
Citation: Xin Nian, Yasuhiro Nagai, Cameron Jeffers, Kara N. Maxwell, Hongtao Zhang. Dietary influence on estrogens and cytokines in breast cancer[J]. AIMS Molecular Science, 2017, 4(3): 252-270. doi: 10.3934/molsci.2017.3.252
[1] | Jinyu Wei, Bin Liu . Global dynamics of a Lotka-Volterra competition-diffusion-advection system for small diffusion rates in heterogenous environment. Mathematical Biosciences and Engineering, 2021, 18(1): 564-582. doi: 10.3934/mbe.2021031 |
[2] | Robert Stephen Cantrell, Chris Cosner, Yuan Lou . Evolution of dispersal and the ideal free distribution. Mathematical Biosciences and Engineering, 2010, 7(1): 17-36. doi: 10.3934/mbe.2010.7.17 |
[3] | Ling Xue, Sitong Chen, Xinmiao Rong . Dynamics of competition model between two plants based on stoichiometry. Mathematical Biosciences and Engineering, 2023, 20(10): 18888-18915. doi: 10.3934/mbe.2023836 |
[4] | Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin . Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068 |
[5] | Azmy S. Ackleh, Keng Deng, Yixiang Wu . Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences and Engineering, 2016, 13(1): 1-18. doi: 10.3934/mbe.2016.13.1 |
[6] | Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135 |
[7] | Meng Zhao, Wan-Tong Li, Yang Zhang . Dynamics of an epidemic model with advection and free boundaries. Mathematical Biosciences and Engineering, 2019, 16(5): 5991-6014. doi: 10.3934/mbe.2019300 |
[8] | Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari . Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences and Engineering, 2016, 13(4): 631-652. doi: 10.3934/mbe.2016012 |
[9] | Honghua Bin, Daifeng Duan, Junjie Wei . Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay. Mathematical Biosciences and Engineering, 2023, 20(7): 12194-12210. doi: 10.3934/mbe.2023543 |
[10] | Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283 |
Biological dispersal means that organisms move from one location to another within a habitat. This phenomenon has an important impact on population dynamics, disease and distribution of species [1,2,3,4]. It is well known that the interactions between dispersal and spatial heterogeneity could create very interesting phenomena. Moreover, a number of ecologists and evolutionary biologists have been interested in this issue for many years.
Over the last few decades, researchers from both biology and mathematics have used some specific mathematical models to describe population dynamics in spatial ecology and evolution. On one hand, taking into account the effects of random perturbations, some researchers investigated some models which described more realistically ecosystems through stochastic approaches, see [5,6,7,8,9]. On the other hand, some researchers considered the impacts of the random diffusion on the population dynamics and used the reaction-diffusion equations to model population dynamics. Among these models, the two species Lotka-Volterra competition-diffusion system perhaps is the most salient example. The following classical Lotka-Volterra competition-diffusion system
{ut=d1Δu+u[m(x)−u−bv],inΩ×R+,vt=d2Δv+v[m(x)−cu−v],inΩ×R+,∂νu=∂νv=0,on∂Ω×R+,u(x,0)=u0,v(x,0)=v0,inΩ, | (1.1) |
models two competing species. Here u(x,t) and v(x,t) represent the densities of two species at location x and time t. d1 and d2 are random diffusion rates of species u and v, respectively, which are therefore assumed to be non-negative. The habitat Ω is a bounded domain in RN with smooth boundary ∂Ω. ∂ν=ν⋅∇, where ν denotes the unit outer normal vector on ∂Ω and the no-flux boundary condition means that no individual can move in or out through the boundary of the habitat. m represents their common intrinsic growth rate which also reflects the environmental richness of the resources at location x. b and c represent inter-specific competition coefficients with (b,c)∈(0,1)×(0,1) and both intra-specific competition coefficients are normalized to 1.
Let g belong to Cγ(¯Ω) for some γ satisfying 0<γ<1 with ∫Ωgdx≥0 and g≢0. It is well known that there is a unique positive solution to the following problem
{dΔϖ+ϖ(g(x)−ϖ)=0,inΩ,∂νϖ=0,on∂Ω, | (1.2) |
which is denoted by ϖd,g (see, e.g., [1]). This implies that system (1.1) admits two semi-trivial steady states, denoted by (ωd1,0) and (0,ωd2), for every d1>0 and d2>0.
