Random dispersal vs. non-local dispersal
Chiu-Yen Kao Yuan Lou Wenxian Shen
Random dispersal is essentially a local behavior which describes the movement of organisms between adjacent spatial locations. However, the movements and interactions of some organisms can occur between non-adjacent spatial locations. To address the question about which dispersal strategy can convey some competitive advantage, we consider a mathematical model consisting of one reaction-diffusion equation and one integro-differential equation, in which two competing species have the same population dynamics but different dispersal strategies: the movement of one species is purely by random walk while the other species adopts a non-local dispersal strategy. For spatially periodic and heterogeneous environments we show that (i) for fixed random dispersal rate, if the nonlocal dispersal distance is sufficiently small, then the non-local disperser can invade the random disperser but not vice versa; (ii) for fixed nonlocal dispersal distance, if the random dispersal rate becomes sufficiently small, then the random disperser can invade the nonlocal disperser but not vice versa. These results suggest that for spatially periodic heterogeneous environments, the competitive advantage may belong to the species with much lower effective rate of dispersal. This is in agreement with previous results for the evolution of random dispersal [9, 13] that the slower disperser has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.
keywords: Reaction-diffusion Competition Non-local dispersal Random dispersal Integral kernel.
On the Benilov-Vynnycky blow-up problem
Marina Chugunova Chiu-Yen Kao Sarun Seepun
We study an initial-boundary value problem for a fourth-order parabolic partial differential equation with an unknown velocity. The equation originated from the linearization of a two-dimensional Couette flow model, that was recently proposed by Benilov and Vynnycky. In the case of a $180^{\circ}$-- contact angle between liquid and a moving plate Benilov and Vynnycky conjectured that the speed of the contact line blows up to infinity in finite time. In this paper we present numerical simulations and qualitative analysis of the model. We show that depending on the initial data and parameter values different long time behaviors of velocity can be observed. The speed of the contact line may blow up to infinity or converge to a constant.
keywords: contact line singularity contact angle fourth-order parabolic equation. blow-up Couette flow
Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains
Chiu-Yen Kao Yuan Lou Eiji Yanagida
This paper is concerned with an indefinite weight linear eigenvalue problem in cylindrical domains. We investigate the minimization of the positive principal eigenvalue under the constraint that the weight is bounded by a positive and a negative constant and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. Both our analysis and numerical simulations for rectangular domains indicate that there exists a threshold value such that if the total weight is below this threshold value, then the optimal favorable region is a circular-type domain at one of the four corners, and a strip at the one end with shorter edge otherwise.
keywords: principal eigenvalue local minimizer cylindrical domain.
Evolution of mixed dispersal in periodic environments
Chiu-Yen Kao Yuan Lou Wenxian Shen
Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between non-adjacent spatial locations and is thus non-local. We propose to study a mixed dispersal strategy, which is a combination of random dispersal and non-local dispersal. More specifically, we assume that a fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes non-local dispersal. We investigate how such mixed dispersal affects the invasion of a single species and also how mixed dispersal strategy will evolve in spatially heterogeneous but temporally constant environment.
keywords: reaction-diffusion integral kernel. random dispersal competition Non-local dispersal

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