until equilibrium is achieved. Table II shows some results of this simulation for the continuous-input model, after 4000 rounds. Several different starting conditions were tried, but this did not affect the eventual solutions. Since the simulations were deterministic, fitnesses could be calculated even when the numbers of individuals in a patch approach zero; hence 'mutant' fitnesses could be estimated that parallel the extrapolation lines in Fig. 5. Table III shows an analogous set of simulations for the interference model. In all cases, the mutant fitnesses (where number of individuals recorded in the tables = 0) are lower than the ESS fitnesses, and the boundary phenotypes (which occur in more than one patch type) have the same fitness in each patch in which they occur. In simulations A and B, there was rather little difference in competitive ability between phenotypes, and the results correspond closely to the predic-

tions of the previous section: the phenotype distribution is truncated between the patch types. Increasing the disparity in patch quality (simulation B) tends to reduce the value of S for the boundary phenotypes. Simulation C shows the effect of introducing a marked discontinuity into the set of phenotypes. If one phenotype ($10) is overwhelmingly dominant, it alone occupies the best two patch types j and n; the set of weaker phenotypes is restricted to the poorest patch type i. The dominant phenotype may (Table II, simulation C) or may not (Table III, simulation C) occur in the poorest patch. These results are anticipated by the predictions of the two patch-two phenotype models described earlier. As expected, the dominant phenotype achieves the same payoff wherever it occurs, and the weak phenotypes will always do worse if they m o v e out of the poorest patch type.

KLEPTOPARASITISM

MODELS

We investigated one other type of competitive interaction, in which dominant individuals steal

Animal Behaviour, 34, 4

1232

Table III. Interference model simulation with 10 phenotypes Simulation Patch

Sl

$5

86

$7

$8

89

Sl0

J

0.046 0.303 0.568 0.778 0.940

n

0.029 0.249 0.507 0.723 0.895

i

100

0 0 100

0

0

0

0

0

0

0 0 0

0

0

0

23.36

28-81

1.066 1.166 1.247 1.314 1.371 0

0

0

0

71.19

1.032 1.142 1.232 1.307 1.371 0

0

0

0

0

0.452 0.672 0.767 0.820 0.853 0.876 0"893 0.905 0.915 0.924 0

0

100

100

100

81.53

0

0

0

0.144 0-536 0.831

n

0.019 0.238 0.555 0-846 1.089 1.290 1.455 1.593 1.709 1'808

i

0.033 0-183 0.322 0.427 0.506 0.567 0"615 0.654 0.685 0-997

J n

100

0 100

1.036 1.182 1.290 1.373 1.439 1'493

0

J

0

C

$4

100 100 100 100 100 100 100 100 76.64 0 0"069 0"362 0-629 0'829 0"979 1"093 1.183 1.255 1'314 1.364

0

B

$3

i

0

A

$2

0 100

0 100

0 100

18-47 100

100 100

100 100

100 100

1.537 100 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

98'30

0

0

0

0

0

0

0

0

0

1.896

0

0

1.70

0.003 0"005 1.896

Values in bold type are the numbers of each phenotype; values in ordinary type are the fitnesses achieved by an individual in a given place (see Table II for further details), The model is for random search with interference constant m = 0' 1. In simulation A, Qi = 1,9, Qj = 2,0, Qn = 2.1 and K(S) = S, where $1 = 1, 5'2 = 2 . . . . $10 = 10. In B, Qi = I, Qj ~ 2, Qn = 3, other values are as in A. In C, values are the same as in B except that one phenotype is overwhelmingly dominant ($1o = 1000). resource items from subordinates. In a given patch, each c o m p e t i t o r has equal prospects o f finding a resource item, b u t is less likely to keep it if s u r r o u n d e d by more d o m i n a n t individuals. F o r a given group, payoffs therefore increase with the n u m b e r ns o f subordinates to self, a n d decrease with the n u m b e r nd o f d o m i n a n t s to self. Gaines o f this type were simulated in a similar vein to the c o m p u t a t i o n s o f the last section. The p o p u l a t i o n consisted o f 100 individuals of each o f 10 ' d o m i n a n c e r a n k ' phenotypes, D~ (least domin a n t ) to Dl0 (most d o m i n a n t ) . These are distributed within three p a t c h types i, j, n. A strategy is again a choice o f p a t c h in a given p h e n o t y p i c state, a n d strategies again replicate in p r o p o r t i o n to payoffs at the previous round. W e assumed for simplicity t h a t no interaction takes place between individuals of the same p h e n o t y p e (or alternatively, that gains = losses within a d o m i n a n c e rank). Two cases were investigated as follows.

