Hopf bifurcation in a generalized Logistic reaction–diffusion population model with instantaneous and delayed feedback

This article considers a generalized Logistic reaction–diffusion population model with mixed instantaneous and delayed feedback control and subject to the homogeneous Dirichlet boundary condition on the one-dimensional bounded spatial domain (0,π). Existence and asymptotic expression of the small enough positive steady state bifurcating from the zero solution are given in virtue of the implicit function theorem. By choosing the delay τ as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation of the small enough positive steady state are obtained. Particularly, the normal form for Hopf bifurcations is derived according to the normal form method and the center manifold theory for partial functional differential equations (PFDEs). It is demonstrated that when the delayed feedback is dominant, the model can undergo a forward Hopf bifurcation at the positive steady state as the delay crosses through monotonically increasing a sequence of critical values and all the bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. Specifically, the bifurcating periodic solutions from the first bifurcation value are orbitally asymptotically stable in the entire phase space. The reasonability of the main analytical conclusions is also verified by carrying out some numerical simulations.