The Kinetic Model In Patch Environment

Matapopulation is an important frontier in the recent research of mathematic biology, theoretical ecology and conservation biology. The research provides new theoretic bases for the investigation of populations and species which are in severe danger, and makes forecasts and measurement for the populations' damage due to global habitat destruction and environment deteriorism.The aim of this work is to construct some tri-trophic systems, to analyze the asymptotic behavior of these models and to study the effect of the diffusion on the stability of equilibria.In the first part, in the cycle of invasion-consumption (damage)-recovery of resource, the number of resource patches does not change in the absence of prey population. Under the assumption that the depleted patches will recover from the damage and become new resource patches instantaneously, we formulate models and analyze the asymptotic behavior. Furthermore, considering that the recovery needs time and the time varies according to the degree of the damage, we construct a model by using Holt Weisser and Hassell's formulating method and analyze the persistence, existence and stability of equilibria. The global stability of the unique equilibrium is obtained by means of conpound matrices.In the second part, in the cycle of invasion-consumption (damage)-recovery of resource, we assume that the number of resource patches grows logistically in the absence of prey population. Considering that some resource patches are destroyed too badly to recover we formulate the model. Sufficient conditions of non-existence of positive equilibria, existence of one equilibrium and two equilibria are obtained, and the stability of equilibria is analyzed. Bistability ,with a stable positive equilibrium is possible when there are two positive equilibria. Lastly, we make a numerical analysis.