Stability for a competitive Lotka-Volterra system with delays
The authors consider the following Lotka-Volterra type competitive system with discrete delays \[ \dot x(t) =x(t)[b_1-a_{11}x(t-\tau_{11})-a_{12}y(t-\tau_{12})],\quad \dot y(t) =y(t)[b_2-a_{21}x(t-\tau_{21})-a_{22}y(t-\tau_{22})], \] with the initial conditions \[ \begin{aligned} x(t)&=\phi_1(t)\geq 0, \quad t\in[-\tau,0],\quad\phi_1(0)>0,\\ y(t)&=\phi_2(t)\geq 0,\quad t\in[-\tau,0],\quad\phi_2(0)>0,\\ \tau&=\max\{\tau_{ij}\},\end{aligned} \] where \(x(t), y(t)\) stand for densities of both the population at time \(t\), respectively, \(b_i, a_{ij}\) are all positive constants and \(\tau_{ij}\) are nonnegative. Conditions for local and global asymptotic stability of the positive equilibrium \(Z^*=(x^*,y^*)\) are obtained.
scientific article; zbMATH DE number 1836635