具有B-D反应项与毒素影响的捕食系统的共存解

Coexistence solution of a predator-prey system with B-D functional response and toxin effects

研究了一类具有B-D反应项及毒素影响的捕-食饵系统在齐次Dirichlet边界条件下的平衡态问题。首先利用极大值原理及特征值比较原理给出了系统共存解的先验估计与无共存解的必要条件;其次利用Leray-Schauder度理论,通过计算锥映不动点指标,结合极值原理、上下解方法,阐明了共存解存在的充分条件;最后利用线性化算子及Riesz-Schauder理论证明了平衡态问题的平凡解和半平凡解的局部渐近稳定性。

The steady state of a predator-prey system with B-D reaction term and toxin effect under the homogeneous Dirichlet boundary condition is investigated. First,using the principle of maximum value and the comparison principle of eigenvalues,we give some prior estimates of coexistence solution on the system and obtain the necessary condition of non-coexistence solution. Secondly,by using the Leray-Schauder degree theory,the calculation of the fixed point index,the maximum principle and the method of upper and lower solutions,the sufficient condition for the existence of the coexistence solution is established. Finally,the local asymptotic stability of the trivial solution and the semi-trivial solution of the steady state system is proved by using the linearization operator and the Riesz-Schauder theory.