Evolution of Complex Density-Dependent Dispersal Strategies

Abstract

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.

Appendix: Relatedness and the Benefit of Dispersal

According to Hamilton’s rule (Hamilton 1964a, 1964b), a costly altruistic behavior can evolve if the cost to the actor, C, is less than R _g ⋅B, where R _g is the genetic relatedness between the actor and the recipient, and B is the fitness benefit to the recipient. To study how kin-cooperation and kin-competition act as evolutionary forces driving dispersal rates in the metapopulation, we therefore need to define a relatedness measure. Costs and benefits can be measured in terms of the fitness proxy (Metz and Gyllenberg 2001), which is the expected amount of successful mutant dispersers emigrated from a clan initiated by a single mutant immigrant over the life span of the colony:

$$ R_{\mathrm{metapop}}=\pi\sum_{n=0}^{K-1}\sum _{n'=1}^{K-n}e'_{n+n'}n'w_{n,n'}, $$

(6)

where w _n,n′ is the expected amount of time that an average clan will spend in a state with n residents and n′ mutants. For a detailed explanation and a method to calculate w _n,n′, see Metz and Gyllenberg (2001) and Parvinen and Metz (2008). Relatedness must therefore also be defined in terms of the clan in the special case that the mutant and resident strategies are the same (note that therefore we are only considering an approximation where we assume that the difference of the mutant and resident is so slight that we can neglect the alteration of the population dynamics brought about by the presence of mutants). Our density-dependent relatedness measure is the probability throughout the life-time of the colony that a randomly chosen individual is a mutant, with the condition that the local population size in the patch at the sampling moment is equal to k:

$$ \rho_{k}=\frac{\sum_{i+j=k}w_{i,j}\frac{j}{i+j}}{\sum_{i+j=k}w_{i,j}}. $$

(7)

We can decompose the invasion fitness proxy into parts

$$ R_{\mathrm{metapop}}=\pi\sum_{k=0}^{K-1}E_{k+1}p_{k}m_{k}, $$

(8)

where p _k is the probability that in a metapopulation-dynamical resident equilibrium, a randomly chosen patch has k residents. Furthermore, the conditional production E _k is the expected amount of successful mutant dispersers emigrated from a clan which initially consists of k−1 residents and a single mutant.

Consider a local population with exactly k individuals, of which at least one is a mutant. Suppose that a specific mutant in this local population is given the option to emigrate immediately. The direct difference in benefits and costs of a positive emigration decision is 1−E _k : 1 for the emigrating mutant, and −E _k for the loss of expected average disperser production of a clan founded by this individual at the decision moment. The indirect effect comes through possible relatives. The expected amount of relatives (i.e., mutants) present in the patch at the decision moment is ρ _k k, of which 1 is the specific individual itself. The fitness difference for each relative is E _k−1−E _k , because the population size in the patch after the emigration decision will be k−1 if the specific mutant emigrates and k if it does not. To summarize, the additional benefits of adult dispersal in a local population with size k compared with the nondispersing case are

$$ 1-E_{k}+(\rho_{k}k-1) (E_{k-1}-E_{k}). $$

(9)

Benefits in natal dispersal goes slightly differently. Consider again a mutant in a local population of k individuals, but now at the moment of birth of one offspring to the mutant. Now the parent (or the offspring) is given the choice of making the offspring to disperse or not. If the offspring disperses, the direct fitness benefits of the parent and offspring together are 1+E _k , and the costs are 2E _k+1. The indirect benefits to the possible relatives are E _k −E _k+1 for each. In total, the additional benefits for a parent that sends its offspring to disperse, compared to parent that does not, are

$$ 1+E_{k}-2E_{k+1}+(\rho_{k}k-1) (E_{k}-E_{k+1}). $$

(10)

Because we assume clonal reproduction, there is no parent-offspring conflict here.

In principle, if these additional benefits are positive in a local population of size k, the fitness gradient for the emigration strategy component e _k should be positive, and vice versa. However, the quantities (9) and (10) do not take into account how w _n,n′, the expected time that a mutant clan spends in a state with n residents and n′ mutants, will change when the strategy of the mutant changes. Thus, the quantities (9) and (10) are only indicative, but can to some extent explain observed dispersal behavior in the model. The complete calculation is taken into account in the actual calculation of the fitness gradient, derived by Parvinen (2011).