Dynamics Of Periodic Impulsive Diffusion System With Resource Competition

Competitive relationship is one of the important relationships among biological populations in interspecific effect,and it has important practical significance.Biological research has shown that among closely related species,many competitive interactions occur that are likely to involve both resource competition and mating interference.Therefore,this paper considered the effects of the resource competition and mating competition,studied the temporal and spatial dynamic of two closely related species in a finite domain.In this paper,an impulsive reaction-diffusion system with competition has been studied,which describes the interaction between the two closely related species in the periodic environment.First of all,by constructing the system of upper and lower solution,proved the existence and uniqueness of the solution.We further proved that there existed a positive invariant set.Secondly,using the comparison principle of impulsive ordinary differential equations,we obtained that the trivial solution and the semi-trivial periodic solution were global asymptotic stable.This conclusion shows that the pulse leads to competitive exclusion.In other words,the pulse makes two species extinct or makes one of species live and another extinct.Again,by using comparison principle of reaction-diffusion equation,the permanent of the system was finally proved,and a sufficient condition for the survival of the two species was obtained.In addition,through the construction of a suitable auxiliary function,we proved that the periodic solution of the system was existence,and was globally asymptotically stable.That is to say,the two species can reach the dynamic equilibrium.At last,the main theoretical results were verified by numerical simulation.