The Three-patch Model With Random Dispersal And Time Delay

The use of delay differential equations has been playing increasingly important role in the modelling of population dynamics in recent years.Models including some of the past states of these systems are more realistic and complicated.In this paper,we focus on the three-patch model with random dispersal and time delay considering the situation when the environment is heterogeneous.We prove the existence and unique-ness of the global solution of this system,together with its positivity when the initial condition is nonnegative and continuous.We show that there exists a unique positive equilibrium when the dispersal rate is in some explicit range and there is τ₀>0 such that the equilibrium is locally asymptotically stable for 0≤τ<τ₀.Furthermore,this conclusion can be extended to n-patch model.