Yuan Lou

Evolution of dispersal and the ideal free distribution

A general question in the study of the evolution of dispersal is what kind of dispersal strategies can convey competitive advantages and thus will evolve. We consider a two species competition model in which the species are assumed to have the same population dynamics but different dispersal strategies. Both species disperse by random diffusion and advection along certain gradients, with the same random dispersal rates but different advection coefficients. We found a conditional dispersal strategy which results in the ideal free distribution of species, and show that it is a local evolutionarily stable strategy. We further show that this strategy is also a global convergent stable strategy under suitable assumptions, and our results illustrate how the evolution of conditional dispersal can lead to an ideal free distribution. The underlying biological reason is that the species with this particular dispersal strategy can perfectly match the environmental resource, which leads to its fitness being equilibrated across the habitats.

Does movement toward better environments always benefit a population

Abstract We study the effects of advection along environmental gradients on logistic reaction–diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.

Evolution of conditional dispersal: a reaction-diffusion-advection model

To study evolution of conditional dispersal, a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment is proposed and investigated. The two species are assumed to be identical except their dispersal strategies: both species disperse by random diffusion and advection along environmental gradients, but one species has stronger biased movement (i.e., advection along the environmental gradients) than the other one. It is shown that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; if both species have such strong biased movements, it can lead to overcrowding of the whole population at places of locally most favorable environments, which causes the extinction of the species with stronger biased movement. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients, and an intermediate biased movement rate may evolve.

On several conjectures from evolution of dispersal

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17–36 [9]] concerning the dynamics of a diffusion–advection–competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [9]. It was shown in [9] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [9] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [9], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.

The Evolution of Conditional Dispersal Strategies in Spatially Heterogeneous Habitats

To understand the evolution of dispersal, we study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The two species are assumed to be identical except for their dispersal strategies: both species disperse by random diffusion and advection along environmental gradients, but with slightly different random dispersal or advection rates. Two new phenomena are found for one-dimensional habitats and monotone intrinsic growth rates: (i) If both species disperse only by random diffusion, i.e., no advection, it was well known that the slower diffuser always wins. We show that if both species have the same advection rate which is suitably large, the faster dispersal will evolve; (ii) If both species have the same random dispersal rate, it was known that the species with a little advection along the resource gradient always wins, provided that the other species is a pure random disperser and the habitat is convex. We show that if both species have the same random dispersal rate and both also have suitably large advection rates, the species with a little smaller advection rate always wins. Implications of these results for the habitat choices of species will be discussed. Some future directions and open problems will be addressed.

Evolutionary stability of ideal free dispersal strategies in patchy environments

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244–251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt’s (Am Nat 140:1010–1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.

Single Phytoplankton Species Growth with Light and Advection in a Water Column

We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coefficient, water column depth, and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. Under a general reproductive rate which is an increasing function of light intensity, we establish the existence of a critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate is a strictly monotone decreasing function of the sinking or buoyant coefficient and water column depth, and it is also a strictly monotone decreasing function of the turbulent diffusion rate for buoyant species. In contrast to the critical death rate, a critical sinking or buoyant velocity, a critical water column depth, and a critical turbulent diffusion rate ma...

Evolution of dispersal in open advective environments.

We consider a two-species competition model in a one-dimensional advective environment, where individuals are exposed to unidirectional flow. The two species follow the same population dynamics but have different random dispersal rates and are subject to a net loss of individuals from the habitat at the downstream end. In the case of non-advective environments, it is well known that lower diffusion rates are favored by selection in spatially varying but temporally constant environments, with or without net loss at the boundary. We consider several different biological scenarios that give rise to different boundary conditions, in particular hostile and “free-flow” conditions. We establish the existence of a critical advection speed for the persistence of a single species. We derive a formula for the invasion exponent and perform a linear stability analysis of the semi-trivial steady state under free-flow boundary conditions for constant and linear growth rate. For homogeneous advective environments with free-flow boundary conditions, we show that populations with higher dispersal rate will always displace populations with slower dispersal rate. In contrast, our analysis of a spatially implicit model suggest that for hostile boundary conditions, there is a unique dispersal rate that is evolutionarily stable. Nevertheless, both scenarios show that unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rate can evolve.

Evolution of conditional dispersal: evolutionarily stable strategies in spatial models

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.

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.