PERSISTENCE IN MODELS OF PREDATOR PREY POPULATIONS WITH DIFFUSION
PERSISTENCE IN MODELS OF PREDATOR PREY POPULATIONS WITH DIFFUSION
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DOI：
10.1016/0022-0396(86)90044-6
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发表时间：
1986-10-01
影响因子：
2.4
通讯作者：
SCHMITT, K
中科院分区：
文献类型：
--
作者：
DUNBAR, SR;RYBAKOWSKI, KP;SCHMITT, K
关于常微分方程模型的持久性有大量的文献（参见[4]中的许多参考文献）。我们的工作是由文件的动机[4，53，其中美丽的几何参数被用来推断在某些三维系统的持久性。我们提出了这样一个问题：一旦人口不再在空间上均匀分布，而是允许在空间上扩散，类似的系统是否仍然会持续存在。本文针对这一问题，并提出了一个allirmative答案的一大类模型。本文的组织方式如下：我们首先提出了一些术语和事实，从理论的动力系统，并使用这些推导出一个抽象的结果不变集。这个结果可以用来证明某些反应扩散系统的持久性。作为应用，我们考虑了文献[4]中所研究的一类三维非线性系统。最后，我们以图形的形式给出了一些数值计算的结果。Butler和P.Waltman（“Persistence in dynamic systems，”J.Differential Equations，in press），其中建立了与我们的定理2.2类似的结果。由于那里采用的证明方法，Butler和Waltman的结果不适用于具有扩散的系统。我们要感谢Oberwolfach Forschungsinstitut使我们三人聚集在一起。
A wealth of literature exists on persistence in ordinary differential equations models (see, eg,[4] for many references). Our work was motivated by the papers [4, 53, where beautiful geometric arguments are used to deduce persistence in certain three-dimensional systems. We posed the question whether similar systems will still be persistent once populations are no longer spatially homogeneously distributed but are allowed to diffuse in space. This paper addresses this question and presents an allirmative answer for a large class of models. The paper is organized in the following way: We first present some terminology and facts from the theory of dynamical systems and use these to deduce an abstract result about invariant sets. This result can be used to show persistence in certain reaction-diffusion systems. As an application we consider a nonlinear three-dimensional system of the type studied in [4]. Finally, we present graphically the results of some numerical calculations.During a recent meeting at Oberwolfach, West Germany, we learned of a paper by G. Butler and P. Waltman (“Persistence in dynamical systems,” J. Differential Equations, in press) where a result similar to our Theorem 2.2 is established. Because of the method of proof employed there, the result of Butler and Waltman is not applicable to systems with diffusion. We wish to thank the Oberwolfach Forschungsinstitut for bringing the three of us together.