中文摘要：

讨论了一类捕食-食饵-互惠反应扩散系统的非常数正平衡解.首先分析了常数正平衡解的稳定性,其次;利用最大值原理和Harnack不等式给出了正解的失验估计.在此基础上,利用积分性质进一步讨论了非常数正解的不存在性,相应地证明了当扩散系数d2 d3大于特定正常数且扩散系数d1有界时此模型没有非常数正解.同时利用度理论证明了当模型的线性化算子的正特征值的代数重数是奇数且扩散系数d3不小于给定正常数时此模型至少存在一个非常数正解,最后研究了非常数正解的分歧.

英文摘要：

The Non-constant positive steady state is discussed for a predator-prey-mutualist reaction-diffusion system.First,the stability of the unique positive constant steady state is proved.Second,By means of maximum principle and Harnack inequality,the prior-estimate to the positive solutions of the model is given.Based on this,it is further discussed that the non-existence of the non-constant positive solutions is considered by using the integral property.and it is proved that the model has no non-constant positive solution when the diffusion coefficient d2 and d3 are larger than the special positive constants and diffusion coefficient d1is bounded.And,the degree theory is utilized for discussing the existence of the non-constant positive solutions,therefore,we will obtain that the model has at least one non-constant positive solution if the algebraic multiplicity of the positive eigenvalue of the linearized operator of the model is odd and diffusion coefficient d3 is not less than some given positive constant,Finally,bifurcation of non-constant positive steady state are studied.