Existence of periodic solutions of a delayed predator-prey system with general functional response (Q856094)

The authors consider the delayed predator-prey system with general functional response, where the prey population growth satisfies Gilpin's model: \[ \begin{aligned} \dot N_1(t)& =N_1(t)\left[r(t)-b(t)N_1\bigl(t-\tau_1(t)\bigr)-\frac{\alpha(t) N_1^{p-1}(t)}{1+mN_1^p(t)}N_2\bigl(t-\sigma(t)\bigr)\right],\\ \dot N_2(t)& = N_2(t)\left[-d(t)-a(t)N_2\bigl(t-\tau_2(t)\bigr)+\frac{\beta(t)N_1^p\bigl(t-\tau_3(t)\bigr)}{1+mN^p_1 \bigl(t-\tau_3(t)\bigr)}\right],\end{aligned} \tag{*} \] where \(N_1(t)\) and \(N_2(t)\) represent the densities of the prey population and predator population at time \(t\), respectively. The functions \(a(t)\), \(b(t)\), \(\alpha(t)\), \(\beta(t)\), \(\sigma(t)\), \(\tau_i(t)\) \((i=1,2, 3):R\to[0,+\infty)\) are continuous, positive and periodic with period \(w\) and \(\alpha(t)\neq 0\), \(\beta(t)\neq 0\); \(r(t)\), \(d(t):R\to R\) are continuous periodic functions with period \(w\) and \(\int^\omega_0d(t)dt>0\), \(\int_0^\omega r(t)dt>0\), \(p\) and \(\theta\) are positive constants and \(p\geq 1\); \(m\) is a nonnegative constant. By using the continuation theorem of coincidence degree theory, the authors obtain a sufficient condition on the existence of positive periodic solutions of system (*).