In this paper, semi-tensor product of real matrices is extended to reduced biquaternion matrices, and then some new conclusions of the reduced biquaternion matrices under the vector operator are proposed using semi-tensor product of reduced biquaternion matrices, so that the reduced biquaternion matrix equation l∑p=1ApXBp=C can be transformed into a reduced biquaternion linear equations, then the expression of the least squares solution of the equation is obtained using the LC-representation and Moore-Penrose inverse. The necessary and sufficient conditions for the compatibility and the expression of general solutions of the equation are obtained, and the minimal norm solutions are also given. Finally, our proposed method of solving the reduced biquaternion matrix equation is applied to color image restoration.
Citation: Jianhua Sun, Ying Li, Mingcui Zhang, Zhihong Liu, Anli Wei. A new method based on semi-tensor product of matrices for solving reduced biquaternion matrix equation l∑p=1ApXBp=C and its application in color image restoration[J]. Mathematical Modelling and Control, 2023, 3(3): 218-232. doi: 10.3934/mmc.2023019
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In this paper, semi-tensor product of real matrices is extended to reduced biquaternion matrices, and then some new conclusions of the reduced biquaternion matrices under the vector operator are proposed using semi-tensor product of reduced biquaternion matrices, so that the reduced biquaternion matrix equation l∑p=1ApXBp=C can be transformed into a reduced biquaternion linear equations, then the expression of the least squares solution of the equation is obtained using the LC-representation and Moore-Penrose inverse. The necessary and sufficient conditions for the compatibility and the expression of general solutions of the equation are obtained, and the minimal norm solutions are also given. Finally, our proposed method of solving the reduced biquaternion matrix equation is applied to color image restoration.
Nowadays predator-prey models have been widely applied in biological and ecological phenomena. The most general prey-predator population model is represented by
{˙x(t)=xG(x)−yP(x,y),˙y(t)=yH(x,y), |
where x(t) and y(t) denote the density of the prey and predator at time t, respectively. G(x) is the per capita growth rate of the prey in the absence of predator, P(x,y) represents the functional response of predators and H(x,y) measures the growth rate of predators.
A prototype of G(x) is the logistic growth pattern of G(x)=r(1−xN), where r>0 denotes the prey intrinsic growth rate and N means the carrying capacity in the absence of predator [1]. One of known growth rate of predators is the Leslie-Gower type: H(x,y)=α(1−kyx) [2,3], where α is the intrinsic growth rates of predator and k is the conversion factor of prey into predators.
Lotka-Volterra response was used by Lotka [4] in studying a hypothetical chemical reaction and by Volterra [5] in modeling a predator-prey interaction. Lotka-Volterra response function is a straight line through the origin and is unbounded. The solutions of Lotka-Volterra model are not structural stable, thus a small perturbation can have a very marked effect [6]. The Holling-type Ⅱ functional responses function is P(x,y)=cxa+bx, where c is the maximum number of prey consumed per predator per unit time [7,8]. When a=1 and b=0, the functional response is of Lotka-Volterra type. In 1975, Beddington [9] and DeAngelis et al. [10] developed a predator-prey model of the mutual interference effects, in which the relationship between predators' searching efficiency and both prey and predator is presented. The Beddington-DeAngelis (B-D) functional response is defined by
P(x,y)=sx1+ax+by, |
where s,a,b>0, s is the consumption rate, a means the saturation constant for an alternative prey and b stands for the predator interference. The predator-prey models with the B-D functional response have been well-studied in the literature, for example, see [11,12,13,14] and references therein.
From the view of human needs, the exploitation of biological resources and harvest of population are commonly practised in the fields of fishery, wildlife, and forestry management. Many mathematical models have been proposed and developed to better describe the relationship between predator and prey populations by taking into account the harvesting, for instance, see [14,15,16,17,18]. In a very general way, harvesting for predator-prey models can be divided into three types. If the harvesting function h(t) is a constant, it is called constant-rate or constant yield harvesting. It arises when a quota is specified (for example, through permits, as in deer hunting seasons in many areas, or by agreement as sometimes occurring in whaling) [19,20]. If the function h(t) is a linear function of population size, it is called proportional or constant-effort harvesting [16,17,18]. The harvesting function h(t) can be of nonlinear form, for example, one of which is the so-called Michaelis-Menten type harvesting used in ecology and economics [21,22].
Movements of some individuals usually cannot be restricted to a small area, and they are often free, so integral operators have been widely applied to model the long-distance dispersal problem [23]. That is, the diffusion process depends on the distance between two niches of population, such as the model:
∂u∂t(x,t)=∫RJ(x−y)(u(y,t)−u(x,t))dy+f(u), |
where ∫RJ(x−y)(u(y,t)−u(x,t))dy represents the nonlocal dispersal process [24,25]. Such model arises not only in biological phenomena, but also in many other fields, such as phase transition modelling [25,26,27,28].
There is, however, considerable evidence that time delay should not be neglected in biological and ecological phenomena. The growth rate of population of species and the response of one species to the interactions with other species are mediated by some time delay. Other causes of response delays include differences in resource consumption with respect to age structure, migration and diffusion of populations, gestation and maturation periods, delays in behavioral response to environmental changes, and dependence of a population on a food supply that requires time to recover from grazing [15,25]. Hence, in order to make the modeling of interactions between predator and prey more realistic, time delay is often necessarily incorporated into predator-prey models [22,28,29,30,31].
The purpose of this paper is to study the existence and nonexistence of traveling wave solution of a nonlocal dispersal delayed predator-prey model with the B-D functional response and harvesting:
{∂u∂t=d1((J∗u)(x,t)−u(x,t))+ru(x,t)(1−u(x,t)K)−su(x,t)v(x,t−τ)1+au(x,t)+bv(x,t−τ)−qu(x,t),∂v∂t=d2((J∗v)(x,t)−v(x,t))+v(x,t)(α−βv(x,t)u(x,t−τ)), | (1.1) |
where
(J∗w)(x,t)=∫RJ(y)w(x−y,t)dy, |
q represents the prey harvesting, τ denotes the time delay, and a,b,r,d1,d2,s,K,α and β are positive real constants. To reduce the number of parameters in system (1.1), we make the following transformations:
ˉt=rt, ˉτ=rτ, ˉu=uK, ˉv=svr, ¯d1=d1r, ¯d2=d2r, |
ˉa=aK, ˉb=rbs, ˉα=αr, ˉβ=βsK, ˉq=qr. |
For the sake of convenience, we ignore the bars on u,v and other parameters, then system (1.1) can be re-expressed as
{ut=d1(J∗u−u)+u(1−u)−uv(x,t−τ)1+au+bv(x,t−τ)−qu,vt=d2(J∗v−v)+v(α−βvu(x,t−τ)). | (1.2) |
Biologically, we require 0<q<1. It is easy to see that system (1.2) has two spatially constant equilibria (1−q,0) and (u∗,v∗), where u∗=(1−q)κ−β−α+√(κ−β−α)2+4βκ2κ, v∗=αu∗β and κ=(aβ+bα)(1−q).
In biology and ecology, traveling wave solutions are often used to describe the spatial-temporal process where the predator invades the territory of prey and they eventually coexist [25]. A solution of system (1.2) is called a traveling wave with the speed c>0 if there exist positive function ϕ1 and ϕ2 defined on R such that
u(x,t)=ϕ1(z),v(x,t)=ϕ2(z),z=x+ct. |
Here ϕ1 and ϕ2 represent the wave profiles and (ϕ1, ϕ2) satisfies the resultant system:
{cϕ′1(z)=d1(J∗ϕ1(z)−ϕ1(z))+ϕ1(z)(1−ϕ1(z))−ϕ1(z)ϕ2(z−cτ)1+aϕ1(z)+bϕ2(z−cτ)−qϕ1(z),cϕ′2(z)=d2(J∗ϕ2(z)−ϕ2(z))+ϕ2(z)(α−βϕ2(z)ϕ1(z−cτ)), | (1.3) |
and
J∗ϕ(z)=∫RJ(y)ϕ(z−y)dy. |
Our primary interest lies in the traveling wave solution of system (1.3) connecting (1−q, 0) and (u∗, v∗) with the asymptotic behavior:
limz→−∞(ϕ1(z),ϕ2(z))=(1−q,0), limz→+∞(ϕ1(z),ϕ2(z))=(u∗,v∗). | (1.4) |
The asymptotic behavior of traveling wave solution plays an important role in dispersion models of biological populations, because it describes the propagation processes of different species and enables us to understand how some species migrate from one area into another area until the density attains a certain value.
Recently, the existence of traveling wave solution for the nonlocal dispersal systems with the time delay has been extensively studied [28,29,30,31,32,33]. We can see that system (1.3) is non-monotone system and Schauder's fixed point theorem is a quiet powerful technique for constructing a suitable invariant set (see, for example [31,33,34,35,36]). To explore the existence of traveling wave solution of nonlocal dispersal systems with c>c∗, we need to construct an invariant cone in a large bounded domain with the initial functions [33,34,35], where the nonlocal dispersal kernel function J is assumed to be compactly supported. For the existence of traveling wave solution at the critical point c=c∗, Corduneanu's theorem and the limiting method are useful techniques [33,36].
Throughout this paper, for the nonlocal dispersal kernel function J of system (1.3), we make the following assumptions:
(G1) J is a smooth function in R, Lebesgue measurable with J∈C1(R) and
J(x)=J(−x)≥0, ∫RJ(x)dx=1. |
(G2) ∫RJ(x)eλxdx<+∞, λ∈R.
For convenience, we assume the parameters of system (1.3) satisfying
0<d1≤d2,0<q<1,b>1,a>1q,0<bα≤β. |
The rest of this paper is structured as follows. We construct an appropriate pair of upper-lower solutions of system (1.3) for c>c∗ in Section 2. We apply Schauder's fixed point theorem to investigate the existence of traveling wave solution for c>c∗ and develop the contracting rectangles method to study the asymptotic behavior of system (1.3) in Section 3. The existence of traveling wave solution for c=c∗ is discussed by means of Corduneanu's theorem and Lebesgue's dominated convergence theorem in Section 4. Section 5 is dedicated to the nonexistence of traveling wave for 0<c<c∗. A brief conclusion is given in Section 6.
Definition 2.1. Assume that Z:={z1,z2,⋯,zm}∈R contains finite points of R. We say that the functions (¯ϕ1, ¯ϕ2) and (ϕ_1, ϕ_2) are a pair of upper-lower solutions of system (1.3), if for any z∈R∖Z, ¯ϕ′i(z) and ϕ_′i(z) (i=1,2) are bounded and continuous such that
{F(¯ϕ1,ϕ_2)(z)=d1(J∗¯ϕ1(z)−¯ϕ1(z))−c¯ϕ′1(z)+¯ϕ1(z)(1−¯ϕ1(z)) −¯ϕ1(z)ϕ_2(z−cτ)1+a¯ϕ1(z)+bϕ_2(z−cτ)−q¯ϕ1(z)≤0,F(ϕ_1,¯ϕ2)(z)=d1(J∗ϕ_1(z)−ϕ_1(z))−cϕ_′1(z)+ϕ_1(z)(1−ϕ_1(z)) −ϕ_1(z)¯ϕ2(z−cτ)1+aϕ_1(z)+b¯ϕ2(z−cτ)−qϕ_1(z)≥0,F(¯ϕ1,¯ϕ2)(z)=d2(J∗¯ϕ2(z)−¯ϕ2(z))−c¯ϕ′2(z)+¯ϕ2(z)(α−β¯ϕ2(z)¯ϕ1(z−cτ))≤0,F(ϕ_1,ϕ_2)(z)=d2(J∗ϕ_2(z)−ϕ_2(z))−cϕ_′2(z)+ϕ_2(z)(α−βϕ_2(z)ϕ_1(z−cτ))≥0. | (2.1) |
Define
fσ(d,c,λ)=d(∫RJ(y)e−λydy−1)−cλ+σ, |
where σ≥0. By a direct calculation, for c>0 and λ>0 we have
(F1) f0(d1,c,0)=0 and fα(d2,c,0)>0;
(F2) ∂fσ∂c=−λ<0, ∂fσ∂λ|λ=0=−c<0 and ∂fσ∂d=∫RJ(y)e−λydy−1>0;
(F3) ∂2fσ∂λ2>0.
From (F1)–(F3), it follows that there exist c∗>0 and λ∗>0 such that [35]
fα(d2,c∗,λ∗)=0and∂fα(d2,c,λ)∂λ|(c∗,λ∗)=0. |
Lemma 2.1. There exist c>c∗ and positive constants 0<λ2<λ∗<λ3<λ1 such that
f0(d1,c,λ){=0λ=0, λ=λ1>0λ∈(λ1,+∞)<0λ∈(0,λ1),fα(d2,c,λ){=0λ=λ2, λ=λ3>0λ∈(0,λ2)∪(λ3,+∞)<0λ∈(λ2,λ3). |
Proof. We only need to show λ1>λ3. It is easy to see fα(d2,c,λ3)=f0(d1,c,λ1)=0 and f0(d1,c,λ1)<fα(d1,c,λ1). Due to d2≥d1 and (F2), we have fα(d1,c,λ1)≤fα(d2,c,λ1). It indicates fα(d2,c,λ3)<fα(d2,c,λ1), i.e., λ1>λ3.
Now, we will construct an appropriate pair of upper-lower solutions for system (1.3). We fix c>c∗. For any given constant m>1, it is easy to check that the function
g(z)=eλ2z−meθz |
has a unique zero point at z0=−lnmθ−λ2 where θ∈(λ2,min{2λ2,λ3}), and a unique maximum point at zM=−lnmλ2(θ−λ2)<z0. Clearly, g is continuous on R and positive on (−∞,z0). For any given y∈R we let
Θ(z)=∫z−∞J(y−x)g(x)dx−g(z)2 |
with z∈[zM,z0]. Since Θ(z) is nondecreasing for z∈[zM,z0] and Θ(z0)>0, we can find a sufficiently small δ∈(0,α(b−1)β) and z2∈(zM,z0) such that
δ=g(z2),Θ(z2)>0. |
Let p and m satisfy the following conditions:
(A1) p>−1f0(d1,c,λ2).
(A2) m>−β(b−1)fα(d2,c,θ).
Then, we introduce ¯ϕ1(z),¯ϕ2(z),ϕ_1(z),ϕ_2(z) as follows:
¯ϕ1(z)=1−q z∈R, | (2.2) |
ϕ_1(z)={(1−q)(1−1b)z≥z1,(1−q)(1−1b(eλ1z+peλ2z))z≤z1, | (2.3) |
¯ϕ2(z)={1−qb z≥0,1−qbeλ2z z≤0, | (2.4) |
ϕ_2(z)={1−qbδz≥z2,1−qb(eλ2z−meθz)z≤z2, | (2.5) |
where z1<0 is defined by eλ1z1+peλ2z1=1.
Lemma 2.2. Assume c>c∗. Then (¯ϕ1, ¯ϕ2) and (ϕ_1, ϕ_2) defined by (2.2)–(2.5) are a pair of upper-lower solutions of system (1.3).
Proof. Firstly, we show that
F(¯ϕ1,ϕ_2)(z)≤0 |
holds for z∈R. For any z∈R, we have ¯ϕ1(z)=1−qE and
F(¯ϕ1,ϕ_2)(z)=(1−q)q−(1−q)ϕ_2(z−cτ)1+a(1−q)+bϕ_2(z−cτ)−q(1−q)=−(1−q)ϕ_2(z−cτ)1+a(1−q)+bϕ_2(z−cτ)≤0. |
For z≠z1, we would like to show that
F(ϕ_1,¯ϕ2)(z)≥0. |
When z>z1, we have ϕ_1(z)=(1−q)(1−1b),¯ϕ2(z)≤1−qb and
F(ϕ_1,¯ϕ2)(z)≥(1−q)b−1b[1−(1−q)b−1b−1−qb+a(1−q)(b−1)+b(1−q)−q]≥0. |
In view of f0(d1,c,λ2)<f0(d2,c,λ2)<fα(d2,c,λ2)=0 and (A1), for z<z1<0 we have ϕ_1=(1−q)[1−1b(eλ1z+peλ2z)], ¯ϕ2=1−qbeλ2z and
F(ϕ_1,¯ϕ2)(z)≥d1(∫RJ(y)(1−q)[1−1b(eλ1(z−y)+peλ2(z−y))]dy−(1−q)[1−1b(eλ1z+peλ2z)])+(1−q)cb(λ1eλ1z+pλ2eλ2z)+(1−q)2[1−1b(eλ1z+peλ2z)]−(1−q)2[1−1b(eλ1z+peλ2z)]2−ϕ_1¯ϕ2(z−cτ)1+aϕ_1+b¯ϕ2=−1−qbeλ1z[d1(∫RJ(y)e−λ1ydy−1)−cλ1]−1−qbpeλ2z[d1(∫RJ(y)e−λ2ydy−1)−cλ2]+(1−q)2b(eλ1z+peλ2z)[1−1b(eλ1z+peλ2z)]−ϕ_1¯ϕ2(z−cτ)1+aϕ_1+b¯ϕ2(z−cτ)>−1−qbpeλ2z[d1(∫RJ(y)e−λ2ydy−1)−cλ2]−¯ϕ2(z−cτ)=−1−qbpeλ2z[d1(∫RJ(y)e−λ2ydy−1)−cλ2]−1−qbeλ2(z−cτ)=1−qbeλ2z[(−p)(d1(∫RJ(y)e−λ2ydy−1)−cλ2)−e−λ2cτ]>1−qbeλ2z[(−p)f0(d1,c,λ2)−1]>0. | (2.6) |
Now, we show
F(¯ϕ1,¯ϕ2)(z)≤0 |
for z≠0. In the case of z>0, we have ¯ϕ1=1−q and ¯ϕ2=1−qb. Then
F(¯ϕ1,¯ϕ2)(z)≤1−qb[α−β1−qb1−q]=1−qb[α−βb]≤0. |
For z<0, we obtain ¯ϕ2=1−qbeλ2z and
F(¯ϕ1,¯ϕ2)(z)≤d2(∫RJ(y)1−qbeλ2(z−y)dy−1−qbeλ2z)−1−qbcλ2eλ2z+1−qbeλ2z[α−βbeλ2z]=1−qbeλ2z[d2(∫RJ(y)e−λ2ydy−1)−cλ2+α]−β(1−q)b2e2λ2z=−β(1−q)b2e2λ2z≤0. |
Finally, to show
F(ϕ_1,ϕ_2)(z)≥0 |
for z≠z2, we use the inequality ϕ_1≥(1−q)(1−1b) and ϕ_2=1−qbδ if z>z2. Then
F(ϕ_1,ϕ_2)(z)≥d2(1−q)b[∫+∞z−z2J(y)(eλ2(z−y)−meθ(z−y))dy+∫z−z2−∞J(y)δdy−δ]+1−qbδ[α−β(1−q)(1−1b)⋅1−qbδ]≥d2(1−q)b(∫z2−∞J(z−y)(eλ2z−meθz)dy−δ2)+1−qbδ[α−βδb−1]=d2(1−q)bΘ(z2)+1−qbδ[α−βδb−1]≥1−qbδ[α−βδb−1]≥0, |
due to 0<δ<α(b−1)β.
On the other hand, if z<z2, we have ϕ_2=1−qb(eλ2z−meθz) and thus
F(ϕ_1,ϕ_2)(z)≥d2(1−q)b[∫RJ(y)(eλ2(z−y)−meθ(z−y))dy−(eλ2z−meθz)]−c(1−q)b(λ2eλ2z−mθeθz)+1−qb(eλ2z−meθz)[α−β(1−q)(1−1b)1−qb(eλ2z−meθz)]=1−qbeλ2z[d2(∫RJ(y)e−λ2ydy−1)−cλ2+α]−m1−qbeθz[d2(∫RJ(y)e−θzdy−1)−cθ+α]−β(1−q)b(b−1)(eλ2z−meθz)2>−m1−qbeθz[d2(∫RJ(y)e−θzdy−1)−cθ+α]−β(1−q)b(b−1)e2λ2z=1−qbeθz[(−m)(d2(∫RJ(y)e−θzdy−1)−cθ+α)−βb−1e(2λ2−θ)z]>1−qbeθz[(−m)(d2(∫RJ(y)e−θzdy−1)−cθ+α)−βb−1]>1−qbeθz[(−m)fα(d2,c,θ)−βb−1]>0. |
The last inequality holds due to θ∈(λ2,min{2λ2,λ3}) and condition (A2).
In this section, we start with discussing the existence of traveling wave solution for system (1.3) with condition (1.4) by using the upper-lower solutions of system (1.3), which is defined in the preceding section, to construct an invariant set.
Let C be a set of bounded and uniformly continuous functions from R to R2 and
Γ={(ϕ1,ϕ2)∈C:ϕ_i(z)≤ϕi≤¯ϕi(z), z∈R, i=1,2}, |
where ¯ϕi(z) and ϕ_i(z) (i = 1, 2) are defined by (2.2)–(2.5). Thus for any (ϕ1,ϕ2)∈Γ, we have (1−q)b−1b≤ϕ1(z)≤1−q and 0≤ϕ2(z)≤1−qb.
For Φ=(ϕ1,ϕ2)∈Γ, we define
{H1(ϕ1,ϕ2)(z):=d1J∗ϕ1(z)+F1(ϕ1(z),ϕ2(z−cτ)),H2(ϕ1,ϕ2)(z):=d2J∗ϕ2(z)+F2(ϕ1(z−cτ),ϕ2(z)), |
where
{F1(y1,y2)=(γ−d1)y1+y1(1−y1−y21+ay1+by2−q),F2(y1,y2)=(γ−d2)y2+y2(α−βy2y1), |
for some constant γ. For any fixed γ>max{d1+(1−q)(1+1b),d2+2βb−1−α}, it follows that F1 is nondecreasing in y1 and is decreasing in y2 for y1∈[(1−q)b−1b, 1−q] and y2∈[0, 1−qb]. Also, F2 is nondecreasing with respect to y1 and y2 for y1∈[(1−q)b−1b, 1−q] and y2∈[0, 1−qb].
Define an operator P=(P1,P2):Γ→C by
{P1(ϕ1,ϕ2)(z)=1c∫z−∞e−γ(z−y)cH1(ϕ1,ϕ2)(y)dy,P2(ϕ1,ϕ2)(z)=1c∫z−∞e−γ(z−y)cH2(ϕ1,ϕ2)(y)dy. |
Apparently, a fixed point of P is a solution of system (1.3). Let ρ∈(0,γc) and ||⋅|| denote the Euclidean norm in R2. We define
Bρ(R,R2)={Φ∈C:supz∈R||Φ(z)||e−ρ|z|<∞} |
and
|Φ|ρ:=supz∈R||Φ(z)||e−ρ|z|. |
It is easy to see that (Bρ(R,R2)),|⋅|ρ) is a Banach space. Clearly, Γ is nonempty, bounded, convex and closed in Bρ(R,R2).
Lemma 3.1. P:Γ→Γ.
Proof. For any Φ(z)=(ϕ1,ϕ2)(z)∈Γ, owing to the monotonicity of F1 and F2 we have
{H1(ϕ1,ϕ2)(z)≥d1J∗ϕ_1(z)+F_1(z)=:H_1(z), z∈R,H1(ϕ1,ϕ2)(z)≤d1J∗¯ϕ1(z)+¯F1(z)=:¯H1(z), z∈R, |
and
{H2(ϕ1,ϕ2)(z)≥d2J∗ϕ_2(z)+F_2(z)=:H_2(z), z∈R,H2(ϕ1,ϕ2)(z)≤d2J∗¯ϕ2(z)+¯F2(z)=:¯H2(z), z∈R, |
in which ¯F1, F_1, ¯F2 and F_2 are defined by
{¯F1(z)=(γ−d1)¯ϕ1(z)+¯ϕ1(z)(1−q−¯ϕ1(z)−ϕ_2(z−cτ)1+a¯ϕ1(z)+bϕ_2(z−cτ)),F_1(z)=(γ−d1)ϕ_1(z)+ϕ_1(z)(1−q−ϕ_1(z)−¯ϕ2(z−cτ)1+aϕ_1(z)+b¯ϕ2(z−cτ)), |
and
{¯F2(z)=(γ−d2)¯ϕ2(z)+¯ϕ2(z)(α−β¯ϕ2(z)¯ϕ1(z−cτ)),F_2(z)=(γ−d2)ϕ_2(z)+ϕ_2(z)(α−βϕ_2(z)ϕ_1(z−cτ)). |
Let
P_1(z)=1c∫z−∞e−γ(z−y)cH_1(y)dy, ¯P1(z)=1c∫z−∞e−γ(z−y)c¯H1(y)dy, z∈R, |
P_2(z)=1c∫z−∞e−γ(z−y)cH_2(y)dy, ¯P2(z)=1c∫z−∞e−γ(z−y)c¯H2(y)dy, z∈R. |
Obviously, P_i(z)≤Pi(z)≤¯Pi(z) (i=1,2). It suffices to prove that
ϕ_i(z)≤P_i(z),¯Pi(z)≤¯ϕi(z), z∈R, i=1,2. |
We denote z0=−∞ and zm+1=∞. For any z∈R∖Z, there exists a k∈{0,1,2,...,m} such that z∈(zk,zk+1), and
¯P1(z)=1c∫z−∞e−γ(z−y)c¯H1(y)dy=(k∑i=11c∫zizi−1+1c∫zzk)e−γ(z−y)c¯H1(y)dy≤(k∑i=11c∫zizi−1+1c∫zzk)e−γ(z−y)c[c¯ϕ′1(y)dy+γ¯ϕ1(y)]=¯ϕ1(z). |
Due to the continuity of both ¯P1(z) and ¯ϕ1(z), we get
¯P1(z)≤¯ϕ1(z), z∈R. |
Similarly, we have
ϕ_1(z)≤P_1(z), z∈R, |
and
ϕ_2(z)≤P2(ϕ)(z)≤¯ϕ2(z), z∈R. |
Consequently, we obtain P(Γ)⊂Γ.
Lemma 3.2. P: Γ→Γ is continuous with respect to |⋅|ρ.
Proof. For any Φ=(ϕ1,ϕ2) and Ψ=(ψ1,ψ2)∈Γ, we have
|H1(ϕ1,ϕ2)(z)−H1(ψ1,ψ2)(z)|≤d1∫RJ(z−y)|ϕ1(y)−ψ1(y)|dy+(γ−d1+1−q)|ϕ1(z)−ψ1(z)|+|ϕ1(z)+ψ1(z)||ϕ1(z)−ψ1(z)|+|ϕ1(z)ϕ2(z−cτ)1+aϕ1(z)+bϕ2(z−cτ)−ψ1(z)ψ2(z−cτ)1+aψ1(z)+bψ2(z−cτ)| |
and
|ϕ1(z)ϕ2(z−cτ)1+aϕ1(z)+bϕ2(z−cτ)−ψ1(z)ψ2(z−cτ)1+aψ1(z)+bψ2(z−cτ)|<1−qb(2−q)(1+a(1−q)b−1b)2|ϕ1(z)−ψ1(z)|+(1−q)(1+a(1−q))(1+a(1−q)b−1b)2|ϕ2(z−cτ)−ψ2(z−cτ)|<1−qb(2−q)|ϕ1(z)−ψ1(z)|+(1−q)(1+a(1−q))|ϕ2(z−cτ)−ψ2(z−cτ)|<1b(1−q)(2−q)|ϕ1(z)−ψ1(z)|+a(1−q)(2−q)|ϕ2(z−cτ)−ψ2(z−cτ)|. |
A straightforward calculation yields
|P1(ϕ1,ϕ2)(z)−P1(ψ1,ψ2)(z)|e−ρ|z|≤d1e−ρ|z|c∫z−∞e−γ(z−s)c(∫RJ(s−y)|ϕ1(y)−ψ1(y)|dy)ds+(γ−d1+1−q)e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ1(y)−ψ1(y)|dy+2(1−q)e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ1(y)−ψ1(y)|dy+(1−q)(2−q)e−ρ|z|cb∫z−∞e−γ(z−y)c|ϕ1(y)−ψ1(y)|dy+a(1−q)(2−q)e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ2(y−cτ)−ψ2(y−cτ)|dy=d1e−ρ|z|c∫z−∞e−γ(z−s)c(∫RJ(s−y)|ϕ1(y)−ψ1(y)|dy)ds+[γ−d1+(1−q)(3+2−qb)]e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ1(y)−ψ1(y)|dy+a(1−q)(2−q)e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ2(y−cτ)−ψ2(y−cτ)|dy. |
We further have
e−ρ|z|c∫z−∞e−γ(z−s)c(∫RJ(s−y)|ϕ1(y)−ψ1(y)|dy)ds=e−ρ|z|c∫z−∞e−γ(z−s)c(∫RJ(s−y)eρ|y||ϕ1(y)−ψ1(y)|e−ρ|y|dy)ds≤|Φ−Ψ|ρc∫z−∞e−(γc−ρ)(z−s)(∫RJ(y)eρ|y|dy)ds≤2∫RJ(y)eρydyγ−cρ|Φ−Ψ|ρ |
and
e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ1(y)−ψ1(y)|dy=e−ρ|z|c∫z−∞e−γ(z−y)ceρ|y||ϕ1(y)−ψ1(y)|e−ρ|y|dy≤|Φ−Ψ|ρc∫z−∞e−(γc−ρ)(z−y)dy≤1γ−cρ|Φ−Ψ|ρ. |
Processing in an analogous manner, we can derive
e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ2(y−cτ)−ψ2(y−cτ)|dy≤eρcτγ−cρ|Φ−Ψ|α. |
We now choose
L1=2d1∫RJ(y)eρydy+γ−d1+(1−q)[3+(1b+aeρcτ)(2−q)]γ−cρ |
such that
|P1(ϕ1,ϕ2)(z)−P1(ψ1,ψ2)(z)|e−ρ|z|≤L1|Φ−Ψ|α. | (3.1) |
On the other hand, we have
|P2(ϕ1,ϕ2)(z)−P2(ψ1,ψ2)(z)|e−ρ|z|≤d2e−ρ|z|c∫z−∞e−γ(z−s)c(∫RJ(y−s)|ϕ2(y)−ψ2(y)|dy)ds+(γ−d2+α+2bβ(b−1)2)e−ρ|z|c∫z−∞e−γ(z−y)c|ϕ2(y)−ψ2(y)|dy+βe−α|z|(b−1)2c∫z−∞e−γ(z−y)c|ϕ1(y−cτ)−ψ1(y−cτ)|dy≤L2|Φ−Ψ|ρ, | (3.2) |
where
L2=2d2∫RJ(y)eρydy+γ−d2+α+β(2b+eρcτ)(b−1)2γ−cρ. |
In view of (3.1)–(3.2), there exists some constant L∗>0 such that
|P(ϕ)−P(Ψ)|ρ≤L∗|Φ−Ψ|ρ. |
Hence, P is a continuous operator from Γ to Γ.
For any given N∈R, let R−N:=(−∞,N] and consider the domain of the functions of the space Bρ on R−N:
Bρ(R−N,R2)={Φ∈C|R−N: supz∈R−N||Φ(z)||e−|ρ|z<∞}. |
Then (Bρ(R−N,R2),|⋅|Nρ) is a Banach space equipped with the norm |⋅|Nρ defined by
|Φ|Nα:=supz∈R−N||Φ(z)||e−|ρ|z. |
Let us recall Corduneanu's Theorem [37,§2.12].
Lemma 3.3. Let F⊂Bα(R−N,R2) be a set satisfying the following conditions:
(1) F is bounded in Bα(R−N,R2));
(2) the functions belonging to F are equicontinuous on any compact interval of R−N;
(3) the functions in F are equiconvergent, i.e., for any given ε>0, there is a corresponding Z(ε)<0 such that ‖ for z\leq Z(\varepsilon) and \Phi\in F .
Then F is compact in B_{\alpha}\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) .
Lemma 3.4. P(\Gamma) is compact in B_{\rho}.
Proof. For any \Phi = (\phi_{1}, \phi_{2})\in \Gamma and n\in \mathbb{N}, we define
\begin{equation} P^{n}(\Phi)(z) = \begin{cases} P(\Phi)(n) &z \gt n, \\ P(\Phi)(z) &z\in(-\infty, n]. \end{cases} \end{equation} | (3.3) |
Clearly, P^{n}(\Gamma) is compact if P(\Gamma)(z)|_{\mathbb{R}^-_{n}} is compact. We will show that the functions belonging to P(\Gamma)(z)|_{\mathbb{R}^-_{n}} satisfy all three conditions (1)–(3) in Lemma 3.3. Since P(\Gamma)\subset\Gamma , it is easy to see that P(\Gamma)(z)|_{\mathbb{R}^-_{n}} is bounded. Indeed, for any z_1, z_2\in (-\infty, n] we deduce
\begin{align*} &\left|P_{1}\left(\phi_{1}, \phi_{2}\right)\left(z_1\right)e^{-\rho |z_1|}-P_{1}\left(\phi_{1}, \phi_{2}\right) \left(z_2\right)e^{-\rho |z_2|}\right|\\ &\quad = \frac{1}{c}\left|e^{-\rho |z_1|}\int_{-\infty}^{z_1}e^{-\frac{\gamma\left(z_1-y\right)}{c}}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y-e^{-\alpha |z_2|}\int_{-\infty}^{z_2}e^{-\frac{\gamma\left(z_2-y\right)}{c}}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad = \frac{1}{c}\left|e^{-\left(\rho|z_1|+\frac{\gamma}{c}z_1 \right)}\int_{-\infty}^{z_1}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y- e^{-\left(\rho|z_2|+\frac{\gamma}{c}z_2 \right)}\int_{-\infty}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad\leq \frac{1}{c}e^{-\frac{\gamma}{c} z_1} \left|\int_{z_1}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right) \left(y\right)\mathrm{d}y\right|\\ &\qquad +\frac{1}{c} \left(e^{-\rho |z_2|}\left|e^{-\frac{\gamma}{c}z_1}-e^{-\frac{\gamma}{c}z_2 }\right| +e^{-\frac{\gamma}{c} z_1}\left|e^{-\rho |z_1|}-e^{-\rho |z_2|}\right|\right) \cdot \left|\int_{-\infty}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad\leq (1-q)e^{\frac{\gamma}{c}|z_2-z_1|} \left[\left(1+\frac{\gamma}{c}\right)|z_2-z_1|+1\right]. \end{align*} |
Similarly, we have
\begin{align*} \left|P_{2}\left(\phi_{1}, \phi_{2}\right)\left(z_1\right)e^{-\rho |z_1|}-P_{2}\left(\phi_{1}, \phi_{2}\right) \left(z_2\right)e^{-\rho |z_2|}\right| \leq \frac{1-q}{b\gamma}\left(\gamma+\alpha -\frac{\beta}{b}\right)e^{\frac{\gamma}{c}|z_2-z_1|} \left[\left(1+\frac{\gamma}{c}\right)|z_2-z_1|+1\right]. \end{align*} |
This implies that \left.P(\Gamma)(z)\right|_{\mathbb{R}^-_n} is equicontinuous on any compact interval of \mathbb{R}^-_{n} .
For any \Phi(z) = (\phi_1(z), \phi_2(z))\in \Gamma , we find
\begin{align*} (1-q)\left(1-\frac{1}{b}\left(e^{\lambda_{1}z}+pe^{\lambda_{2}z}\right)\right)\leq \phi_1(z)\leq 1-q, \ \ \frac{1-q}{b}\left(e^{\lambda_{2}z}-me^{\theta z}\right)\leq \phi_1(z)\leq\frac{1-q}{b}e^{\lambda_{2}z} \end{align*} |
for z < \min\{z_1, \ z_2\} . That is,
\lim\limits_{z\rightarrow-\infty}\phi_1(z) = 1-q, \quad \lim\limits_{z\rightarrow-\infty}\phi_2(z) = 0. |
Then
|\phi_1(z)-(1-q)|e^{-\rho|z|} \lt \frac{1-q}{b}\left(e^{(\lambda_{1}+\rho) z}+pe^{(\lambda_{2}+\rho) z}\right), \ \ |\phi_2(z)-0|e^{-\rho|z|} \lt \frac{1-q}{b}e^{(\lambda_{2}+\rho) z} |
for z < \min\{z_1, \ z_2\} . That is, condition (3) is satisfied. According to Lemma 3.3, P^n(\Gamma)(z) is compact in the sense of the norm |\cdot|_{\rho} . Note that
|P^{n}(\Phi)(z)-P(\Phi)(z)|e^{-\rho |z|}\leq 2(1-q)\sqrt{1+\left(\frac{\gamma+\alpha -\frac{\beta}{b}}{b\gamma}\right)^2}e^{-\rho n}\rightarrow0, \ \, as \ n\rightarrow \infty. |
Hence, P^{n}(\Phi)(z) converge to P(\Phi)(z) with respect to the norm |\cdot|_{\rho} , and P(\Gamma) is compact.
From Lemmas 3.1–3.4 and Schauder's fixed point theorem, we can see that P has a fixed point \Phi\in\Gamma such that P(\Phi) = \Phi , which is a solution of system (1.3). Hence, we obtain the following theorem immediately.
Theorem 3.5. Assume that conditions (G1)–(G2) hold. Then for any fixed c > c^* , system (1.3) has a positive solution (\phi_1(z), \phi_2(z))\in \Gamma . That is, \underline{\phi}_{i}(z)\leq{\phi_{i}(z)}\leq\overline{\phi}_{i}(z) (i = 1, 2) , where \overline{\phi}_{i} and \underline{\phi}_{i} (i = 1, 2) are defined by (2.2)–(2.5).
We now discuss the asymptotic behavior of traveling wave solution described in Theorem 3.5. For z\rightarrow-\infty , it is easy to see that
\lim\limits_{z\rightarrow -\infty} \phi_1(z) = 1-q, \quad \lim\limits_{z\rightarrow -\infty} \phi_2(z) = 0. |
By applying the contracting rectangles method, we analyze the asymptotic behavior of traveling wave solution as z\rightarrow\infty . We define
\begin{eqnarray} \left\{ \begin{aligned} E_{1}(\xi, \eta)&: = \xi\left(1-q E-\xi-\frac{\eta}{1+a\xi+b\eta}\right), \\ E_{2}(\xi, \eta)&: = \eta(\alpha-\beta\frac{\eta}{\xi}), \end{aligned} \right. \end{eqnarray} | (3.4) |
and
\begin{equation} \begin{split} &u_{1}(\theta): = u^*\theta, \qquad\qquad\qquad\qquad\qquad u_{2}(\theta): = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^*, \\ &v_{1}(\theta): = \begin{cases} v^*\frac{\theta}{2(1-\epsilon)} &\theta \lt 2\epsilon\\ v^*\frac{\theta-\epsilon}{1-\epsilon} &\theta\geq2\epsilon \end{cases}, \qquad\quad\ v_{2}(\theta): = v^*+\frac{a}{b}u^*(1-\theta), \end{split} \end{equation} | (3.5) |
for \theta\in[0, 1] , where (u^*, v^*) is the equilibrium point of system (1.3) and 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} .
Theorem 3.6. The following three statements are true.
(C1) u_1(\theta) and v_1(\theta) are continuous and strictly increasing while u_{2}(\theta) and v_2(\theta) are continuous and strictly decreasing for \theta\in[0, 1] .
(C2) For \theta\in[0, 1] , we have
\begin{equation*} \left\{ \begin{aligned} u_{1}(0)\leq u_1(\theta)\leq u_{1}(1)& = u^{\ast} = u_{2}(1)\leq u_{2}(\theta)\leq u_{2}(0), \\ v_{1}(0)\; \leq v_{1}(\theta)\leq v_{1}(1)& = v^{\ast} = v_{2}(1)\leq v_{2}(\theta)\leq \; v_{2}(0). \end{aligned} \right. \end{equation*} |
(C3) If \xi_{1} = u_{1}(\theta_{0}) , \eta_{1} = v_{1}(\theta_{0}) for any \theta_{0}\in(0, 1) and
u_1(\theta_0))\leq \xi \leq u_{2}(\theta_{0}), \ \ v_1(\theta_0)\leq \eta \leq v_{2}(\theta_{0}), |
then E_{1}(\xi_{1}, \eta) > 0 and E_{2}(\xi, \eta_{1}) > 0 .
If \xi_{2} = u_2(\theta_0) , \eta_{2} = v_2(\theta_0) for any \theta_{0}\in(0, 1) and
u_1(\theta_0))\leq \xi \leq u_2(\theta_0), \ \ v_1(\theta_0)\leq \eta \leq v_2(\theta_0), |
then E_{1}(\xi_{2}, \eta) < 0 and E_{2}(\xi, \eta_{2}) < 0.
Proof. It is easy to see that (C1)–(C2) are true.
To prove (C3), we claim that for any \theta_0\in(0, 1) , there holds E_{1}(\xi_{1}, \eta) > 0 with \xi_{1} = u_1(\theta_0) = u^*\theta_0 and v_1(\theta_0)\leq \eta\leq v_2(\theta_0) . As E_{1}(\xi, \eta) is decreasing in \eta , we only need to show that E_{1}(u^*\theta_0, v_2(\theta_0)) > 0 .
Let
\tilde{v}(\theta) = -\frac{a}{b}u^*\theta+\frac{a-b}{b^{2}}+ \frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*\theta}. |
Then E_{1}\left(u^*\theta, \tilde{v}(\theta)\right)\equiv0 for \theta\in[0, 1] . In view of a > \frac{1}{q} , it follows that
\begin{equation*} \begin{aligned} v_2(\theta_0)& = v^*+\frac{a}{b}u^*(1-\theta_0)\\ & = -\frac{a}{b}u^*\theta_0+\frac{a-b}{b^{2}}+\frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*}\\ & \lt -\frac{a}{b}u^*\theta_0+\frac{a-b}{b^{2}}+\frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*\theta_0}\\ & = \tilde{v}(\theta_0). \end{aligned} \end{equation*} |
Thus, E_{1}(u^*\theta_0, v_2(\theta_0)) > E_{1}(u^*\theta_0, \tilde{v}(\theta_0)) = 0 .
To show E_{2}(\xi, \eta_{1}) > 0 for any \theta_0\in(0, 1) , \eta_{1} = v_1(\theta_0) and u_1(\theta_0))\leq \xi\leq u_2(\theta_0) , we know that E_2(\xi, \eta) is nondecreasing in \xi . So it is equivalent to prove E_2(u_1(\theta_0), v_1(\theta_0)) > 0 . When 2\epsilon\leq \theta_0 < 1 , in view of v_1(\theta_0) = v^*\frac{\theta_0-\epsilon}{1-\epsilon} we have
\begin{align*} E_{2}(u_1(\theta_0), v_1(\theta_0)) = v_1(\theta_0)\left[\alpha-\beta\frac{v^*\frac{\theta_0-\epsilon}{1-\epsilon}}{u^*\theta_0}\right] = \alpha v_1(\theta_0)\frac{\epsilon(1-\theta_0)}{\theta_0(1-\epsilon)} \gt 0. \end{align*} |
For 0 < \theta_0 < 2\epsilon , using v_1(\theta_0) = v^*\frac{\theta_0}{2(1-\epsilon)} we have
\begin{align*} E_{2}(u_1(\theta_0), v_1(\theta_0)) = v_1(\theta_0)\left[\alpha -\beta \frac{v^*\frac{\theta_0}{2(1-\epsilon)}}{u^*\theta_0}\right] = \alpha v_1(\theta_0)\frac{1-2\epsilon}{2(1-\epsilon)} \gt 0. \end{align*} |
For any \theta_0\in(0, 1) , to show E_{1}(\xi_{2}, \eta) < 0 , where \xi_{2} = u_2(\theta_0) and v_1(\theta_0)\leq \eta\leq v_2(\theta_0) , it suffices to prove that E_{1}(u_2(\theta_0), v_1(\theta_0)) < 0 . Let \varphi(\theta): = E_{1}(u_{2}(\theta), v_1(\theta))/u_{2}(\theta) . Then \varphi(1) = 0 . We proceed by considering two cases.
\underline{\mathrm{Case\ 1}} : for 2\epsilon\leq \theta < 1 , from (3.5) we have v_1(\theta) = v^*\frac{\theta-\epsilon}{1-\epsilon} , u_{2}(\theta) = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^* for \theta\in[2\epsilon, 1) , and then
\begin{align*} \frac{\mathrm{d}\varphi}{\mathrm{d}\theta}& = \frac{\mathrm{d}}{\mathrm{d} \theta}\left(1-q-u_{2}(\theta)-\frac{v_1(\theta)}{1+au_{2}(\theta)+bv_1(\theta)}\right)\\ & = \frac{\rho_{1}(\theta)}{(1+au_{2}(\theta)+bv_{1}(\theta))^{2}}, \end{align*} |
where
\begin{equation*} \begin{aligned} \rho_{1}(\theta)& = -\frac{\mathrm{d}u_{2}}{\mathrm{d}\theta}(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{\mathrm{d}v_{1}}{\mathrm{d}\theta}(1+au_{2}(\theta))+av_{1}(\theta)\frac{\mathrm{d}u_{2}}{\mathrm{d}\theta}\\ & = \frac{\beta a}{\alpha b}\left(1-\epsilon\right)u^*(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{v^*}{1-\epsilon}(1+au^*)-\frac{\beta a^2}{\alpha b}u^*v^*(1-\epsilon). \end{aligned} \end{equation*} |
Since \alpha b\leq\beta and 0 < \epsilon < 1-\frac{1}{a} , we get
\frac{\mathrm{d}}{\mathrm{d} \theta}(1+au_{2}(\theta)+bv_{1}(\theta)) = \frac{bv^*}{1-\epsilon}-\frac{\beta a^2}{\alpha b}u^*(1-\epsilon) = \frac{bv^*}{1-\epsilon}\left[1-\left(\frac{\beta a}{\alpha b}\right)^2(1-\epsilon)^2\right] \lt 0. |
It is easy to see that \inf\limits_{\theta\in[2\epsilon, 1)}\rho_{1}(\theta) = \rho_{1}(1) . In view of 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} , we have
\begin{align*} \rho_{1}(1) & = \frac{\beta a}{\alpha b}(1-\epsilon)u^* \left[(1+au^*+bv^*)^{2}-av^*\right]-\frac{v^*}{1-\epsilon}(1+au^*)\\ & \gt u^*\left[(1+au^*+bv^*)^{2}-av^*-\frac{1+au^*}{(1-\epsilon)}\right]\\ & \gt u^*\left[1+2au^*+2bv^*-av^*-(1+2\epsilon)(1+au^*)\right]\\ & \gt 2u^*\left[bv^*-\epsilon(1+au^*)\right]\\ & \gt 0. \end{align*} |
This implies that for \theta\in[2\epsilon, 1) , \rho_{1}(\theta) > 0 holds and \varphi(\theta) is nondecreasing. That is, \varphi(\theta) < 0 . Moreover, E_{1}(u_2(\theta), v_1(\theta)) = \varphi(\theta)u_2(\theta) < 0 for \theta\in[2\epsilon, 1) .
\underline{\mathrm{Case\ 2}} : for 0 < \theta < 2\epsilon , from (3.5) we have v_1(\theta) = v^*\frac{\theta}{2(1-\epsilon)} and u_{2}(\theta) = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^* for \theta\in(0, 2\epsilon) . Then we get
\begin{align*} \frac{\mathrm{d}\varphi}{\mathrm{d}\theta} = \frac{\rho_2(\theta)}{(1+au_{2}(\theta)+bv_{1}(\theta))^{2}}, \end{align*} |
where
\rho_{2}(\theta) = \frac{\beta a}{\alpha b}\left(1-\epsilon\right)u^*(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{v^*}{2(1-\epsilon)}(1+au^*)-\frac{\beta a^2}{2\alpha b}u^*v^*, |
and \inf\limits_{\theta\in(0, 2\epsilon)}\rho_{2}(\theta) = \rho_{2}(2\epsilon) . In view of 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} , there holds
\begin{align*} \rho_{2}(2\epsilon) & \gt u^*\left[1+au^*\left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-2\epsilon)\right)+\frac{\epsilon bv^*}{2(1-\epsilon)}\right]^{2}-\frac{v^*}{2(1-\epsilon)}(1+au^*)-\frac{\beta a^2}{2\alpha b}u^*v^*\\ & \gt u^*\left[1+2au^*+2u^*\frac{\beta a^2}{\alpha b}(1-3\epsilon)-\frac{1+2\epsilon}{2}(1+au^*) -\frac{\beta a^2}{2\alpha b}u^*\right]\\ & = u^*\left[\left(1-\frac{1+2\epsilon}{2}\right)+\left(2-\frac{1+2\epsilon}{2}\right)au^* +\left(2(1-3\epsilon)-\frac{1}{2}\right)\frac{\beta a^2}{\alpha b}u^* \right]\\ & \gt 0. \end{align*} |
Since \varphi(2\epsilon) < 0 , for \theta\in(0, 2\epsilon) we have \rho_{2}(\theta) > 0 and \varphi(\theta) < 0 . This leads to E_{1}(u_2(\theta), v_1(\theta)) = \varphi(\theta)u_2(\theta) < 0 for \theta\in(0, 2\epsilon) . Hence, E_{1}(u_2(\theta), v_1(\theta)) < 0 for \theta\in(0, 1) .
To prove E_{2}(\xi, \eta_{2}) < 0 for \theta_0\in(0, 1) , \eta_{2} = v_2(\theta_0) and u_1(\theta_0))\leq \xi\leq u_2(\theta_0) , from (3.5) we deduce
\begin{align*} E_{2}(u_2(\theta_0), v_2(\theta_0)) & = v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)}{u^*+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta_0)u^*}\right]\\ & \lt v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)}{u^*+\frac{\beta a}{\alpha b}u^*(1-\theta_0)}\right]\\ & = v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)} {\frac{\beta}{\alpha}(v^*+\frac{a}{b}u^*(1-\theta_0))}\right]\\ & = 0. \end{align*} |
Hence, E_2(u_2(\theta), v_2(\theta)) < 0 for any \theta\in(0, 1) .
Theorem 3.7. Assume that conditions (G1)–(G2) hold and \Phi = (\phi_{1}, \phi_{2})\in \Gamma is a solution of system (1.3). Then we have
\begin{equation} \lim\limits_{z\rightarrow\infty}(\phi_{1}(z), \phi_{2}(z)) = (u^{\ast}, v^{\ast}). \end{equation} | (3.6) |
Proof. From (3.5), we observe
\begin{align*} &u_{1}(0) = 0, \qquad u_2(0) = u^*+\frac{\beta a}{\alpha b}(1-\epsilon)u^* \gt 2u^* \gt 1-q, \\ &v_1(0) = 0, \qquad v_2(0) = v^*+\frac{a}{b}u^* \gt \frac{v^*+u^*}{b} \gt \frac{1-q}{b}. \end{align*} |
In view of (\phi_{1}, \phi_{2})\in \Gamma for z > > 0 , it follows that
(1-q)\left(1-\frac{1}{b}\right)\leq \phi_{1}(z)\leq 1-q, \quad \frac{1-q}{b}\delta\leq \phi_2(z)\leq \frac{1-q}{b}. |
So we have
\begin{equation} \begin{aligned} u_1(\theta_0))\leq\liminf\limits_{z\rightarrow\infty}\phi_{1}(z)\leq \limsup\limits_{z\rightarrow\infty}\phi_{1}(z)\leq u_2(\theta_0), \\ v_1(\theta_0)\leq\liminf\limits_{\xi\rightarrow\infty}\phi_{2}(z)\leq \limsup\limits_{z\rightarrow\infty}\phi_{2}(z)\leq v_2(\theta_0), \end{aligned} \end{equation} | (3.7) |
for some \theta_0\in(0, 1) .
Denote
\theta^*: = \sup\{\theta\in\left[\theta_{0}, 1\right)| \ (3.7)\ \mathrm{ hold}\}. |
Then, \theta^* = 1 . Otherwise, we have \theta^* < 1 in (3.7). Namely, at least one of the following equalities is true:
u_{1}\left(\theta^*\right) = \liminf\limits_{z\rightarrow\infty}\phi_{1} \left(z\right), \ \, u_{2}\left(\theta^*\right) = \limsup\limits_{z\rightarrow\infty}\phi_{1} \left(z\right), |
v_{1}\left(\theta^*\right) = \liminf\limits_{z\rightarrow\infty}\phi_{2} \left(z\right), \ \, v_{2}\left(\theta^*\right) = \limsup\limits_{z\rightarrow\infty}\phi_{2} \left(z\right). |
Without loss of generality, we assume that
u_{1}(\theta^*) = \liminf\limits_{z\rightarrow\infty}\phi_{1}(z). |
It follows from Lebesgue's dominated convergence theorem that
\begin{align*} \liminf\limits_{z\rightarrow\infty}\phi_{1}(z) = &\liminf\limits_{z\rightarrow\infty} \frac{1}{\gamma}\left[\gamma\phi_{1}(z)+ E_1(\phi_{1}(z), \phi_{2}(z-c\tau) )\right]\\ \geq& \liminf\limits_{z\rightarrow\infty}\phi_{1}(z)+\frac{1}{\gamma} E_{1}(\liminf\limits_{z\rightarrow\infty}\phi_{1}(z), \limsup\limits_{z\rightarrow\infty}\phi_{2}(z)). \end{align*} |
That is,
E_{1}(\liminf\limits_{z\rightarrow\infty}\phi_{1}(z), \ \limsup\limits_{z\rightarrow\infty}\phi_2(z))\leq0. |
This implies that E_{1}(u_{1}(\theta^*), \ \eta)\leq0 with v_{1}\left(\theta^*\right)\leq \eta\leq v_{2}\left(\theta^*\right) , which yields a contradiction to (C3) of Theorem 3.6. The other three cases can be proceeded in an analogous manner.
Let z\in\mathbb{R}^-_N with N\in \mathbb{R} . We define
C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) = \left\{(\phi_1, \phi_2)\in \mathcal{C}|_{\mathbb{R}^-_{N}}:\ \lim\limits_{z\rightarrow-\infty}\phi_{1}(z) = \phi_{1}(-\infty), \quad \lim\limits_{z\rightarrow-\infty}\phi_{2}(z) = \phi_{2}(-\infty)\right\}. |
It is not difficult to see that C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is isomorphic to C\left(\left[\frac{N}{N-1}, 1\right], \mathbb{R}^2\right) . Indeed, if x(s)\in C\left(\left[\frac{N}{N-1}, 1\right], \mathbb{R}^2\right) , then y(t) = x(s) for t = \frac{s}{s-1} , s\in \left[\frac{N}{N-1}, 1\right) , and y(t)\in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) . That is, C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is a Banach space equipped with the superemum norm.
Theorem 4.1. When c = c^* , system (1.3) has a positive traveling wave solution satisfying (1.4).
Proof. Let \{c_n\} be a decreasing sequence with c_n < c^*+1 and \lim\limits_{n\rightarrow\infty} c_n = c^* . Then for each c_n , system (1.3) has a positive traveling wave solution \left(\phi_{1n}\left(z\right), \, \phi_{2n}\left(z\right)\right) satisfying (1.4) and
(1-q)\frac{b-1}{b}\leq\phi_{1n}(z)\leq1-q, \ \ 0\leq\phi_{2n}(z)\leq\frac{1-q}{b}. |
Since a traveling wave solution is invariant in the sense of phase shift, we can assume that
\phi_{1n}(0) = (1-q)\iota_1, \ \phi_{1n}(z) \gt (1-q)\iota_1\ \text{for}\ z \lt 0 \ \text{and}\ \phi_{2n}(0) = \iota_2, \ \phi_{2n}(z) \lt \iota_2\ \text{for}\ z \lt 0, |
with \frac{b-1}{b} < \iota_1 < 1 and 0 < \iota_2 < \frac{1-q}{b} . From (1.4), we know that the above expressions are admissible.
For n\in\mathbb{N} , it is evident that (\phi_{1n}(z), \, \phi_{2n}(z)) are equipcontinuous, bounded and equipconvergent in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) . According to Lemma 3.3, \{(\phi_{1n}(z), \, \phi_{2n}(z))\} has a subsequence, still denoted by \{(\phi_{1n}(z), \, \phi_{2n}(z))\} , such that
\phi_{1n}(z)\rightarrow\phi_{1}(z), \ \, \phi_{2n}(z)\rightarrow\phi_{2}(z), \ \mathrm{as} \ n\rightarrow\infty |
and
\lim\limits_{z\rightarrow-\infty}\phi_{1}(z) = 1-q, \quad \lim\limits_{z\rightarrow-\infty}\phi_{2}(z) = 0. |
Here, (\phi_{1}(z), \, \phi_{2}(z))\in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is continuous and the above limits converge uniformly on \mathbb{R}^-_N . It follows from Lebesgue's dominated convergence theorem that
\lim\limits_{n\rightarrow\infty} J*\phi_{in}(z) = \phi_{i}(z) , \quad i = 1, 2 |
on z\in \mathbb{R}^-_N . Thus, (\phi_{1}(z), \, \phi_{2}(z)) is a solution to system (1.3) which satisfies
\phi_{1}(0) = (1-q)\iota_1, \ \phi_{1}(z) \gt (1-q)\iota_1 \ \text{for}\ z \lt 0\ \text{and}\ \phi_{2}(0) = \iota_2, \ \phi_{2}(z) \lt \iota_2 \ \text{for}\ z \lt 0, |
and
(1-q)\frac{b-1}{b}\leq\phi_{1}(z)\leq1-q, \ \, \ 0\leq\phi_{2}(z)\leq\frac{1-q}{b}. |
From \phi_{2}(0) = \iota_2 > 0 , \liminf\limits_{z\rightarrow-\infty}\phi_{2}(z) > 0 holds. By virtue of Theorem 3.7, we obtain
\lim\limits_{z\rightarrow+\infty}\phi_1(z) = u^*, \ \ \lim\limits_{z\rightarrow+\infty}\phi_2(z) = v^*. |
Consider the Cauchy problem:
\begin{equation} \left\{ \begin{aligned} &\frac{\partial u\left(x, \, t\right)}{\partial t} = d\left(J \ast u\left(x, \, t\right)-u\left(x, \, t\right)\right)+u\left(x, \, t\right)\left(1-ru\left(x, \, t\right)\right), \\ &u\left(x, \, 0\right) = u_0\left(x\right), \ \, x\in\mathbb{R}, \end{aligned} \right. \end{equation} | (5.1) |
where J satisfies condition (G1), r > 0 is constant and the initial value u_0\left(x\right) is uniformly continuous and bounded for x\in\mathbb{R} .
Lemma 5.1. [32] Assume that 0\leq u_0\left(x\right)\leq\frac{1}{r} . Then system (5.1) admits a solution for x\in\mathbb{R} and t > 0 . If \omega\left(x, \, 0\right) is uniformly continuous and bounded, and \omega\left(x, \, 0\right) satisfies
\begin{equation*} \left\{ \begin{aligned} &\frac{\partial\omega\left(x, \, t\right)}{\partial t}\geq\left(\leq\right)d\left(J \ast\omega\left(x, \, t\right)-\omega\left(x, \, t\right)\right) +\omega\left(x, \, t\right)\left(1-r\omega\left(x, \, t\right)\right), \\ &\omega\left(x, \, 0\right)\geq\left(\leq\right)u_0\left(x\right), \ \, x\in\mathbb{R}, \\ \end{aligned} \right. \end{equation*} |
then we have
\omega\left(x, \, t\right)\geq\left(\leq\right)u\left(x, \, t\right), \ \, x\in\mathbb{R}, \ t \gt 0. |
Lemma 5.2. [32] Assume that u_0\left(x\right) > 0 . Then for any 0 < c < c^* there holds
\liminf\limits_{t\rightarrow{\infty}}\inf\limits_{|x| \lt ct} u\left(x, \, t, \ u_0\left(x\right)\right) = \limsup\limits_{t\rightarrow{\infty}}\sup\limits_{|x| \lt ct} u\left(x, \, t, \ u_0\left(x\right)\right) = \frac{1}{r}. |
Theorem 5.3. For any speed 0 < c < c^* , there is no nontrivial positive solution \left(\phi_{1}\left(z\right), \, \phi_{2}\left(z\right)\right) of system (1.3) satisfying condition (1.4).
Proof. Suppose on the contrary that there exists some 0 < c_{1} < c^*, such that system (1.3) has a positive solution (\phi_{1}(z), \phi_{2}(z)) satisfying condition (1.4). Then \phi_{1}(z) is bounded on \mathbb{R} and we can find a positive constant K such that \psi(x, t) = \phi_2(x+ct) satisfies
\begin{equation*} \left\{ \begin{aligned} &\frac{\partial \psi\left(x, t\right)}{\partial t}\geq d_2\left(J \ast \psi\left(x, t\right)-\psi\left(x, t\right)\right)+\alpha\psi\left(x, t\right)\left(1-K\psi \left(x, t\right)\right), \\ &\psi\left(x, 0\right) = \phi_{2}\left(x\right) \gt 0. \end{aligned} \right. \end{equation*} |
Let x(t) = -\frac{c_1+c^*}{2}t . From Lemmas 5.1 and 5.2 it follows that
\liminf\limits_{t\rightarrow{\infty}}\inf\limits_{2|x| = (c_1+c^*)t}\psi(x, t)\geq\frac{1}{K}. |
Meanwhile, in view of x(t)+c_1t = \frac{c_1-c^*}{2}t , we see z = x(t)+c_1t\rightarrow -\infty as t\rightarrow +\infty , and
\limsup\limits_{t\rightarrow{\infty}}\psi(x(t), t) = \lim\limits_{z\rightarrow{-\infty}}\phi_2(z) = 0. |
This yields a contradiction.
In this paper, we have studied the existence and nonexistence of traveling wave solution of a nonlocal delayed predator-prey model with the B-D functional response and harvesting. As we see, model (1.3) is nonmonotone or not quasimonotone. We employed Schauder's fixed point theorem and the upper-lower solutions method to discuss the existence of traveling wave solution for the speed c > c^* . Then, we investigated the asymptotic behavior of traveling wave solution by construction of the upper-lower solutions at -\infty and by developing the contacting rectangles technique at +\infty . For the special case of c = c^* , one usually can not establish the existence of traveling wave solution directly by constructing a pair of upper-lower solutions. One of available methods is the limiting argument together with the Arzela-Ascoli Theorem [33,36,39]. In this study we have presented not only the existence of traveling wave solution but also the asymptotic behavior of traveling wave solution at -\infty by Corduneanu's theorem. The nonexistence of traveling wave solution of system (1.3) with condition (1.4) was investigated by applying the comparison principle of nonlocal dispersal equations.
It is remarkable that for the parameters of system (1.3), we only need b > 1 and 0 < b\alpha\leq \beta to prove Theorem 3.5. These conditions were used to construct a pair of suitable upper-lower solutions of system (1.3). For a > 1 and 0 < a\alpha\leq \beta , we could also construct the appropriate upper-lower solutions of system (1.3) in a similar way. To obtain the asymptotic behavior of traveling wave solution as z\rightarrow\infty , we additionally needed a > \frac{1}{q} .
When q = 0 in model (1.3), it means that there does not have any prey harvesting. By assuming b > 1 , 0 < b\alpha\leq \beta and a > \frac{b\alpha}{\beta} , we can derive the same results as Theorems 3.5 and 3.7 in an analogous manner.
We are grateful to the anonymous referees for their valuable comments. This work is supported by National Science Foundation of China under 11601029. All authors declare no conflicts of interest in this paper.
The authors declare there is no conflicts of interest.
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