Diffusion in the heterogeneous environments

Abstract

In the past decades, the effect of dispersal size in evolutionary selection has been intensively studied. The movement of organisms is divided into two main categories: active and passive dispersal. Firstly, we consider the active dispersal with a patch model. Instead of considering constant dispersal, we consider the simple but useful and practical dispersal. The starvation driven dispersal is introduced and applied to the patch model. This starvation driven dispersal runs according to starvation measure which is evaluated both by population and by resources. The main interest is that which dispersal strategy will be benefitial to individuals. Therefore, we discuss an ideal free distribution in the viewpoint of the fitness. In addition, in two species-two patch model, two distinct species compete each other and coefficients of competition are different. Our results predict that two species can coexist instead of extinction of inferior species if the inferor species migrate with the starvation driven dispersal.

Secondly, we investigate the passive dispersal which is affected by external factors such as winds or the environment structure in which organisms live. We derive diffusion model in heterogeneous space by using random walks and apply it to vineyard environment. If we consider the environment which is aligned vertically such as vineyard, then we are able to obtain an anisotropic diffusion. In this stratified media, the diffusivities are different depending on the direction and thus we need to consider an anisotropic diffusion. We calculate the anisotopic diffusion using wind frequency distribution and anisotropy of environment simultaneously. With the anisotropy diffusion, we propose two different models to explain the movement of pathogens. Since the spreading speed of pathogen is the most important part to control or manage the disease, we find the spreading speed from dispersion relations and figure out the envelope of the leading edge. In particular, we can observe the pushed front instead of pulled front in some cases.