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Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

  • Received: 01 August 2021 Revised: 01 September 2021 Published: 26 October 2021
  • Primary: 92C17, 35K35; Secondary: 35B35, 35A01

  • This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

    {ut=Δuχ1(uv)+μ1u(1ua1w),xΩ,t>0,0=Δvv+w,xΩ,t>0,wt=Δwχ2(wz)χ3(wv)+μ2w(1wa2u),xΩ,t>0,0=Δzz+u,xΩ,t>0,

    under homogeneous Neumann boundary conditions in a bounded domain ΩRn with n1, where the parameters a1,a2, χ1,χ2,χ3, μ1,μ2 are positive constants. We first showed some conditions between χ1μ1, χ2μ2, χ3μ2 and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.

    Citation: Rong Zhang, Liangchen Wang. Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop[J]. Electronic Research Archive, 2021, 29(6): 4297-4314. doi: 10.3934/era.2021086

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  • This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

    {ut=Δuχ1(uv)+μ1u(1ua1w),xΩ,t>0,0=Δvv+w,xΩ,t>0,wt=Δwχ2(wz)χ3(wv)+μ2w(1wa2u),xΩ,t>0,0=Δzz+u,xΩ,t>0,

    under homogeneous Neumann boundary conditions in a bounded domain ΩRn with n1, where the parameters a1,a2, χ1,χ2,χ3, μ1,μ2 are positive constants. We first showed some conditions between χ1μ1, χ2μ2, χ3μ2 and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.



    In this paper, we consider the two-species chemotaxis- competition system with two chemicals

    {ut=Δuχ1(uv)+μ1u(1ua1w),xΩ,t>0,0=Δvv+w,xΩ,t>0,wt=Δwχ2(wz)χ3(wv)+μ2w(1wa2u),xΩ,t>0,0=Δzz+u,xΩ,t>0,uν=vν=wν=zν=0,xΩ,t>0,(u,w)(x,0)=(u0(x),w0(x)),xΩ, (1.1)

    where ΩRn (n1) is a bounded domain with smooth boundary Ω and /ν denotes the derivative with respect to the outer normal of Ω; a1,a2, χ1,χ2,χ3, μ1 and μ2 are positive constants.

    Chemotaxis describes oriented movement of cells along the concentration gradient of a chemical signal produced by the cells, which is important in a large variety of fields within the life cycle of most multicellular organisms. Based on a well-known chemotaxis model for the chemotactic movement of one specie [11], a generalization of chemotaxis model for multi-species or multi-chemical-signal chemotaxis system was proposed [9,24,42]. System (1.1) describes the communication between breast tumor cells and macrophages in close proximity via a short-ranged chemical signaling loop according to the classical Lotka-Volterra dynamics [22], which was proposed by Knútsdóttir et al. [12]. The key physical variables in (1.1) are assumed to be the density of the macrophages (denoted by u) and the tumor cells (denoted by w), the concentration of the colony-stimulating factor 1 (CSF-1, denoted by v) and the epidermal growth factor (EGF, denoted by z). In their experiment, it was shown that in a short-term chemical signalling loop, the process is promoted by the macrophages. More precisely, the tumor cells w secrete CSF-1 v, which can bind to CSF-1 receptors on the macrophages w. This activates the macrophages to increasing concentrations of CSF-1 gradient and to secrete EGF z. Then the EGF can bind to receptors on tumor cells, continuing the chain of activation of them in return. Furthermore, activated tumor cells release more CSF-1 and partially direct their movement toward both concentration gradients of the EGF and the CSF-1, respectively.

    From a mathematical point of view, the system (1.1) contains mainly two subsystems. When u=z0 and a20, (1.1) is reduced to the chemotaxis-only system [11]:

    {wt=Δwχ3(wv)+μ2w(1w),xΩ,t>0,0=Δvv+w,xΩ,t>0, (1.2)

    which describe the aggregation of the paradigm species Dictyostelium discoideum. Model (1.2) has been extensively studied during past four decades. Boundedness of global solutions and unbounded solutions for (1.2) have been extensively investigated (see the surveys [2,6,8] and the references therein). When χ30, (1.1) becomes the two-species chemotaxis system with two chemicals:

