Stability of Traveling Wavefronts for a Spatially Nonlocal Population Model with Delay

(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

Abstract:The stability of traveling wavefronts for a spatially nonlocal population model with quasi-monotonicity and delay is discussed in this article.It is shown that all monostable wavefronts are exponentially stable for large speed with the help of weightedenergy method and comparison principle.The proper selection of the weighted function is necessary to overcome the difficulty caused by the nonlocal nonlinearity for establishing the energy estimates of solutions.

Keywords:Stability;Wavefronts;Weighted-energy method

§1.Introduction

Population dynamics models are always the investigated objects as a kind of important models in reaction-diffusion equations[1,3,7,8,18].For example,Hadeler and Lewis[8]presented the classical population model

which models the Verhulst type population growth with immobile and mobile subpopulations,whereu1(t,x)andu2(t,x)denote the densities of mobile and immobile subpopulations at timetand positionx,respectively,F(u1(t,x))is the reproduction function,d>0 is the diffusion rate ofu1(t,x),andβ1>0,β2>0 are the convert rates between the two states respectively.Consider that population growth is not instantaneous,it takes time for infant to turn into maturity,and thus the maturationτ≥0 appearing in the model(1.1)is more practical,i.e.,the delayed population dynamic model

with a quiescent stage.In addition,the drift of individuals not only depends on the position at present but on all other possible positions at a period in the past,which can be shown by the spatially weighted average idea[3],and(1.2)is naturally optimized to the spatially nonlocal and delayed model

As is well-known,the classical Laplace diffusion(local diffusion)only reflects the flow of individuals at the present position and the moment,whereas the movement of individuals is not only dependent on the same one point but subjects to other points around it because of their random movement in the space.Hence,the spatially long dispersal of individuals is modeled naturally by the nonlocal diffusion.Thereby,the nonlocal diffusion model with delay corresponding to(1.2)is

Generally,there are extensive application background in ecology,epidemiology and many fields[12,22,23]for traveling waves which is a kind of steady-state solution to reactiondiffusion equations.With the deep research on the existence,uniqueness,stability and other qualitative properties of traveling waves,the related theory has been gradually developed and matured[5,6,11,19,20].However,a difficult topic to investigate in traveling waves theory is the stability of traveling waves,especially for the monostable waves due to the fact that there is an unstable equilibrium in the two equilibria.The frequently used methods for the stability of traveling waves are the spectral analysis method and semigroup theory[11,19],the squeezing technique[6],weighted energy method[13-17]and the method of monotone semiflows[21],and so on.Here,the developed weighted energy method by Mei[13-17]is a very efficient tool to deal with the stability of monostable waves and there are many meaningful outcomes[12,22-24].For example,motivated by the work of Mei[13-17],Yang[22]extended the globally exponential stability of(non)monotone traveling waves for scalar equations to systems.Hereafter,Zhang[24]also generalized the results for Laplace diffusion equations to the nonlocal dispersal counterpart of the epidemic model with delay in[22].

As result,this paper intends to discuss the following spatially nonlocal population model

andGis an nonnegative and unit kernel function.By the help of the weighted energy method combining with comparison principle,we put our concentration on establishing the globally exponential stability of traveling wavefronts for(1.5)under the quasi-monotone condition.Note that(1.5)includes the Nicholson’s blowflies equations with nonlocal diffusion and quiescent stage ifF(u,v)=−D(u)+B(v)e−µ0τ,whereB(·)andD(·)are respectively the birth and death function,µ0is the mortality of juvenile.In addition,(1.5)can be reduced to(1.4)whenG(·)is the Dirac function andS(u)=u.Therefore,we complement the stability of traveling waves,which perfects Zhou’s work[25]for the existence of traveling waves for(1.5).Furthermore,compared with Zhang’s work[24],the piecewise weighted function is not suitable for establishing the stability of traveling waves for(1.5)in this article since the reaction term contains an integral.We choose an non-piecewise weighted function derived from the idea of Mei in[16]to overcome the difficulty caused by the nonlocal nonlinearity for establishing the energy estimates of solutions.Finally,the comparison principle is no longer holding if the quasi-monotonicity of the system(1.5)is scarce,and the effective anti-weighted energy method combining Fourier transform[17]will be applied to investigate the stability of monstable waves for the non-quasi-monotone case of(1.5),which is our continuous work.

The remainder of this article is divided into two sections.In Sect.2,some preliminaries and the main result in this paper are stated.In Sect.3,the result of exponential stability for traveling wavefronts is proved.

§2.Preliminaries and main result

LetC([0,T0];X)andL2([0,T0];X)be the spaces ofX-valued continuous functions andL2-functions on[0,T0],respectively,whereT0>0 is a constant andXis a Banach space.The corresponding spaces ofX-valued functions on[0,+∞)are defined similarly.

whereξ=x+ct.

Well, he used to live in this peach seed, but now that the peach has been harvested and sold, and I have eaten half of it, it looks as if he s out of house and home.

To investigate the stability of traveling wavefronts for(1.5),it is necessary to establish the existence of traveling waves of(1.5).Similar to Zhou’s work[25],it is trivial to establish the following two results:

Lemma 2.1.If(A1)−(A4)hold,then there are two positive numberλ∗and c∗such that

Moreover,

Lemma 2.2.(Existence of traveling wavefronts)Suppose that(A1)−(A4)hold.Then thesystem(1.5)has a non-decreasing traveling wavefront for each c≥c∗,and has no traveling wavefront for0

Define a weight function as

whereλ∗is defined in Lemma 2.1.

Now,the main result in this article is stated as followings.

Theorem 2.1.(Exponential stability of traveling wavefronts)Let(A1)−(A4)hold andτ<τ0withτ0>0is a real number.Provided that the traveling wavefront(ψ1(x+ct),ψ2(x+ct))for(1.5)with the speed

then the solution(u1(t,x),u2(t,x))to the initial-value problem(1.5)and(2.5)exists global ly,unique in time,and satisfies

In addition,the solution(u1(t,x),u2(t,x))converges to the traveling wavefront(ψ1(x+ct),ψ2(x+ct))exponential ly in time,namely,

for t>0,where C andγare some positive constants.

§3.Proof of main result

We first consider the initial-value problem(1.5)and(2.5),and the following relevant result is trivial as shown in[15,16,22].

In the followings,we prove our main result Theorem 2.1 by using weighted-energy method combining with the comparison principle.

Denote