Hopf bifurcation in a Volterra prey--predator model with strong kernel
In the prey-predator model with distributed delays and strong kernels \(F_i(u)=a_i^2u\text{e}^{-a_iu}\), \(a_i>0\), \(i=1, 2\), \[ \begin{aligned} \dot{N}_1 &=b_1N_1 \biggl((1-c_{11}N_1-c_{12}\int_0^{+\infty}N_1(t-u)F_1(u)\,du\biggr),\\ \dot{N}_2 &=b_2N_2\biggl(-1+c_{21}\int_0^{+\infty}N_1(t-u)F_2(u)\,du\biggr), \end{aligned} \] \(b_i>0\), \(c_{ij}>0\), suggested by \textit{J. M. Cushing} [J. Math. Biol. 3, 369--380 (1976; Zbl 0342.92012)], the stability of the bifurcating periodic solutions is investigated taking the density coefficient \(c_{11}\) of the prey as bifurcation parameter. For the determination of the direction and stability of the Hopf bifurcation, the Nyquist criterion and graphical Hopf bifurcation theorem are used. Numerical simulation results are presented.
scientific article; zbMATH DE number 2160091