(1) F o o d - f o r a g i n g C a s e

I n d i v i d u a l s search in patches for food items. Each individual obtains the same rate o f capture of

prey items, i.e. Qi in p a t c h type i. Simultaneously, for each individual a certain p r o p o r t i o n of the prey captured m a y be stolen by d o m i n a n t s , but some items m a y in t u r n be stolen f r o m subordinates. The fitness payoff of a given individual in p a t c h i is therefore Wi(D) = Qi(1 + G(ns) --L(nd)) in which the 1 represents each capture by self, G(ns) represents the gains from subordinates, a n d L(nd) represents losses to d o m i n a n t s . T h e total losses to all individuals in i must equal the total gains. The simplest model obeying this r e q u i r e m e n t is where G(nO =ans a n d L(nd)= and, in which a is a positive c o n s t a n t t h a t is sufficiently small to ensure that payoffs are never negative. G a i n s a n d losses to self therefore increase linearly with the n u m b e r of individuals respectively a b o v e a n d below self in the hierarchy. T h e use o f the same c o n s t a n t for b o t h functions also gives the s y m m e t r y properties t h a t the highest-ranking individual achieves a gain t h r o u g h kleptoparasitism t h a t equals the losses o f the lowest-ranking individual, a n d the m i d - r a n k i n g individual experiences equal gains a n d losses. This case is analogous to the interference models

Parker & Sutherland: Phenotypic idealfree models

1233

where ni = the total n u m b e r o f competitors in i. The assumption is that each individual has a densityd e p e n d e n t capture rate o f items equal to Qi/ni; the items are then reallocated in relation to dominance as in the food-foraging case (1). The same forms for G(n~) and L(nd) were used as in case 1.

in t h a t an individual's rate o f gain is set by the density o f prey items in the patch. However, in the present model, there is no reduction in the prey capture rate due to interference f r o m competitors; interaction between the competitors results in a reallocation o f the captured prey but does n o t affect the average intake rate.

Results Providing that p a t c h qualities were sufficiently similar, neither o f these kleptoparasitism models generated stable distributions o f phenotypes. We obtained stability in the food-foraging model only in conditions when patch qualities were so disparate as to favour all individuals colonizing the best patch.

(2) Continuous-input Case This follows the concept o f a fixed input Qi o f resource items that are shared between the competitors in the patch. Payoffs to a given individual in p a t c h i were therefore taken as Wi(D) = Qi(1

+G(ns)-L(nd))/ni

Table IV. Kleptoparasitism model simulation with I0 phenotypes Patch

D1

D2

D3

D4

D5

D6

D7

D8

D9

Dlo

Food-foraging case (500 rounds) 0.20 0.12 15.38 64.64 i 144.6 144.6 147.8 163.8 98.85 99.87 84.61 35"35 J 85.9 125.8 162.4 185.6 0.95 0 . 0 1 0.01 0 n 1 9 1 . 0 191.1 191.1 191.1

2,96 177.3 97.04 211.4 0 191.2

23.63 183.5 76.37 244.9 0 191.2

3.09 189.9 96.91 278.1 0 191.2

99-70 209.7 0.30 297.3 0 191,2

13.93 232.0 86.07 313.7 0 191-2

0 235.5 14.35 332.1 85.65 210-1

Food-foraging 0 i 165.1 0 J 161.0 100 n 89.5

14.66 1 4 . 3 6 5'70 3.33 201.94 208.1 212.2 214.0 0 0 94'30 0 161'0 161.0 180.0 199.0 85.34 85.64 0 96"67 216.8 252.0 269.4 289.5

0 214.9 99-99 219,0 0-01 309.6

0 214'9 0'03 239"0 99,97 330.6

case (2000 rounds) 2.03 0 . 0 1 88.27 165.1 165.1 182.1 0 0 0 161.0 161"0 161.0 100 99.99 11'73 131.5 173.5 196.8

Continuous-input case 77.76 29.14 i 0.389 0.437 9.92 64.66 J 0 . 3 8 7 0.423 12'31 6 . 2 0 n 0'389 0'411

(500 rounds) 38,45 39.60 0,468 0,503 9.77 44.78 0.473 0.505 51"78 15'62 0-451 0,499