    {ut=Δuχ1(uv)+μ1u(1ua1w),xΩ,t>0,0=Δvv+w,xΩ,t>0,wt=Δwχ2(wz)+μ2w(1wa2u),xΩ,t>0,0=Δzz+u,xΩ,t>0, (1.3)

    which describes the spatio-temporal evolution of two populations which on the one hand proliferate and mutually compete with Lotka-Volterra kinetics and on the other hand the individuals shall move according to random diffusion and migrate toward a chemical signal produced by the opposing species. Without kinetic terms, that is, when μ1=μ2=0, if χ1,χ2{1,1}, global boundedness and finite time blow-up was constructed by Tao and Winkler [32]. In particular, when n=2, global bounded solutions are proved if max{m1,m2}<C with some C>0, where m1:=Ωu0 and m2:=Ωw0; if min{m1,m2}>4π, the solution may be blow up in finite time. These results were improved in [45]. When μi>0 (i=1,2), global bounded solution and stabilization have been addressed by Tu et al. [36]. Recently, this results were improved by Wang et al. [38]. There are some another results on various the system (1.3) ([3,5,10,18,23,26,29,27,43,47,46]).

    Compared with the chemotaxis-only system and the two-species chemotaxis system with two chemicals, the coupled a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop the system (1.1) is much less understood. For the simplified version (1.1) in the unit disk Ω=BR(0)R2 with μi=0 (i=1,2) and the second and fourth equation replaced by

    0=Δvm2|Ω|+wand0=Δzm1|Ω|+u,

    respectively, where m1:=Ωu0 and m2:=Ωw0, global boundedness and blow-up of solutions were constructed in [16]. In particular, a critical mass phenomenon has been found: solutions remain bounded if 2m1+χ3χ2m2<8πχ2, whereas blow-up may occur if 8πχ1m1+8πχ2m2<2m1m2+m22χ3χ2 and Ωu0|x|2 and Ωw0|x|2 are sufficiently small. Recently, (1.1) with μi=0 (i=1,2) and ΩR2 has been considered in [17]: global boundedness was constructed when χ1χ2m1m2<(πχ3m2)π, where π=8π when Ω=BR(0) is a disk, otherwise, π=4π; for the fully parabolic system (1.1) with μi=0 (i=1,2) and ΩR2, global bounded solution was considered when χ1χ2m1m2<14χ3C4GNm24C8GN, where CGN>0 is a constant. Moreover, gradient estimates, blow-up in finite time and asymptotic behavior have also been established. When n=3 and μi>0 (i=1,2) are sufficiently large, for the fully parabolic system (1.1), global bounded solutions were proved by Pan et al.[25]. Very recently, a symmetric model of (1.1) has been investigated [34,35], i.e., replacing the first equation in (1.1) by ut=Δuχ1(uv)χ4(uz)+μ1u(1ua1w).

    In order to better understand (1.1), we should mention two biological species which compete for the resources and migrate towards a higher concentration of a chemical produced by themselves was proposed by Tello and Winkler [33]

    {ut=Δuχ1(uw)+μ1u(1ua1v),xΩ,t>0,vt=Δvχ2(vw)+μ2v(1a2uv),xΩ,t>0,0=Δww+u+v,xΩ,t>0, (1.4)

    this model has been extensively studied. When a1,a2(0,1), global existence and the large time behavior were established under some conditions by Tello and Winkler [33], which was partially improved in [4], and by Stinner et al. [30] for the case of a11>a2. For all a1,a2>0, global boundedness solutions were derived by Mizukami [20], which covers the case that a1,a21. Moreover, the convergence rates has been also obtained in the case that a1,a2(0,1) and a11>a2. For the fully parabolic version of model (1.4), which is obtained by replacing the equation with wt=Δww+u+v, global existence and boundedness has been established for the space dimension does not exceed two by Bai and Winkler [1] and the n-dimensional setting by [14,15,44]. Moreover, the convergence rates has been established [1], this conditions were improved by Mizukami [19]. Recently, the conditions for asymptotic behavior in the case of a1,a2(0,1) were once more improved by Mizukami [21].

    The focus of this paper is to establish the global existence, large time behavior, and the rates of convergence of solution to (1.1). The first of our result asserts global existence of a bounded solution based on the comparison methods in [38,37,4,30].

