中文摘要：

考虑了一类齐次Neumann边界条件下两物种竞争同一种食饵的捕食模型的反应扩散方程组及其对应的平衡态问题,其功能响应函数分别是Holling-Ⅱ型和Beddington-DeAngelis型.利用比较原理研究了解的大时间性态,其中包括耗散性、持久性、半平凡解的稳定性.借助于最大值原理给出了正确的先验估计,并讨论了随着参数的变化非常数正平衡解的不存在性.在一维情形下利用线性化方法,分歧理论研究了半平凡解处的局部分歧,而且得出局部分歧可以延拓到全局分歧即给出了一定条件下正确的存在性.

英文摘要：

In this paper, a predator-prey model with two consumers and one resource under the homogeneous Neumann boundary condition is considered. Two consumers with Holling-Ⅱ functional response and Beddington- DeAngelis functional response compete for the one resource. The large time behavior of the solutions including dissipation, pesistence, stability are first studied with comparison principle. Then, a priori-estimates of positive steady states is given by maximum principle, and the non-existence of noncontant positive steady states is studied as some parameters are varied. Finally, in the one dimensional case, the local bifurcation of the semi-trivial solution is considered by linearization methods and the bifurcation theory. Moreover, the local bifurcation can extend to the global bifurcation. In other words, the existence of positive steady-states is proved in some special conditions.