The existence of periodic solutions for some models with delay
Consider the system \[ \begin{aligned} \frac{dv_1(t)}{dt}& = \tau\bigl(v_1^*+v_1(t)\bigr)\biggl[-a_1v_1(t)+a_2g\bigl(v_2 (t)\bigr)\biggr],\\ \frac{dv_2(t)}{dt} & = \tau \bigl(v_2^*+v_2(t)\bigr)\biggl[-a_1v_1(t)+ a_2g\bigl(v_2(t)\bigr)-a_3g\bigl(v_2 (t-1)\bigr)\biggr],\tag{*} \end{aligned} \] where \(a_1,a_2,a_3,v_1^*,v_2^*\) are constants and \(\tau\) is the bifurcation parameter. The authors derives conditions such that it holds: (i) There exists a sequence \(\{\tau_n\}\) with \(\tau_{n+1}> \tau_n\) such that (*) has a Hopf bifurcation at \(\tau_n\), \(n=0,1,1, \dots\) (ii) For \(\tau>\tau_1\), system (*) has at least one nonconstant periodic solution. Finally, the results are applied to a predator-prey model with Michaelis-Menten type functional response.
scientific article; zbMATH DE number 1880136