    Theorem 1.1. Suppose that ΩRn (n1) is a bounded domain with smooth boundary, and a1,a2, χ1,χ2,χ3, μ1,μ2 are positive constants. If one of the following conditions holds:

    (i) χ1μ1a1 and χ3μ2<1;

    (ii) χ2μ2a2 and χ3μ2<1;

    (iii) χ1μ1>a1, χ2μ2>a2 and χ3μ2<1 as well as χ1χ2μ1μ2+a1a2a1χ2μ2a2χ1μ1+χ3μ2<1.

    Then for the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω), the model (1.1) possesses a unique global classical solution (u,v,w,z) which is uniformly bounded in Ω×(0,). Moreover, the solutions u,v,w,z are the Höder continuous functions, i.e., there exists some θ(0,1) and M>0 such that

    uC2+θ,1+θ2(¯Ω×[t,t+1])+vC2+θ,1+θ2(¯Ω×[t,t+1])+wC2+θ,1+θ2(¯Ω×[t,t+1])+zC2+θ,1+θ2(¯Ω×[t,t+1])Mfor allt1. (1.5)

    The large time behavior and convergence rates of solutions to (1.1) is mathematically and biologically interesting. We first give the result of competitive coexistence case a1,a2(0,1).

    Theorem 1.2. Let a1,a2(0,1) and ΩRn (n1) be a bounded domain with smooth boundary. Assume that the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω) with u00w0 and (u,v,w,z) is a global bounded classical solution of (1.1). Let χ1,χ2,χ3, μ1 and μ2 are positive constants and satisfy

    μ1μ2a1(1a1a2)a2μ2χ21u16wμ1a1χ238>0 (1.6)

    and

    μ1μ22a1a2(1a1a2)a22μ22χ21u16μ1μ2a1a2χ23w8μ1μ2a21χ22w8+μ2a1a2χ21χ22uw128+μ1a21χ22χ23w64>0, (1.7)

    then one can find C>0 and μ>0 such that

    uuL(Ω)+vvL(Ω)+wwL(Ω)+zzL(Ω)Ceμt

    for all t>0, where

    u=z=1a11a1a2,v=w=1a21a1a2. (1.8)

    Remark 1. Since a1,a2(0,1), it is easy to check f(μ1,μ2):=μ1μ22a1a2(1a1a2)a22μ22χ21u16μ1μ2a1a2χ23w8μ1μ2a21χ22w8+μ2a1a2χ21χ22uw128+μ1a21χ22χ23w64 satisfies

    limμ1+,μ2+f(μ1,μ2)μ1μ22=a1a2(1a1a2)>0.

    Hence, there exist some constants μ10,μ20>0 such that f(μ1,μ2)>0 for all [μ10,)×[μ20,). Thanks to a continuity argument implies (1.7) holds for all [μ10,)×[μ20,).

    The following result is on the competitive exclusion case a11>a2>0.

    Theorem 1.3. Let a11>a2>0, χ1,χ2,χ3>0, μ1,μ2>0, and let ΩRn (n1) be a bounded domain with smooth boundary. Assume that the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω) with w00 and (u,v,w,z) is a global bounded solution of (1.1).

    (i) If a1>1 and assume that for some a1(1,a1] such that a1a2<1. Moreover, let

    μ22a2(1a1a2)a2μ2χ238+a1χ22χ2364μ2a1χ228>0andμ2>χ238(1a1a2), (1.9)

    there exist C>0 and λ>0 such that

    uL(Ω)+v1L(Ω)+w1L(Ω)+zL(Ω)Ceλtfor allt>0.

    (ii) If a1=1 and (1.9) holds with a1=1, there exist C>0 and κ>0 such that

    uL(Ω)+v1L(Ω)+w1L(Ω)+zL(Ω)C(1+t)κfor allt>0.

    This paper is organized as follows. In Section 2, we show local existence of a solution to (1.1) and use comparison methods to prove global existence and boundedness of (1.1) (Theorems 1.1). Section 3 is devoted to the proof of asymptotic stability to (1.1) (Theorems 1.2 and 1.3).

    The local existence of solutions to (1.1) which can be achieved similarly by using well-established methods in [30,31,41].

    Lemma 2.1. Let ΩRn (n1) be a bounded domain with smooth boundary, a1,a2>0, χ1,χ2,χ3>0, μ1,μ2>0. Then for the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω), there exists Tmax(0,] such that (1.1) has a unique local non-negative classical solution

    u,v,w,zC(¯Ω×[0,Tmax))C2,1(¯Ω×(0,Tmax)), (2.1)

    and which is such that either Tmax= or Tmax< and

    u(,t)L(Ω)+w(,t)L(Ω)astTmax.

