A reaction-diffusion model is investigated to understand infective environments in a man-environment-man epidemic model. The free boundary is introduced to describe the expanding front of an infective environment induced by fecally-orally transmitted disease. The basic reproduction number R F 0 (t) for the free boundary problem is introduced, and the behavior of positive solutions to the reaction-diffusion system is discussed. Sufficient conditions for the bacteria to vanish or spread are given. We show that, if R 0 ≤ 1, the bacteria always vanish, and if R F 0 (t 0 ) ≥ 1 for some t 0 ≥ 0, the bacteria must spread, while if R F 0 (0) < 1 < R 0 , the spreading or vanishing of the bacteria depends on the initial number of bacteria, the length of the initial habitat, the diffusion rate, and other factors. Moreover, some sharp criteria are given. MSC: primary: 35R35; secondary: 35K60

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In this paper, we study the dynamics of predator-prey interaction systems between two species with ratio-dependent functional responses. First we provide sufficient and necessary conditions for positive steady-state solutions, and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain. Furthermore, we deal with the uniqueness and the stability of positive steady-states solutions with some assumptions. In addition, we discuss the extinction and the persistence results of time-dependent positive solutions to the system.

a b s t r a c tWe study a ratio-dependent reaction-diffusion system incorporating one prey and two competing predator species under homogeneous Neumann boundary conditions. In this paper, we examine the global attractor and persistence of the system, which characterize the long time behavior of the time-dependent solution, and the stability of all non-negative equilibria of the system. We classify the relations between two competing predators into four categories via domination of one predator over another and weak/strong competition states. These competition states will be the criteria which influence the outcomes of the system. The results include total extinction, competitive exclusion, behavior of a predator driving the extinction of another predator and its prey, behavior of a predator saving its competing predator and prey on the verge of extinction.