Analysis of a mathematical predator-prey model with delay (Q1007014)

Consider the system of autonomous delay differential equations \[ \begin{aligned} {dN_1(t)\over dt} &= N_1(t)(\varepsilon_2- \gamma_1N_2(t- \tau)),\\ {dN_2(t)\over dt} &= -N_2(t) (\varepsilon_2- \gamma_2 N_1(t-\tau)),\end{aligned}\tag{\(*\)} \] where \(\varepsilon_1\) and \(\gamma_i\) are positive constants, \(\tau> 0\). The authors prove that \((*)\) has an unstable \(\tau\)-periodic solution for all \(\tau\) satisfying \(2\pi< \tau\sqrt{\varepsilon_1\varepsilon}\). The proof is based on an associated system of ordinary differential equations.