    Moreover, if the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω) with u00w0, then the solution of (1.1) satisfies u>0, v0 w>0 and z0 in Ω×(0,Tmax).

    According to comparison methods in [38,37,4,30], we will prove boundedness of solution to model (1.1). First we let the parabolic operator Li=Li(x,t) (i=1,2) satisfy

    L1u:=Δuχ1uv,(x,t)Ω×(0,Tmax) (2.2)

    and

    L2w:=Δwχ2wzχ3wv,(x,t)Ω×(0,Tmax). (2.3)

    Thus from the model (1.1) shows that

    utL1u=u{χ1Δv+μ1(1ua1w)}=u{μ1μ1uχ1v+(χ1a1μ1)w} (2.4)

    and

    wtL2w=w{χ2Δzχ3Δv+μ2(1wa2u)}=w{μ2(μ2χ3)w+(χ2a2μ2)uχ2zχ3v} (2.5)

    for all (x,t)Ω×(0,Tmax).

    We first consider the conditions χ1μ1a1 and χ3μ2<1.

    Lemma 2.2. Let τ=0 and the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω). Assume that χ1μ1a1 and χ3μ2<1. Then Tmax= and u,v,w,z are bounded in Ω×(0,).

    Proof. Since χ1μ1a1 and u,v,w,z are nonnegative, in view of (2.4) and (2.5), we have

    p1u:=utL1uu{μ1μ1u}0,(x,t)Ω×(0,Tmax) (2.6)

    and

    p2w:=wtL2ww{μ2(μ2χ3)w+(χ2a2μ2)u}0,(x,t)Ω×(0,Tmax), (2.7)

    where L1u and L2w are given in (2.2) and (2.3), respectively. Next, in order to prove Lemma 2.2, we have two cases to proceed as below:

    Case 1. χ2a2μ20. We choose A>0 and let ξ>0 sufficiently large such that

    ξmax{u0L(Ω)A,1A,w0L(Ω),μ2μ2χ3}. (2.8)

    Hence, we can define constant functions ¯u and ¯w as follows

    ¯u=¯u(x,t):=Aξand¯w=¯w(x,t):=ξ,(x,t)¯Ω×[0,Tmax), (2.9)

    which satisfy

    ¯u(x,0)=Aξu0(x)and¯w(x,0)=ξw0(x)for allxΩ. (2.10)

    Due to the definition of p1 in (2.6), using (2.8) and (2.9), we have

    p1¯u=Aξ{μ1μ1Aξ}0,(x,t)Ω×(0,Tmax). (2.11)

    According to (2.6), (2.10) and (2.11), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that

    u(x,t)¯u(x,t)=Aξfor all(x,t)Ω×(0,Tmax). (2.12)

    Using the definition of p2 in (2.7), we deduce from (2.8), χ3μ2<1 and χ2a2μ20 that

    p2¯w=ξ{μ2(μ2χ3)ξ+(χ2a2μ2)u}ξ{μ2(μ2χ3)ξ}0,(x,t)Ω×(0,Tmax). (2.13)

    According to (2.7), (2.10) and (2.13), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that

    w(x,t)¯w(x,t)=ξfor all(x,t)Ω×(0,Tmax). (2.14)

    Case 2. χ2a2μ2>0. Due to χ3μ2<1 and χ2a2μ2>0, we can find A fulfilling

    0<A<μ2χ3χ2a2μ2. (2.15)

    Let ξ large enough such that

    ξmax{u0L(Ω)A,1A,w0L(Ω),μ2μ2χ3(χ2a2μ2)A}. (2.16)

    We again let ¯u=¯u(x,t):=Aξ and ¯w=¯w(x,t):=ξ for all (x,t)¯Ω×[0,Tmax), as in (2.9), we deduce from (2.16)

    ¯u(x,0)=Aξu0(x)and¯w(x,0)=ξw0(x)for allxΩ. (2.17)

    Similar to (2.11) and (2.12), we infer

    u(x,t)¯u(x,t)=Aξfor all(x,t)Ω×(0,Tmax). (2.18)

    Hence, using χ2a2μ2>0, (2.7), (2.16) and (2.18) yields

    p2¯w=ξ{μ2(μ2χ3)ξ+(χ2a2μ2)u}ξ{μ2(μ2χ3)ξ+(χ2a2μ2)Aξ}=ξ{μ2[μ2χ3(χ2a2μ2)A]ξ}>0,(x,t)Ω×(0,Tmax), (2.19)

    thus by (2.7) and (2.17) we have

    w(x,t)¯w(x,t)=ξfor all(x,t)Ω×(0,Tmax). (2.20)

    Using (2.12), (2.14), (2.18) and (2.20) along with extensibility criterion in Lemma 2.1, this entails that Tmax= and u,v,w,z are globally bounded.

    Next, we consider the conditions χ2μ2a2 and χ3μ2<1.

    Lemma 2.3. Let τ=0 and the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω). Assume that χ2μ2a2 and χ3μ2<1. Then Tmax= and u,v,w,z are bounded in Ω×(0,).

    Proof. Since χ2μ2a2, χ3μ2<1 and u,v,w,z are nonnegative, in view of (2.4) and (2.5), we have

    p3u:=utL1uu{μ1μ1u+(χ1a1μ1)w}0,(x,t)Ω×(0,Tmax) (2.21)

    and

    p4w:=wtL2ww{μ2(μ2χ3)w}0,(x,t)Ω×(0,Tmax), (2.22)

    where L1u and L2w are given in (2.2) and (2.3), respectively. In the case of χ1a1μ10 and χ3μ2<1, Lemma 2.2 showed (1.1) has global bounded solution. Hence, here we need only consider the case χ1a1μ1>0. We can find A>0 and ξ>0 fulfilling

    A>χ1a1μ1+μ1ξμ1 (2.23)

    and

    ξmax{u0L(Ω)A,w0L(Ω),μ2μ2χ3}. (2.24)

    We again let ¯u=¯u(x,t):=Aξ and ¯w=¯w(x,t):=ξ for all (x,t)¯Ω×[0,Tmax), as in (2.9), we deduce

    ¯u(x,0)=Aξu0(x)and¯w(x,0)=ξw0(x)for allxΩ. (2.25)

    Using (2.22) and (2.24) we conclude

    p4¯w=ξ{μ2(μ2χ3)ξ}>0,(x,t)Ω×(0,Tmax), (2.26)

    so according to (2.22) and (2.25) imply

    w(x,t)¯w(x,t)=ξfor all(x,t)Ω×(0,Tmax). (2.27)

    Hence, using χ1a1μ1>0, (2.21), (2.23) and (2.27) yields

    p3¯u=Aξ{μ1μ1Aξ+(χ1a1μ1)w}Aξ{μ1μ1Aξ+(χ1a1μ1)ξ}0,(x,t)Ω×(0,Tmax). (2.28)

    From (2.21), (2.25) and (2.28) and the comparison principle, we infer

    u(x,t)¯u(x,t)=Aξfor all(x,t)Ω×(0,Tmax). (2.29)

    Then using (2.27) and (2.29) along with extensibility criterion in Lemma 2.1, this entails that Tmax= and u,v,w,z are globally bounded.

    Finally, we consider the conditions χ1μ1>a1, χ2μ2>a2 and χ3μ2<1.

    Lemma 2.4. Let τ=0 and the nonnegative initial data (u0,w0)C0(¯Ω)×C1(¯Ω). Assume that χ1μ1>a1, χ2μ2>a2 and χ3μ2<1 as well as (χ1μ1a1)(χ2μ2a2)+χ3μ2<1. Then Tmax= and u,v,w,z are bounded in Ω×(0,).

    Proof. The conditions χ1μ1>a1, χ2μ2>a2, χ3μ2<1 and (χ1μ1a1)(χ2μ2a2)+χ3μ2<1 imply that

    χ1a1μ1μ1<μ2χ3χ2a2μ2. (2.30)

    Hence, we can choose ξ>0 large enough such that

    ξmax{μ1χ1a1μ1u0L(Ω),w0L(Ω)} (2.31)

    and

    χ1a1μ1+μ1ξμ1<μ2χ3μ2ξχ2a2μ2,

    which enables us to choose A>0 satisfying

    χ1a1μ1+μ1ξμ1<A<μ2χ3μ2ξχ2a2μ2. (2.32)

    Hence, (2.32) implies

    {Aξ{μ1μ1Aξ+(χ1a1μ1)ξ}>0,ξ{μ2(μ2χ3)ξ+(χ2a2μ2)Aξ}>0. (2.33)

    We again let ¯u=¯u(x,t):=Aξ and ¯w=¯w(x,t):=ξ for all (x,t)¯Ω×[0,Tmax), as in (2.9), we deduce

    ¯u(x,0)=Aξ>χ1a1μ1μ1ξu0(x)and¯w(x,0)=ξw0(x)for allxΩ. (2.34)

    Due to u,v,w,z are nonnegative, from (2.4) and (2.5), and in view of (2.33), (2.34) imply that

    {utL1uu{μ1μ1u+(χ1a1μ1)w}¯utL1¯u¯u{μ1μ1¯u+(χ1a1μ1)¯w},wtL2ww{μ2(μ2χ3)w+(χ2a2μ2)u}¯wtL2¯w¯w{μ2(μ2χ3)¯w+(χ2a2μ2)¯u}

    in Ω×(0,Tmax), by the comparison principle for this cooperative systems [28,Proposition 52.22], it follows that

    u(x,t)¯u(x,t)=Aξandw(x,t)¯w(x,t)=ξfor all(x,t)Ω×(0,Tmax).

    Therefore, using the extensibility criterion in Lemma 2.1, we have Tmax= and u,v,w,z are globally bounded.

    Proof of Theorem 1.1. We only need to use Lemmas 2.2-2.4 to obtain the global bounded solutions of (1.1). Finally, the Hölder continuity of the solution (u,v,w,z) comes from standard parabolic regularity theory [13].

    In this section, we will derive the asymptotic behavior of the solutions to the model (1.1), the ideas mainly come from [1,21,38,39]. To achieve our goals, we first recall the following lemma which is important for the proof of Theorems 1.2 and 1.3 (see [7,Lemma 4.6] or [20,Lemma 3.1]).

    Lemma 3.1. Suppose that φ(x,t)C0(¯Ω×[0,)) and there exist constant C>0 and σ>0 such that

    φ(x,t)Cσ,σ2(¯Ω×[t,t+1])Cfor allt1.

    Moreover, assume that there exists some constant M>0 such that

    0Ω(φ(x,t)M)2dxdt<.

    Then

    φ(,t)MinC0(¯Ω)ast.

    This following lemma is a straightforward result from [40,Lemma 5.1].

    Lemma 3.2. Let a11,a22,a33,a44,a13,a14,a23R and satisfy

    a11>0,a22>0,a11a22a33a22a2134a11a2234>0

    and

    a11a22a33a44a22a44a2134a11a44a2234a22a33a2144+a214a22316>0.

    Then there exists ε>0 such that

    a11x21+a22x22+a33x23+a44x24+a13x1x3+a14x1x4+a23x2x3ε(x21+x22+x23+x24)

    for all x1,x2,x3,x4R.

    In order to prove stabilization of solutions to (1.1), we will divide the proof into two cases.

    In this subsection, we will study the asymptotic behavior of the solution of (1.1) with a1,a2(0,1) based on the following energy functional

    E1(t):=a2μ2Ω(uuulnuu)+a1μ1Ω(wwwlnww), (3.1)

    where (u,v,w,z) is the constant steady state defined by (1.8).

    Lemma 3.3. Under the assumptions of Theorem 1.2, then there exists ε>0 such that

    E1(t)0andddtE1(t)εF1(t)for allt>0, (3.2)

    where the function E1(t) defined by (3.1) and F1(t) satisfies

    F1(t):=Ω(uu)2+Ω(vv)2+Ω(ww)2+Ω(zz)2. (3.3)

    Proof. From Taylor's formula we see that E1(t)0 for t>0 (for more details, see [1,Lemma 3.2]). From straightforward calculations we infer that

    ddtE1(t)=uμ2a2Ω|u|2u2+uμ2a2χ1Ωuvuwμ1a1Ω|w|2w2+wμ1a1χ2Ωwzw+wμ1a1χ3Ωwvw+μ1μ2a1Ω(ww)(1wa2u)+μ1μ2a2Ω(uu)(1ua1w)=μ1μ2a2Ω(uu)2μ1μ2a1Ω(ww)2uμ2a2Ω|u|2u22μ1μ2a1a2Ω(uu)(ww)+uμ2a2χ1Ωuvuwμ1a1Ω|w|2w2+wμ1a1χ2Ωwzw+wμ1a1χ3Ωwvw. (3.4)

    In order to obtain (3.2), we first deal with the last five parts on the right of (3.4), by a simple computation we conclude

    uμ2a2Ω|u|2u2+uμ2a2χ1Ωuvu=uμ2a2Ω(uuχ12v)2+uμ2a2χ214Ω|v|2uμ2a2χ214Ω|v|2 (3.5)

    and

    wμ1a1Ω|w|2w2+wμ1a1χ2Ωwzw+wμ1a1χ3Ωwvw=wμ1a1Ω(wwχ22zχ32v)2+wμ1a1χ224Ω|z|2+wμ1a1χ234Ω|v|2+wμ1a1χ2χ32Ωvzwμ1a1χ224Ω|z|2+wμ1a1χ234Ω|v|2+wμ1a1χ2χ32Ωvzwμ1a1χ222Ω|z|2+wμ1a1χ232Ω|v|2. (3.6)

    Since v=w, by the second equation in (1.1) we have

    Ω|v|2=Ω(vv)2+Ω(ww)(vv). (3.7)

    Similar to the fourth equation in (1.1) yields

    Ω|z|2=Ω(zz)2+Ω(uu)(zz). (3.8)

    Inserting (3.5)-(3.8) into (3.4) we conclude

    ddtE1(t)a2μ1μ2Ω(uu)2a1μ1μ2Ω(ww)2(a2μ2χ21u4+wμ1a1χ232)Ω(vv)2a1μ1χ22w2Ω(zz)2+a1μ1χ22w2Ω(uu)(zz)2a1a2μ1μ2Ω(uu)(ww)+(a2μ2χ21u4+wμ1a1χ232)Ω(ww)(vv). (3.9)

    To see (3.2), we will show that there exists ε>0 such that

    a2μ1μ2Ω(uu)2a1μ1μ2Ω(ww)22a1a2μ1μ2Ω(uu)(ww)(a2μ2χ21u4+wμ1a1χ232)Ω(vv)2a1μ1χ22w2Ω(zz)2+a1μ1χ22w2Ω(uu)(zz)+(a2μ2χ21u4+wμ1a1χ232)Ω(ww)(vv)ε(Ω(uu)2+Ω(vv)2+Ω(ww)2+Ω(zz)2). (3.10)

    To confirm that the assumptions of Lemma 3.2 are satisfied, let

    a11:=μ1μ2a2,a22:=a2μ2χ21u4+wμ1a1χ232,a33:=μ1μ2a1,a44:=a1μ1χ22w2,
    a13:=2μ1μ2a1a2,a14:=a1μ1χ22w2,a23:=a2μ2χ21u4wμ1a1χ232

    and

    x1:=uu,x2:=vv,x3:=ww,x4:=zz.

    Since μ1,μ2,a2,χ1 and u are positive constants, we obtain

    a11=μ1μ2a2>0,a22=a2μ2χ21u4+wμ1a1χ232>0. (3.11)

    Thanks to (1.6) we have

    a11a22a33a22a2134a11a2234=(μ21μ22a1a2μ21μ22a21a22)(a2μ2χ21u4+wμ1a1χ232)μ1μ2a24(a2μ2χ21u4+wμ1a1χ232)2=μ1μ2a2(a2μ2χ21u4+wμ1a1χ232)(μ1μ2a1(1a1a2)a2μ2χ21u16wμ1a1χ238)>0. (3.12)

    Using (1.7), we obtain

    a11a22a33a44a22a44a2134a11a44a2234a22a33a2144+a214a22316=μ21μ2a1a2χ22w2(a2μ2χ21u4+wμ1a1χ232)(μ1μ2a1(1a1a2)a2μ2χ21u16wμ1a1χ238)+μ21a21χ42w216(a2μ2χ21u4+wμ1a1χ232)(a2μ2χ21u16+wμ1a1χ238μ1μ2a1)=μ21a1χ22w2(a2μ2χ21u4+wμ1a1χ232){μ1μ22a1a2(1a1a2)a22μ22χ21u16μ1μ2a1a2χ23w8+μ2a1a2χ21χ22uw128+μ1a21χ22χ23w64μ1μ2a21χ22w8}>0. (3.13)

    Hence, combining (3.11)-(3.13) and Lemma 3.2 we have (3.10), which concludes the proof of Lemma 3.3.

    Next, we will establish convergence rates for the solution to the model (1.1).