35.71 0.537 0.01 0.530 64.28 0.553

0 0.554 97-13 0.587 2.87 0.599

0 0.544 17.41 0.653 82,59 0-658

0 0.544 99'65 0.721 0,35 0.715

0 0 0.554 0-554 1.13 0 0.780 0.781 98.9 100 0.783 0.920

Continuous-input case 31.05 32.45 i 0,404 0.435 43.98 51.35 J 0 . 3 8 7 0.441 24'96 16.19 n 0"399 0.427

(2000 rounds) 29.98 46,19 0.466 0.504 10.99 48.28 0'476 0.509 59.03 5.52 0,478 0.522

64.64 0.558 0 0.536 35"36 0.550

0 0.590 94.81 0.589 5.19 0'578

0 0.590 5.11 0.645 94'89 0.646

0 0.590 99"07 0,704 0.93 0,712

0 0 0.590 0.590 2'13 0 0.761 0.762 98.87 100 0-779 0.914

Values in bold type are the numbers of each phenotype (DI, D2, D3 . . . . Dl0) in each patch i,j, n. The population consists of 100 individuals of each phenotype. Values in ordinary type are the fitnesses achieved by an individual in a given place. Fitness payoffs were calculated as in the text with constant a=0.001, Qi = 190, Qj=200, Qn = 210 for the food-foraging case; Qi = 100, Qj = 200, Qn = 300 for the continuousinput case,

1234

Animal Behaviour, 34, 4

This result contrasts sharply with the ESSs found typically in the previous models. A sample pair of results at different numbers of rounds or generations is given in Table IV. Note that unlike the results of Tables II and III, fitnesses are not equal across patches when a phenotype is present in more than one patch. The boundary phenotype rule is thus violated and the distribution of these phenotypes is clearly altering. However, there was not an equal degree of cycling in all phenotypes. The most dominant phenotype tended to become confined to the best patch. Oscillations of the next most dominant phenotypes often tended to be restricted to the best two patch types, whilst the least dominant phenotypes tended to switch across all three patch types. Because of the density-dependent gains, the continuous-input case was least sensitive to differences in patch qualities. Increasing the difference between patches in the foodforaging case leads eventually to the total avoidance of one or both of the poorer patch types. The kleptoparasitism model shares one similarity with the two previous types of model. This is, if samples were taken continually, say throughout a season and then averaged, there would be an overall correlation between phenotypic quality and patch quality. However, there will be much variance except in the distribution of the most dominant phenotype, and low-ranking subordinates should be found in all patch types. Unlike the two previous models, kleptoparasitism should be characterized by continual movement of individuals between patches, especially low-ranking individuals, and fitnesses will not be equal for a given phenotype across patch types.

DISCUSSION The present models give quite different results from those expected under the assumptions for ideal free systems with equal competitors. Since the predictions of the various models are diverse and complex, we give a summary in Table V. The diversity of the results makes conclusions difficult to test: pick the right model and almost any pattern of distribution can be explained! A rigorous approach would consist of establishing that the different predictions are fulfilled only under the appropriate sets of assumptions. It is difficult to estimate the robustness of our conclusions until many more types of competitive interaction have been ana-

lysed, but Table V suggests that a considerable range of states is likely to apply in nature, and possibly helps to explain why the simple ideal free solution for identical competitors is unlikely to be the prevalent result in natural systems. Both theory (Table V) and data (Table I) suggest that continuous-input systems give different results from interference systems. We therefore discuss these separately.

Continuous-input Studies Many workers have obtained the result that, despite considerable individual differences in competitive ability, the competitors appear, at least superficially, to conform to the ideal free distribution for equal competitors (see Table I). The average payoff within a patch tends to be equal across patches and the number of competitors is proportional to the input rate (input matching; see equation 3). One interpretation of this effect is that competitive differences are trivial, so that the appropriate model is indeed the ideal free for a single phenotype. A more likely explanation is that the relative competitive weights of the different phenotypes do not differ significantly between patches, so that i f A does twice as well as B in one site, it does twice as well in all other sites. As explained earlier, a strategy that ignores phenotype, but obeys the rule that the number of competitors should be proportional to patch quality, will belong to the set of equilibrium distributions. This strategy obeys the rule required for each element of the equilibrium set: the sum of competitive weights in a patch must balance the input rate to that patch (see Appendix I). If competitors are distributed randomly within the constraints of this rule, there will be a superficial resemblance to the ideal free distribution for equal competitors. On average the same (random) phenotype frequencies will occur in each patch. Thus although payoffs will vary according to competitive weight, the average payoff/individual will be constant for each patch type and competitor densities will match input rates. The results of the various experiments involving two feeding stations (Milinski 1979, 1984; Harper 1982; Godin & Keenleyside 1984) might be attributable to this type of interpretation. In dungflies there is a slight tendency for larger males to search on the dropping and for smaller males to search in the surrounding grass (Borgia

Parker & Sutherland: Phenotypic ideal Jkee models

1235

Table V. Summary of the models and their main predictions Model

Main assumptions

(1) Ideal free distribution

Predictions

(1) Stable distribution of competitors, but random with respect to phenotype (2) Fitness of all individuals equal (1) Stable distribution of competitors, but phenotype distribution not truncated between patches (2) Fitnesses of different phenotypes unequal; fitness of a given phenotype constant across all patches it occurs in Competitive differences (1) Stable distribution of between phenotypes competitors, phenotype Payoffs of a given distribution truncated competitor always reduced by between patches the addition of more (2) Most competitive phenotypes competitors to the patch in highest quality patches Relative payoffs of pheno(3) Fitness correlated with types change between patches both phenotype and patch quality Phenotypes differ in domi(1) No stable distribution; nance status continuous switching of Interactions kleptoparasitic; phenotypes between patches payoffs to dominants can be increased by adding subordinates, (2) Some (but poor) payoffs to subordinates correlation between dominance decrease by adding dominants status and patch quality (3) Fitness of a given phenotype unequal across all patches it occurs in

(1) No competitive differences between phenotypes (2) Payoffs of a given competitor always reduced by the addition of more competitors to the patch (2) Continuous-input model (I) Competitive differences with constant between phenotypes ratios of competitive (2) Payoffs of a given competitor weights in all patches always reduced by the addition or of more competitors to the All patches patch equal in quality: (3) Relative payoffs of phenotypes various models do not change between patches

(3) Interference model

(1)

or

Continuous-input model (2) with changes in ratios of competitive weights across patches

(3) (4) Kleptoparasitism model (1)

(2)

1979, 1980; Sigurjonsdottir 1980). Larger males have an advantage over smaller males in struggles for the possession of females (Sigurjonsdottir & Parker 1981). Take-overs typically occur during oviposition, which occurs on the dropping. Thus although smaller males may have equal prospects of obtaining newly-arrivingfemales (but see Borgia 1981), they suffer a greater chance of loss of the female while guarding her during oviposition. The mating system of common toads appears to be exactly analogous (Davies & Halliday 1979). None of the present models fits the requirements of the dungfly/toad system exactly; suitable models are currently being investigated. Thornhill (1980) found that mating swarms of male lovebugs, Plecia nearctica were structured so that largest males tend to occur at the bottom of

swarms closest to the females, which emerge from the substrate. Further, larger males achieved higher mating success, as might be predicted from their position in the swarm. Although the mechanism by which the dispersion of the phenotypes occurs is not fully understood, Thornhill's results do appear to fit rather well the predictions of the continuousinput model, with changing ratios of competitive abilities at different heights in the swarm. We might expect highest densities of males at the base of the swarm closest to the female input, with the result that competitive differences are greatest at the bottom and least at the top. This should lead to the form of structuring that Thornhill observed. Using computer simulations, Regelmann (1984) examined a continuous-input model for the special case where competitive weights remain constant

1236

Animal Behaviour, 34, 4

between patches. Starting from a random distribution, he found that distributions tended to stabilize when the number of individuals in a patch matched the input rate (cf. the input matching rule, Parker 1978) but that there was a greater tendency for 'good' competitors to occupy the more profitable patch than 'bad' competitors. We believe that this result occurred because of the learning rule used by Regelmann to decide whether an individual stays or switches to the other feeding site. Good competitors decided more quickly where to feed and switched less often. Our results suggest that this distribution is unlikely to be an ESS rather than the mean of the set of equilibrium distributions that are generated in the Regelmann simulations. However, Milinski's (1984) experiments offer support for the type of effect noted by Regelmann; good competitors occurred more frequently at the more profitable feeding station and they switched less than poor competitors. Our continuous-input models indicate that it is possible to have a low density of highly competitive individuals in the best quality sites, especially when individuals differ considerably in their competitive abilities and patch qualities are rather similar. This prediction runs counter to that expected under the ideal free model for equal competitors, where competitor densities will always be highest in the best sites.

