Hopf Bifurcations In A Two-patch Nicholson's Blowflies Model With Random Dispersal And Dirichlet Boundary Condition

Nicholson's experimental data can be well fitted by a time equation,so-called Nicholson's model,proposed by Gurney in 1990.The model has been extensively studied by many scholars,including the existence of the traveling wave solution and the asymptotic spreading speed in infinite space and the existence,stability and bifurcation problem under Neumann boundary condition and Dirichlet boundary condition in finite space for diffusive Nicholson's model.The Hopf bifurcation near the positive solution under the Dirichlet boundary condition is a difficult problem.Applying the Implicit Function Theorem and Liapunov-Schmidt method,some results on the Hopf bifurcation can be obtained for few models.However this method does not apply to the Nicholson model.In order to investigate the existence of Hopf bifurcation near the positive steady solution of Nicholson's model under Dirichlet boundary condition.We discretize the spatial of the diffusive model under Dirichlet boundary condition,obtaining a coupled two patch delay equation.We have proved the existence and uniqueness of the positive equilibrium point.The stability of the positive equilibrium point and the condition of the Hopf bifurcation are analyzed by the method of eigenvalue analysis.Then,the stability of the periodic solution is analyzed using center manifold theory and normal form method.For a certain set of parameter values,numerical simulations are carried out,which are consistent with theoretical results.