Stability of positive steady-state solutions in a delayed Lotka-Volterra diffusion system (Q2898894)

The authors consider the following pray-predator system NEWLINE\[NEWLINE\begin{cases} \displaystyle{\frac{\partial u(x,t)}{\partial t}}=d_1u(x,t)+r_1u(x,t)[1-u(x,t-\tau)-av(x,t-\tau)],&x\in \Omega,t>0,\\ \displaystyle{\frac{\partial v(x,t)}{\partial t}}=d_2v(x,t)+r_2v(x,t)[1+bu(x,t-\tau)-v(x,t-\tau)],&x\in \Omega,t>0,\\ \end{cases}NEWLINE\]NEWLINE with Dirichlet boundary and initial value conditions. Under certain technical conditions, they obtain the existence of positive nontrivial steady state \((u_r,v_r)\) for \(0<r\ll 1\). Furthermore, they show that there is a threshold value \(\tau_0\), such that \((u_r,v_r)\) is asymptotically stable if \(\tau\in [0,\tau_0)\), and \((u_r,v_r)\) is unstable if \(\tau>\tau_0\). In addition, there exists a sequence of values \(\{\tau_n\}_0^{\infty}\) such that the system undergoes a Hopf bifurcation at \((u_r,v_r)\) when \(\tau=\tau_n\).