Optimal selection under satisfaction dependent dispersal strategy

Abstract

Considering how many populations live in certain environments is an important thing in ecology. For this reason, there has been a great interest in the patch model and dispersal. Most of all, they have a remarkable attention in density-dependent dispersal recently. However it is so complicated to study that there are a few mathematical results compared with the result of modeling or simulation. Nevertheless, they continued study because constant-dispersal does not explain the real situations well. In other words, density-dependent dispersal is more suitable for the real situations. Thus, in this paper, we explore the motility dispersal similar to density-dependent dispersal. We consider population and carrying capacity of its patch at the same time as a dispersal. Thus we introduce a concept of satisfaction as a ratio of population and carrying capacity. Our dispersal is ruled by a simple logic of satisfaction. If individuals satisfy their patch, then they stay their patch. On the other hand, if individuals do not satisfy their patch, then they move with high rate. It is easy to understand and more suitable than the others. We call this dispersal as motility function. Secondly, we emphasize the optimal selection which is a meaningful concept of ecology. Why is it so important in ecology? For one thing, it is the best situation for each individual. To put it delicately, if there is no optimal selection, it is easy to be invaded or compete with others in the same patch. We will research the results related to mathematics through this paper. We split it up into two groups to deal with patch model symmetric and non-symmetric case. This paper contains existence of steady state (equilibrium point) and its stability. Moreover, it also has some properties about total population and condition of optimal selection strategy. More precisely, we mainly discuss that optimal selection have intimate connection with dispersal rate.