Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

and Applied Analysis 3 2. The Model System Inmodel (1), the productf(u)g(V) gives the rate atwhich prey is consumed. The prey consumed per predator, f(u)g(V)/V, was termed as the functional response by Solomon [63]. These functions can be defined in differentways. In this paper, following Lotka [64], we adopt f (u) = cu, (4) which is a linear functional response without saturation, where c > 0 denotes the capture rate [65]. And followingHarrison [28, 29], we set g (V) = V mV + 1 , (5) where m > 0 represents a reduction in the predation rate at high predator densities due tomutual interference among the predators while searching for food. The proportionality constant σ is the rate of prey consumption. And the function z(V) is given by z (V) = γV + lV 2 , γ > 0, l ≥ 0, (6) where γ denotes the natural death rate of the predator, and l > 0 can be used tomodel predator intraspecific competition that is not the direct competition for food, such as some type of territoriality [28]. In this paper, we will discuss the case l = 0, which is used in a much more traditional case. Based on the previous discussions, we can establish the following predation model of two partial differential equations with additive Allee effect on prey: