Dynamical behavior of a diffusive prey-predator model with two-predators competing for one-prey (Q6615302)

In this paper, the authors are concerned with the following three-species diffusive predator-prey system in a bounded and smooth domain \(\Omega\):\N\[\N\left\{ \begin{array}{lll} \displaystyle u_t - d_1 \Delta u =au -bu^2 -p_1 uv -p_2 uw ,&x\in\Omega,\, t>0,\\ \displaystyle v_t -d_2\Delta v = k_1 uv -\beta_1 v- h_1 vw,&x\in\Omega,\,t>0,\\\N\displaystyle w_t -d_3 \Delta w =k_2 uw -\beta_2 w- h_2 vw,&x\in\Omega,\,t>0,\\\N\displaystyle \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu}=0,&x\in\partial \Omega,\, t>0,\\\N \displaystyle u=u_0(x)>0, \ v=v_0(x)>0, \ w=w_0(x)>0,&x\in\bar \Omega,\,t=0. \end{array}\right.\N\]\NHere, \(u\) is the prey species, while both \(v\) and \(w\) are predators competing for the prey \(u\); all the parameters are positive constants.\N\NDepending on parameter values, the system may admit up to 5 constant steady states. Their local or global stability are derived in certain parameter ranges by performing linearized analysis and the Lyapunov functional method. The authors also discuss the corresponding model with zero Dirichlet boundary condition.