Abstract
We study reaction-diffusion equations describing population dynamics of single harvested species and of two competing species. The main aim of the thesis is to study the roles of two different diffusion strategies: the regular diffusion and the directed diffusion. In directed diffusion, rather than the population itself, its ratio to either locally available resources (carrying capacity) or to a positive distribution function diffuses. We focus on how directed diffusion, especially, carrying capacity driven dispersion in the habitat influences selection. For single species, we present comparative numerical results between carrying capacity driven diffusion and regular diffusion for Gilpin-Ayala type growth and harvesting.
For two competing species, we study the interaction between different types of dispersal: one of them is subject to a regular diffusion while the other moves in the direction of most per capita available resources. If spatially heterogeneous carrying capacities coincide, and intrinsic growth rates are proportional then competitive exclusion of a regularly diffusing population is inevitable. When the resource function of a regularly diffusing population is higher than of the other species, the two populations may coexist.
For symmetric growth, we consider the case when the ideal free distribution is attained as a combination of the two strategies adopted by the two species. Then there is an ideal free pair, and the relevant coexistence equilibrium is a global attractor. In the event that only one of the diffusion strategies is proportional to the carrying capacity, we prove the competitive exclusion of the other species. In case of weak competition, both species can coexist even only one of the species adopts the ideal free dispersal strategy.
When one of the species is following the directed dispersal strategy and the other is
dispersing regularly, there is a unique coexistence solution if the difference between the carrying capacity and the directed function is a positive constant. Coexistence can be a result of the interplay of different diffusion coefficients or growth rates.
