這篇論文我們研究在兩個或三個破碎環境下捕食者與被捕食者物種密度相互影響的傳播動態分析。在每個破碎棲息地上，擁有兩或三物種的雜食生態系。我們假設在兩物種時的捕食者和三物種時頂端捕食者和中間物種可以在破碎棲息地之間移動。在兩物種兩棲息地的情況下，我們已經研究出所有變數的動態行為，也找出來局部穩定的所有平衡點。甚至一些全域穩定的情況也找到理論判別出來。最後我們提出三物種兩棲息地和三物種三棲息地的數據分析。傳播消耗的影響我們則在生物多樣性下面去討論。

In this work, we consider the population-dispersal dynamics for predator-prey interactions in a two- or three-patchy environment. On each fragmented patch, there is a two-species predator-prey or three-species omnivory ecological system. It is assumed that the predator species of two-species models or the intermediate species and top predator of three-species models are mobile. For the 2-species-2-patch case, we have completely classi_ed all dynamics with respect to all parameters and have showed the local stability of all equilibria. Moreover, some global extinction results are verified analytically. Finally, numerical simulations of 3-species-two-patch and 3-species-3-patch model are performed. The the effect of dispersal rate with respect to the bio-diversity are discussed.

Contents
1 Introduction	1
2 Theoretical Analysis of Predator-Prey Systems in Two Patches	3
2.1 Preliminary Results and Local Stability of Boundary Equilibria	5
2.2 Numerical Study of Global Stability of the EquilibraĒ2 and Ē*	10
3 Some Numerical Simulations of Three-Species Omnivory Model on Two-Patch and Three-Patch Habitat	11
3.1 Numerical Simulations of Three-Species Omnivory Model on Two-Patch Habitat	14
3.2 Numerical Simulations of Three-Species Omnivory Model on Three-Patch Habitat 	20
3.2.1 Simulations of Three-Species Omnivory Model on Three-Patch Habitat with Linear Connection 	20
3.2.2 Simulations of Three-Species Omnivory Model on Three-Patch Habitat with Complete Connection	21
4 Discussion and Biological Interpretations	23
References	24


List of Figures
2.1: The dynamics of (2.3)-(2.4) by parameters ayx and byx. 	10
2.2: Time courses of numerical simulation of each species of (2.3)-(2.4) with parameters and dy1 = 0.1 in Table 2.1.	11
2.3: Time courses of numerical simulation of each species of (2.3)-(2.4) with parameters and dy1 = 0.5 in Table 2.1.	12
3.1: The asymptotical states of the isolated patches are presented with parameters in Table 3.1	15
3.2: The asymptotical states of whole systems with coupled patch 1 and patch 2 are calculated its corresponding bio-diversity indicator .	16
3.3: The asymptotical states of whole systems with coupled patch 1 and patch 2 are calculated its corresponding bio-diversity indicator I for another four cases.	17
3.4: The four cases of the Table 3.3	18
3.5: Another six cases of the Table 3.3	19
3.6: I as a function of symmetric dispersal rates m12 = m21 > 0 and n12 = n21 > 0	21
3.7: Sx ⊕ Sx ⊕ Sxy	23


List of Tables
2.1: Parmeters to simulate the global dynamics of (2.3)-(2.4) numerically.	11
3.1: Parameters of (3.5) with respect to asymptotical states for mij = nij = 0.	14
3.2: The indicator I of bio-diversity of (3.5) with mij > 0 and nij = 0	17
3.3: The indicator I of bio-diversity of (3.5) with mij > 0 and nij > 0	18
3.4: The indicator I of bio-diversity of (3.6)-(3.8) with mij > 0 and nij > 0	22

References
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