In this work, we consider an ecological system of three species with two predators-one prey type without or with diffusion. For the system without diffusion, i.e. a system of three ODEs, we clarify global dynamics of all equilibria and find an exact condition to guarantee the existence and global asymptotic stability of the positive equilibrium. However, when the corresponding condition does not hold, the prey becomes extinct due to the over exploitation. On the other hand, for the system with diffusion, using the cross iteration method we find the minimum speed c∗. The existence of traveling wave front connecting the trivial solution and the coexistence state with some sufficient conditions is verified if the wave speed is large than c∗ and we also prove the nonexistence of such solutions if the wave speed is less than c∗. Finally, numerical simulations of system without or with diffusion are implemented and biological meanings are discussed.
Discrete and Continuous Dynamical Systems-Series B 20(4), pp.1135–1154