Interference Studies The simple ideal free expectation that the average payoff will be equal in all sites is contradicted both by field data (Table I) and by the predictions of our interference model (Table V). We expect the most competitive individuals to aggregate in the best sites, as shown for example in studies of herring gulls (Monaghan 1980) and oystercatchers (Goss-Custard et al. 1984). We also expect the average prey intake rate to be highest in the best sites, which these studies also established. In general the interference studies do not fit the predictions of the simple ideal free model, but fit better the predictions of our phenotype-limited ideal free models. However, in no case was the phenotype distribution truncated perfectly between the different sites as our models would predict. Possible explanations of this lack of perfect truncation are as follows. (1) Perfect truncation w o u l d be observed only after a vast number of 'rounds' of learning. This is impossible and there

will always be some input of naive individuals which will, by random settlement, tend also to disrupt the distribution of non-naive individuals. (2) Relative qualities of alternative sites vary through time, which exacerbates the effects under (1). (3) There are costs associated with moving between alternative sites. (4) Some kleptoparasitism is apparent in most of the systems studied. The relationship between site quality and competitor density is complex under the interference model. As with continuous-input cases, the intuitively appealing prediction of the simple ideal free model (that competitor density will increase with site quality: Sutherland 1983) does not necessarily apply for the phenotype-limited ideal free interference model. The arrangement of competitor densities depends on the distribution and range of phenotypes present in the population, the way in which competitor density affects interference, and on the way in which interference between competitors of different rank affects payoffs (see Sutherland & Parker 1985).

Kleptoparasitic Interactions In nature many competitive interactions involve theft of resource items by dominants from subordinates. As we have shown, kleptoparasitism can readily lead to oscillatory dynamics in animal distributions, since it pays dominants to move to patches containing subordinates but pays subordinates to move away from patches containing dominants (see also Pulliam & Caraco 1984). In the field it may be difficult to distinguish between the dynamics necessary for populations to move by learning to an equilibrium distribution, and the dynamics that arise from endogenous oscillatory processes, especially if patches are short-lived and continuously varying in quality. In studies of kleptoparasitism by black-headed gulls on lapwings, where gulls obtain much of their food intake by theft, Thompson (1984; Barnard & Thompson 1985) found that black-headed gulls tend to build up to an asymptotic frequency in the total flock; there was no obvious indication of oscillations in the frequency of gulls. Thompson (1986) found that the gull intake rate decreased as the gull:lapwing ratio increased. Competitive interactions were most numerous in flocks with more than one gull. Finally we stress that we have regarded as constant the proportions of the different pheno-

P a r k e r & Sutherland: P h e n o t y p i c ideal f r e e mo..dels

types in the population. Selection is assumed always to favour the highest possible competitive ability, so that individuals vary in competitive ability mainly due to chance environmental effects or to age. Where competitive ability reverses between different strategies (A does better than B in one strategy or site; B does better than A in the alternative strategy or site) then selection may adjust the phenotype frequencies in accordance with the opportunities available within each stra-

1237

tegy (Parker 1982). Such circumstances could perhaps most easily arise for alternative mating strategies.

ACKNOWLEDGMENTS We thank D. B. A. Thompson, A. Houston and an anonymous referee for comments and Miss Jane Farrell for typing.

APPENDICES

I. The Continuous-input Model with Two Patch Types and Two Phenotypes

Phenotypes are A and B, with population frequencies PA and PB. The two patch types are i and j, with resource input rates Qi, Qj. Let KAi, KAj be the competitive weights of A in i and j; KBi, KBj are the competitive weights of B in i and j. Pni is the frequency of A in i, PAj ( = 1 - Pai) is the frequency of A in j; similarly for PBi and PBj. We first consider whether it is possible to have a mixed strategy in which both phenotypes are represented in both patch types; i.e. 1 > PAl; PAj; PBi; PBj > 0 F o r this to be so requires the payoff of an A in i to equal that of an A in j, and similarly for B. Hence QiKA~ _ QjKAj for A phenotypes

G

cj

and QiKBi _ QjKBj for B phenotypes

Ci

Cj

where Ci, Cj are the total competitive weights in i and j respectively. This requires that QiKAi QjKAj

QiKBi-----Ci QjKaj Cj

i.e. KAi _ KBi KAj KBj

r (constant)

