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In this paper, we are concerned with positive solutions of a predator–prey model with Crowley–Martin functional response under homogeneous Dirichlet boundary conditions. First, we prove the existence and reveal the structure of the positive solutions by using bifurcation theory. Then, we investigate the uniqueness and stability of the positive solutions for a large key parameter. In addition, we derive some sufficient conditions for the uniqueness of the positive solutions by using some specific inequalities. Moreover, we discuss the extinction and persistence results of time-dependent positive solutions to the system. Finally, we present some numerical simulations to supplement the analytic results in one dimension.

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension * Qu Gai-Zhu(屈改珠) a)b) , Zhang Shun-Li(张顺利) a) † , and Li Yao-Long(李尧龙) b) a)

In this paper, we continue the mathematical study started in (Jang et al. in J. Dyn. Differ. Equ. 16:297-320, 2004; Ni and Tang in Trans. Am. Math. Soc. 357:3953-3969, 2005) on the analytic aspects of the Lengyel-Epstein reaction-diffusion system. First, we further analyze the fundamental properties of nonconstant positive solutions. On the other hand, we continue to consider the effect of the diffusion coefficient d. We obtain another nonexistence result for the case of large d by the implicit function theory, and investigate the direction of bifurcation solutions from (u * , v * ). These results promote the Turing patterns arising from the Lengyel-Epstein reaction-diffusion system.

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