Abstract
In this work, we consider an ecological system of three specieswith two predators-one prey type without or with diffusion. Forthe system without diffusion, i.e. a system of three ODEs, weclarify global dynamics of all equilibria and find an exactcondition to guarantee the existence and global asymptoticstability of the positive equilibrium. However, when thecorresponding 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.
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