(A1)

Thus in order to have an equilibrium where both phenotypes occur in both patches, competitive weights must change in exactly the same proportion (for each phenotype) between patches. Call PAKAi = 0q, PAKAj = ~j = eli/r PBKBi = fli, PBKBj = flj = fli/r Qi/Qj = R

Then

Rr

-

Ci Cj

PAi~i "}-P8ifli (PAj~i + PRfli)/r

Animal Behaviour, 34, 4

1238 Therefore

R - PAi 0q+ PBifli __ PAiCXj+ PBiflj PAjCq+ PBjfli PAjaj+ Pajflj and since PAj = 1 -- PA~; PBj = 1 -- PBi PAi~Xi+ PBifli -- R(~xi+ fli) ( R + 1)

(A2)

PAj~ + PBj/~ -- (~i +/~i) (R + 1)

(A3)

These equations have the characteristic that there is a range of pairs of values PAt, PBi (or PAj, PBj) that can satisfy the equilibrium condition (see Fig. 1). Similar equations apply for ctj,flj, but these reduce to the same equations because aj = c~i/r and flj = [li/r. Note that equations (A2) and (A3) are independent of r, the ratio of the competitive weights of a phenotype in the two patch types. Altering the ratio r does not affect the distribution of strategy frequencies. The equilibrium distributions all have the characteristic that

O~_ Qjr c, G so that the total competitive weight in a patch must balance the patch quality, scaled in terms of the relative effect of competitive weight. If r = 1 (competitive weights o f a phenotype are the same in each patch type), it is easy to see that for several patch types Q_!i = 0i = Q_~k Q__2n Ci Cj Ck . . . . Cn

(A4)

This theorem states simply that the ratio between patch quality and the total competitive weight of the competitors within it must remain constant for all patches. A n y distribution of phenotypes will be an equilibrium distribution provided it obeys rule (A4). The ratio of fitnesses z =

fitness of an A phenotype fitness of a B phenotype = r (ratio of competitive weights)

whatever the equilibrium distribution, even when the distribution has one phenotype unrepresented in a patch. Since the fitness ratio (z) must equal the payoff ratio (r) in the patch where both phenotypes are present, the expectation of a phenotype represented on its own in a patch must be equal to its expectation when coexisting with the alternative phenotype. Now consider the case where equation (A1) is violated so that KAi r Kni KAj KBj Such a case cannot allow the possibility of both phenotypes being represented in both patches (see arguments leading to A1), i.e. one or more of the probabilities PAl, PBi, PAj, PBj must be zero. One phenotype must always be restricted to patch type, though it is possible for the other phenotype to occur in both. There are three ESSs listed in this paper. Defining

KAi

KBi

the conditions for each ESS can be taken from Parker (1982). (I) A occurs in i only, B occurs in both i and j:

Parker & Sutherland: Phenotypic ideal free models R(1 -- PA) -- PAK P,i =

0 - - ~ F )

1239

RK( 1 -- PA) -- PAKL ' where PBi <

(1-PA)(L+RK)

(2) B occurs in j only, A occurs in both i and j: PAl =

R(PAL -+- 1 -- PA) P ~

])) -' where PAl >

R(PAL-}- 1 - PA) PA(R + l)

(3) A occurs in i only, B occurs in j only:

R / ( K + R ) > PA > R / ( L + R ) II. Proof that Truncated Sets of Continuous Phenotypes have Only Neutral Stability if Competitive Weightings Alter in Fixed Proportion between Patches Take two adjacent patch types, i, j, out of the series i, j . . . . n, with resource inputs Qi < Qj < . . . . Qn. Assume that competitive weights alter in fixed proportion for all phenotypes when switched between two patch types, so that competitive weights are IK(S) in i, JK(S) in j, etc., where I and J are positive constants. F r o m boundary phenotype rule

Qi" Ig(Si max)

Qj" Jg(Sj min)

Si max

Sj max

I ~ P ( S ) . K(S)dS

J ~ P(S). K(S)dS

Simin

Sjmin

a n d since Si max: Sj rain

Qi

Qj

Si max

Sj max

I P(S)K(S)dS

I P(S)K(S)dS

Si rain

(A5)

Sj rain

The same relationship will hold for all adjacent patch types, and so calling the integrals in the denominator of (Ah) Mi, Mj, etc., we can see that

Qi _ Qj _ Q. Mi Mj . . . . ~ = q (constant) The payoff to an individual in any patch is therefore

W(S) = K(S).q Thus if K(S) is monotonic, so is W(S), and there can be no discontinuities in the gradient

d W / d S = q. dK/dS. Extrapolations of W3(S) (see Extrapolation rule) must fall on the line W(S), i.e. an individual can achieve the same payoff by a unilateral switch to another patch. An example where K(S) is linear is shown in Fig. 3.