    Lemma 3.4. Under the assumptions of Theorem 1.2, then there exist C,μ>0 such that

    uuL(Ω)+vvL(Ω)+wwL(Ω)+zzL(Ω)Ceμt

    for all t>0.

    Proof. The proof is similar to the corresponding proofs of [20,Lemmas 3.5 and 3.6] or [38,Lemmas 3.4 and 3.5], for the convenience of the readers, we give a sketch the proof. We divide the proof into two steps.

    Step 1. We derive the large time behavior of solution to (1.1).

    Integrating the second part of (3.2) and using (3.3) to see that

    1Ω(uu)2+1Ω(vv)2+1Ω(ww)2+1Ω(zz)2E1(1)ε.

    According to Lemma 3.1 and (1.5), we obtain

    uuL(Ω)+vvL(Ω)+wwL(Ω)+zzL(Ω)0ast. (3.14)

    Step 2. We derive the convergence rates of solution to (1.1).

    Let H(s):=sulns for s>0, by L'Hôpital's theorem to see that

    limsuH(s)H(u)(su)2=limsuH(s)2(su)=12u. (3.15)

    Hence, using (3.14) and (3.15) and the definitions E1(t) in (3.1), there exists t1>0 such that

    min{a2μ24u,a1μ14w}(Ω(uu)2+Ω(ww)2)E1(t)max{a2μ2u,a1μ1w}(Ω(uu)2+Ω(ww)2) (3.16)

    for all . Using (3.16) and (3.2) and the definitions in (3.3), one can find some such that

    which implies

    (3.17)

    with some . Then by the Gagliardo-Nirenberg inequality with some

    (3.18)

    we can find some such that

    (3.19)

    Then by the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) enables us to obtain

    (3.20)

    with some . Hence, combining (3.19) and (3.20) we can complete the proof of this lemma.

    When , we will prove the competitive exclusion will occur based on the following energy functional

    (3.21)

    where satisfies .

    Lemma 3.5. Under the assumptions of Theorem 1.3, then there exists such that

    (3.22)

    where is defined by (3.21) and

    Proof. Using Taylor's formula to see that for all (for more details, see [1,Lemma 3.2]), then using is nonnegative, which implies for all . By the straightforward calculations we infer

    (3.23)

    Using the third equation in (1.1), we obtain

    (3.24)

    Combining (3.23) and (3.24), we have

    (3.25)

    Using Young's inequality we infer that

    (3.26)

    Multiplying the second equation in (1.1) with , we have

    (3.27)

    Testing the fourth equation in (1.1) with , we see that

    (3.28)

    Substituting (3.26)-(3.28) into (3.25), we have

    (3.29)

    In order to prove (3.22), we will show that there exists such that

    (3.30)

    Then using the same argument as in the proof of Lemma 3.4, we put

    and

    Since are positive constants, we obtain

    (3.31)

    Thanks to the second part of (1.9) we have

    (3.32)

    By the first part of (1.9), we infer that

    (3.33)

    Then collecting (3.31)-(3.33) we have (3.30), which implies the end of the proof.

    According ideas come from [20,Lemmas 3.8,3.9 and 3.10] or [38,Lemmas 3.7 and 3.8], we shall give the convergence rates for the case .

    Lemma 3.6. Under the assumptions of Theorem 1.3.

    (i) Let and , then there exist and such that

    (ii) Suppose that and , then there exist and such that

    Proof. (i) This part can be proved by a similar proof in Lemma 3.4.

    (ii) Since , then the second part in (3.22) can be rewritten as

    (3.34)

    Similar to (3.14), integrating (3.34) and using Lemma 3.1 enable us to obtain

    (3.35)

    Hence, by the definition of in (3.21), using L'Hôpital's theorem and Hölder's inequality, we can find such that

    (3.36)

    Combining (3.34) and (3.36), using the definition of in Lemma 3.5, one can find some such that

    which implies there exists such that

    (3.37)

    By L'Hôpital's theorem and (3.37), we can find such that

    By (1.5) and the Gagliardo-Nirenberg inequality, we can find such that

    Using the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) implies

    with some . Hence, the proof of Lemma 3.6 is completed.

    Proof of Theorem 1.2. The proof of Theorem 1.2 follows from Lemma 3.4.

    Proof of Theorem 1.3. We only need to use Lemma 3.6 to obtain the proof of Theorem 1.3.

    The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work.



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