Dynamical Analysis In A Three-patch Nicholson’s Blowflies Model With Random Dispersal And Dirichlet Boundary Condition

In 1980,Gurney et al.proposed a time-delayed population model in Nature magazine to explain the phenomenon observed by biologist Nicholson in blowfly experiments,which is called Nicholson’s blowfly model.It has been widely promoted and studied.For the Nicholson’s blowfly model with diffusion,the positive steady state cannot be represented analytically under Dirichlet boundary conditions.This makes it difficult to study many dynamic properties of this model.A typical challenge is how stability of the positive steady state with respect to parameters changes? In order to study this problem,the ideas proposed by Liao et al.were applied in this article.By discretizing spatial variables,a patchy model is obtained,which approximates the original problem.Although the positive equilibrium of the patchy model cannot be represented analytically,the distribution of the roots of the characteristic equation at the positive equilibrium is obtained through detailed analysis of the dependence of the positive equilibrium on the parameters.The main research contents of this article are as follow: Firstly,the Nicholson’s blowfly model with diffusion and Dirichlet boundary conditions on three patches is derived,then the positiveness and the final consistent boundedness of the solution are proved.Secondly,when parameters change the stability of equilibrium is studied by means such as eigenvalue analysis,the theory of monotonic dynamical system,exponential order method for time delayed differential equations and so on.Finally,a numerical simulation is conducted on the basis of the influence of parameters on stability of the positive equilibrium point,then numerical theoretical and numerical simulation results are compared.