Evolutionary convergence to ideal free dispersal strategies and coexistence

Population dynamics with resource-dependent dispersal: single- and two-species models

In this paper, we consider the population models with resource-dependent dispersal for single-species and two-species with competition. For the single-species model, it is well-known that the total population supported by the environment is always greater than the environmental carrying capacity if the dispersal is simply random diffusion. However, we find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This analytical finding not only well agrees with the yeast experiment observation of Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), but also indicates that resource-dependent dispersal may produce different outcomes compared to the random dispersal. For the two-species competition model, when two competing species use the same dispersal strategy up to a multiplicative constant (i.e. their dispersal strategies are proportional) or both diffusion coefficients are large, we give a classification of global dynamics. We also show, along with numerical simulations, that if the dispersal strategies are resource-dependent, even one species has slower diffusion, coexistence is possible though competitive exclusion may occur under different conditions. This is distinct from the prominent result that with random dispersal the slower diffuser always wipes out its fast competitor. Our analytical results, supported by the numerical simulations, show that the resource-dependent dispersal strategy has profound impact on the population dynamics and evolutionary processes.

Dimorphism by Singularity Theory in a Model for River Ecology

Geritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our finding shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics.

Evolution of dispersal in closed advective environments

We study a two-species competition model in a closed advective environment, where individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. The two species have the same growth and advection rates but different random dispersal rates. The linear stability analysis of the semi-trivial steady state suggests that, in contrast to the case without advection, slow dispersal is generally selected against in closed advective environments. We investigate the invasion exponent for various types of resource functions, and our analysis suggests that there might exist some intermediate dispersal rate that will be selected. When the diffusion and advection rates are small and comparable, we determine criteria for the existence and multiplicity of singular strategies and evolutionarily stable strategies. We further show that every singular strategy is convergent stable.

Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments

We consider a population living in a patchy environment that varies stochastically in space and time. The population is composed of two morphs (that is, individuals of the same species with different genotypes). In terms of survival and reproductive success, the associated phenotypes differ only in their habitat selection strategies. We compute invasion rates corresponding to the rates at which the abundance of an initially rare morph increases in the presence of the other morph established at equilibrium. If both morphs have positive invasion rates when rare, then there is an equilibrium distribution such that the two morphs coexist; that is, there is a protected polymorphism for habitat selection. Alternatively, if one morph has a negative invasion rate when rare, then it is asymptotically displaced by the other morph under all initial conditions where both morphs are present. We refine the characterization of an evolutionary stable strategy for habitat selection from Schreiber (Am Nat 180:17-34, 2012) in a mathematically rigorous manner. We provide a necessary and sufficient condition for the existence of an ESS that uses all patches and determine when using a single patch is an ESS. We also provide an explicit formula for the ESS when there are two habitat types. We show that adding environmental stochasticity results in an ESS that, when compared to the ESS for the corresponding model without stochasticity, spends less time in patches with larger carrying capacities and possibly makes use of sink patches, thereby practicing a spatial form of bet hedging.