Here we apply the above results first to a simple model of evolution of dispersal [21]. Although disruptive selection has been found in other dispersal models (e.g., [22–24]), it is not expected to occur in this model, which serves mainly to compare the present approach with a previous computation of stable strategies. Next we consider two models of resource competition, one of which was previously considered in ref. [25]. These are small patch, limited dispersal versions of a class of models of competition widely considered in previous works: see e.g. [16, 25–29], and references therein.
Dispersal
The trait under selection is the dispersal probability of juveniles, and there is a survival cost c of dispersal: the survival probability of dispersed juveniles is (1 - c) times that of philopatric ones. For mutants with dispersal z + δ, the probability that an individual is the philopatric descendant of a mutant parent is
This expression is obtained as the ratio of the relative number of mutant juveniles that do not disperse [which is 1 - (z + δ) for each of j mutant parents] to the relative number of juveniles which come in competition in the patch. The latter is the number of philopatric juveniles, 1 - (z + δ) for each of the j mutant parents and 1 - z for each of the N - j non-mutants, plus the relative number of immigrant juveniles from other patch, which is Nz(1 - c) as parents in other patches are not mutants and their juveniles pay the cost of dispersal.
Likewise, the number of successful emigrant offspring of each deviant parent is
because each deviant parent contributes a relative number (1 - c)(z + δ) of emigrant juveniles which compete with a relative number N [1 - z + (1 - c)z] = N(1 - zc) of competing juveniles in every other patch, and N adult offspring are sampled out of these juveniles (hence N disappears).
From these expressions and from eq. 4, one obtains
and
so that
= 0 only for z* = (F - c)/(F - c2), a formula known since ref. [21]. Together with eq. 7 and the relationship d = (1 - c)z/(1 - cz) between the immigration probability d and the emigration probability z, this yields
Explicitly evaluating d
/dz at the candidate ESS would be cumbersome, but it is easily verified that
is positive (resp. negative) as z approaches 0 (resp. 1). Since
does not vanish between 0 and z*,
must increase (resp. decrease) from 0 when z decreases (resp. increases) from z*. Hence, z* is locally convergence stable. Further, at z*,
From this formula,
< 0, i.e. the candidate ESS is indeed an ESS, since cK <F2 at the candidate ESS. The latter result is not obvious, but can be verified as follows. Using eq. 40 to eliminate the three-genes relationship K, the sign of
is seen to depend on the sign of the factors F - c (which is positive) and c3 (N2 - 1) + cF - N2F3. The latter is always negative for 0 <c < 1 and N ≥ 1, because the third order polynomial in x, c3 (N2 - 1) + cx - N2x3, has only one real root r, which is such that r < (1 - c + 2cN)/(2N) <F.
The equivalence with the "direct fitness" method previously used to obtain eq. 14 and related results (e.g., [20, 30]) can be verified as follows. According to this method, one considers the fitness function
which describes the expected number of offspring of an individual with dispersal probability z•, in a patch of individuals with mean dispersal z0 in a population with resident dispersal z [20, 30, 31]. Since jw
j
= jw (z + δ, z + j δ/N),
and
are equal to the two partial derivatives of w, yielding
The term in brackets is the inclusive fitness [20]. One would also obtain (with π1(0) = wp (z*, z*)/N)
where w(i,j)is the derivative of w, i times with respect to z•, and j times with respect to z0. Such expressions are convenient when fitness is exactly a function of z0 (i.e. when the phenotypes of all neighbors do not need to be distinguished), but they may not be useful otherwise.
Competition for resource
Here we consider a model where an individual is in stronger competition with other individuals which have phenotypes similar to its own phenotype. Many previous models are phenomenological, effectively assuming that each individual exploits only one type of resource according to its phenotype, yet that it interferes more with individuals with similar phenotypes. For a population subdivided in small patches, such a model was formulated in ref. [25] and fully analyzed in the case N = 2. However, its formulation is relatively complex, so we will analyze here a slightly different model, and the final section of the Appendix gives comparable results for the model of ref. [25]. Both models have similar qualitative outcomes, which converge for large patch size.
Here we assume that each individual effectively exploits a range of resource and that interference on fitness of similar phenotypes results only from scramble competition for acquisition of resource (e.g., [26]). We consider a range of resource, such that resource of type y has abundance ρ(y), which is the same in all patches. We assume that resource follows a normal distribution,
for some constant σρ, and some ym such that resource ym is the most abundant one. σρ describes the width of the distribution of resource over different types.