III. An Explicit Example of an ESS for the Input Model with Continuous Phenotypes There are three patches, i, j, n, with resource input rates Qi < Qj < Qn. Let the size distribution P(S) = b(1 - CS), where b and C are positive constants. The size range lies arbitrarily between minimum limit 0 and the maximum of S', and we assume that P(0) = b and P(S') = O. Then

S' = I/C, since0 = b ( 1 - C S ' )

S" C = b/2, since Sp(S)dS = 1

0 Therefore s'

= 2/b

1240

Animal Behaviour. 34, 4

Competitive weights are

~(s)

= 1+is

~(s)

= 1+ Js

K.(S) = 1+NS

where/, J, N are positive constants with I < J < N. Call the boundary pbenotypes S~ (between i and j) and $2 (between j and n). By the boundary rule Q~(1 + I S , ) _ Q~(1+ JS~) Ci C~

(A6)

and

Qj(1 q- J S 2 )

Qn(1+

Q

NS2)

C.

(A7)

where the total competitive weights Ci, Q, C, in i, j and n are Ci = S b ( 1 - C S ) ( I + I S ) d S 0

cj = s,~ b ( 1 - C S ) ( 1

Cn =

= bS1 1-~

2

+ J S ) d S = bS2 1 +

~b(1-CS)(1 +NS)dS

= 2

4

2

+-~-

-

4

.bS~

-bS 2 1 q

2

1

4

2

4

-

-

$2

With a suitable iterative computer programme, it is easy to solve for St and $2 at given numerical values of constants b, Q~, Qj, Q , , / , J and N. Some numerical results ale shown in Fig. 4. Several iterations were performed with 1< J < N, as is required by the model. In all cases d~(S) dS

d Wjj(S) IO
dS

d Wn(S) IS1
dS

IS2
(A8)

so that the extrapolation rule is obeyed. Reversing the series, with I > J > N, reverses the inequalities in (A8) so that the extrapolation rule is violated. IV. Explicit Examples of ESSs for the Interference Model with Continuous Phenotypes As in Appendix III, there are three patches, i, j, n, with prey densities Qi < Q~< Qn. Assume that competitive weights K(S) do not alter between patches. For simplicity, let K(S) = KS, where Kis a positive constant. From the boundary phenotype rule and equation (8)

-m

Qi " ni

~

--m

Qj " nj

-DI

K(S,)

,

QJ " ni

:_

(A9)

K(S2)

(A 1O)

--m

Qn " nn

K(S2)

,

K(S,)

which are analogous to (A6) and (A7). It is easy to calculate ni, nj, nn from the appropriate integrals of P(S) multiplied by the total population size N. For example s, ni = U ~ P ( S ) d S 0

Parker & Sutherland." Phenotypic ideal free models a n d hence

1241

&

R i = (N S K(S)P(S)dS)/ni o As in A p p e n d i x III, let P ( S ) = b(1 - C S ) , then

Wi(SI) = Q i ' N b S I ( 1 - ~ ' ~ 1 )

/1 CSI~ [" --mt2--"~--)/~ l

c_~@) / /1 cs2\ /1 cs~\\ -mQS~[~-T)- s~[2-T) ) /

/

CS2\

Nb(S2(1 CS2 S " CS1"~'~s'~$211---2-)-s' (1-~-))

Wj(SI) = aj 9

T)

/

/1

CS2\

fl

CSI~

-mtS~t;-T)-s~t2-T) ) wj( s2) = / 2 z/1 CS2\\ -mt~-fi- s 2 t 2 - T ) )

Wn(S2) =

T h e values of $1 a n d $2 c a n again be solved iteratively at given numerical values o f the constants; a typical result is s h o w n in Fig. 5. As in the i n p u t model, rule (A8) was always obeyed p r o v i d e d t h a t Qj < Qj < Q,. Reversing the order o f these inequalities caused violation o f the e x t r a p o l a t i o n rule.

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(Received 10 May 1985," revised 11 July 1985; MS. number: 2699)