Positive periodic solution for a nonautonomous logistic model with linear feedback regulation
The authors considered the following nonautonomous logistic model with linear feedback regulation \[ \begin{aligned} & x_1'(t)=x_1(t)\bigl(r(t)-a_1 (t)x_1(t)-a_2(t) x_1(t-\tau_1)-c(t)x_2(t-\tau_2)\bigr),\\ & x_2'(t)=-a(t) x_2(t)+b(t)x_1(t-\tau_3), \end{aligned}\tag{*} \] with initial condition: \[ x_i(\theta)=\psi_i (\theta),\theta\in[-\tau,0],\;\psi_i(0)\geq 0,\psi_i\in C^1\bigl([-\tau,0],R_+ \bigr),\;i=1,2, \] where \(a(t)\), \(b(t)\), \(c(t)\), \(r(t)\), \(a_1(t)\) and \(a_2(t)\) are continuous \(\omega\)-periodic functions; \(a(t)\), \(b(t)\), \(c(t)\) and \(r(t)\) are strictly positive, \(a_1(t)\) and \(a_2 (t)\) are nonnegative, \(\tau_1\), \(\tau_2\) and \(\tau_3\) are nonnegative constants, \(\tau=\max\{\tau_1,\tau_2,\tau_3\}\). The authors establish sufficient conditions for the uniform persistence of the model and for existence, uniqueness and global asymptotic stability of a positive \(\omega\)-periodic solution of the system (*), respectively.
scientific article; zbMATH DE number 5077205