We assume that each individual cannot exploit all resources with equal efficiency. An individual i with phenotype z
i
exploits resource y with an efficiency given by an efficiency function α(y, z
i
), such that the individual gets a fraction
of resource y, which is the ratio of its own efficiency at exploiting resource y to the sum of efficiencies at exploiting y among all individuals in the population. We assume that the efficiency function is normal-shaped,
for some σα, identical for all individuals. Thus, individual i is best at exploiting resource z
i
, and σα quantifies the diversity of resource that an individual may exploit. An individual strategy is therefore characterized by z
i
and constrained by the range of resources it can efficiently exploit around z
i
.
Below we consider residents with strategy z
i
= z and mutants with strategy z
i
= z + δ. Hence, in a patch with j mutants and N - j non-mutants,
.
The share s
j
(δ) of total resource that individual i obtains is its share of resource y, times abundance of resource y, integrated over the distribution of y. Hence, in a patch with j mutants and N - j non-mutants, it is
We assume that the expected number of juveniles of an individual is proportional to this share s
j
of total resource. The probability that an individual in the offspring generation is the descendant of anyone of j mutant parents in its patch is then
π
j
(δ) = js
j
(δ) (1 - d) (22)
and the expected number of successful emigrant gametes of j mutant parents is
h
j
(δ) = N djs
j
(δ). (23)
The factor N appears here, as in the dispersal example, because N adults settle in each patch. A general, useful check of expressions for π
j
and h
j
is that jw
j
(0) = j = h
j
(0) + N π
j
(0) (from eq. 4).
Here
= -
= (ym - z)/
. Hence
is of the sign of ym - z, which means, as expected, that the population will evolve to exploit the most abundant resource, z* = ym. In other words, ym is locally convergence stable. At z*, similar computations yield
For given d, F and K become negligible as N increases (they are of the order of 1/N and 1/N2, respectively), and in the limit we recover the result for panmictic populations, that disruptive selection occurs when σα > σρ, i.e. when the resource is more broadly distributed than can be efficiently exploited by one individual.
Population structure inhibits branching, a previously noted result [25] for the interference competition model (see Appendix): for N = 2,
is always negative and, unexpectedly, is independent of σρ (because the term in brackets in eq. 25 is then independent of σρ and proportional to 1/
). For any N, the exact condition for disruptive selection is 1/
> 1/
+ 2(F - K)/(1 - F)/
. The latter condition is complex, because
but it implies that branching is inhibited by small patch size and low dispersal. This result can be understood as follows [25]. There is disruptive selection when a deviant individual gains fitness from avoiding competition with the resident strategy. However, in a subdivided population, competition is preferentially with genetically related individuals, i.e. with individuals more likely to be deviant than the average individual in the population. Then, deviant individuals do not avoid competition as much as if the neighbors in the patch were not related.
Since K is of the order of 1/N2, it may seem reasonable to approximate
by setting K = 0. This seems to work well only for N large and d large (Figure 1), because K is not negligible when d → 0 even for large N. Ignoring K will be misleading in this case.
The strategy z* = 0 may be locally stable without being globally stable:
< 0 does not exclude that mutants with large effects could invade (for examples, see [32, 33]). Here this may happen for a narrow range of parameter values in the structured population model, even though it does not happen in a single large patch of infinite size. In a single large patch at the candidate ESS, the fitness of a single mutant is
so that large mutations invade if and only if small mutations invade. In structured populations, global stability was investigated by numerical evaluation of Rm, focusing on threshold combinations of
/
and d such that
= 0, that is on the set of parameter values represented by the lines in Figure 1. By continuity, if for some mutants Rm > 1 in the neighborhood of these threshold values, there must be close values of σρ/σα and d such that Rm > 1 for some mutants while
< 0, which is the sought phenomenon of global instability despite local stability. It was found to happen for some parameters combinations as shown in Figure 2. Note that Rm can be quite large in some cases, e.g. Rm > 182 for N = 36, d = 1/100 (implying the threshold value
/
3.94), and δ = 0.729, so the selective pressures at work could be efficient on a short time scale.
To some extent, global instability could be sought from higher-order derivatives of Rm. In the present case the symmetry of selection on mutants with effect δ and -δ implies that Rm is an even function of δ around the candidate ESS and that all its odd-order derivatives are null, so at least the fourth-order derivatives should be considered. In practice, this would be very complex.