RICH DYNAMICS OF GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM
RICH DYNAMICS OF GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM
复制标题
DOI：
10.1090/fic/021/27
复制
发表时间：
1999
期刊：
影响因子：
--
通讯作者：
Y. Kuang
中科院分区：
文献类型：
--
作者：
Y. Kuang
比率依赖的捕食者-食饵模型越来越受到野外生态学家的青睐，被认为是捕食者-食饵相互作用的一种替代或更合适的模型，其中捕食涉及搜索过程。然而，这样的模型在过去并没有得到很好的数学研究。在我们最近的工作中，我们已经表明，这类模型比传统模型表现出更丰富的动力学。在边界动力学中尤其如此。例如，依赖于比率的模型可以表现出动力学，例如对于某些参数和初始条件，两个物种都可能灭绝。本文考虑了一类相当一般的基于比率的Gaus型捕食者-食饵系统解的整体性态。除了证明Gaus型比率依赖的捕食者-食饵模型具有丰富的边界动力学外，我们还给出了当比率依赖的捕食者-食饵系统的正平衡态局部渐近稳定时，系统不存在非平凡正周期解的非常精确的充分条件。我们还给出了三种可能的稳态都是全局渐近稳定的充分条件。我们注意到，对于比率依赖的体系，不会出现富集悖论。一般而言，正定态的局部渐近稳定性甚至不能保证系统的持久性，因此也不意味着系统的全局渐近稳定性。
Ratio-dependent predator-prey models are increasingly favored by field ecologists as an alternative or more suitable ones for predator-prey interactions where predation involves searching process. However, such models are not well studied mathematically in the past. In our recently work, we have shown that such models exhibit much richer dynamics than the traditional ones. This is especially true in boundary dynamics. For example, the ratio-dependent models can exhibit dynamics such as for some parameters and initial conditions, both species can become extinct. In this paper, we consider the global behaviors of solutions of the rather general Gause-type ratio-dependent predator-prey system. In addition to confirm that Gause-type ratio dependent predator-prey models are rich in boundary dynamics, we shall also present very sharp sufficient conditions to assure that if the positive steady state of the ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial positive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically stable. We note that for ratio-dependent systems, paradox of enrichment can not occur. In general, local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system, and therefore does not imply global asymptotic stability.