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.
Dispersal은 개체수를 증가시키거나 감소시킬 수 있을 뿐만 아니라, 생태계의 stability에도 영향을 미칠 수 있기 때문에 최근 patch model에서 dispersal의 중요성이 커지고 있다. 그 여파로 많은 논문들이 density dependent dispersal 에 관한 연구를 다루고 있다. 하지만 생각보다 많이 복잡할 뿐더러 수학적으로 많은 연구가 되지 못하였다. 하여, 간단하지만 생태계의 섭리를 잘 표현한 불연속인 motility 함수를 도입하여, biological patch 모델을 분석하였다.
본 논문 3장에서는 먼저 가장 간단한 모델에 대하여 steady state, stability와 그 외 다른 성질들을 다루었다. 이를 바탕으로 일반적인 모델에서의 같은 내용을 말하였다. 이 논문에서 가장 주목한 것은 생태계에서 중요한 성질 중 하나인 optimal selection이었다. Optimal selection이 중요한 이유는 개체수가 carrying capacity보다 작으면 침략을 당하기 쉽고, 만약 carrying capacity보다 크면 patch안에서 경쟁이 더 심해지기 때문에 optimal selection이 생태계에서는 가장 이상적인 상태라고 할 수 있다. 이 논문에서는 patch의 carrying capacity의 비율이 dispersal의 비율보다 작다면 항상 optimal selection이 일어남을 밝혔다.