For system (1.1), when b=c=1, one well-known and widely accepted result is due to Hastings [10]. It was shown by Hastings [10] that the species with slower diffusion rate has advantage against its competitor with larger diffusion rate, regardless of the initial values, which is contrary to the case of m≡ constant. When m≡ constant, there is a compact global attractor consisting of a continuum of steady states {((1−s)m,sm)|s∈[0,1]} connecting the two semi-trivial steady states to system (1.1) for every d1,d2>0.
In an attempt to understand this phenomenon, Lou [11] adopted the weak competition approach to study the system (1.1). Lou [11] verified that there exists a critical value for competition coefficient c∗ in (0,1) such that (ⅰ) for every c less than this critical value, if d1,d2>0, the steady state (ϖd1,0) is unstable; (ⅱ) for every c satisfying c∗<c<1, there exists ¯d1>0 and ¯d2>0 such that (ϖd1,0) is unstable. Meanwhile, when c is in (c∗,1), there exists a constant b∗ in (0,1] such that for some (d1,d2) in (0,+∞)×(0,+∞), (ϖd1,0) is globally asymptotically stable provided that b satisfies 0<b≤b∗. He and Ni [12] made a great breakthrough on the estimate of linear stability of any coexistence stead state, and based on this, they provided a complete classification on all possible long time dynamical behaviors of system (1.1). We refer to [13,14,15,16] for more investigations.
In addition to random dispersal, it is quite reasonable to suppose that species could take directed movements due to certain external environment force or self-propelling in some environments. On one hand, species will move downstream due to the unidirectional water flow in a river [17,18,19]. On the other hand, the individuals are very smart so that they can sense and follow gradients in resource distribution [20,21,22,23] which can be modeled by the following competition-diffusion-advection system
{ut=∇⋅(d1∇u−αu∇m1)+u[m1(x)−u−bv],inΩ×R+,vt=∇⋅(d2∇v−βv∇m2)+v[m2(x)−cu−v],inΩ×R+,(d1∇u−αu∇m1)⋅ν=(d2∇v−βv∇m2)⋅ν=0,on∂Ω×R+,u(x,0)=u0,v(x,0)=v0,inΩ, | (1.3) |
where d1 and d2 are random diffusion rates of species u and v, respectively. α and β measure the speed of movement upward along the gradient of resources. m1 and m2 are intrinsic growth rates of u and v, respectively. The boundary condition means that the environment is isolated. This system has been frequently used as a standard model to study the evolution of conditional dispersal and a number of related works have been done, see, e.g., [24,25,26] for α,β>0, [27,28,29] for α>0=β when m1=m2 and b=c=1. For more investigations, we refer to [30,31,32,33,34].
Recently, Wang [21] studied system (1.3), and he established Σu and Σv (which can be seen in the following definition) for m1 and m2 are non-constant under d1,d2,α and β satisfying certain conditions and (b,c) in (0,1]×(0,1]. Xu and Jiang [22] investigated the dynamics of system (1.3) on the condition that m1 is not a constant, m2=¯m is a constant (∇¯m≡0) and b=c=1. In this case, advection rate β has no effect on system (1.3). They showed that there exists a critical value for the diffusion rate d∗2 in (0,+∞), and their results can be interpreted biologically in the following statements:
(i) The species u can always invade when the species v is rare for every d1,d2>0, and α≥0.
(ii) For α/d1≤1/max¯Ωm1, the species v can invade when the species u is rare, and the two species will coexist if d2 is larger than d∗2.
(iii) If m1 is positive in ¯Ω and α/d1≥1/min¯Ωm1, then the two species always coexist for every d2>0. In this case, the species u has relatively strong advection, and then it leaves sufficient habitat for v to evolve.
How does the inter-specific competition coefficients b and c affect the dynamics of system (1.3) when m1 is not a constant and m2 is constant? What values of the inter-specific competition coefficient b and c are favorable for the coexistence of the two species? In this paper, we shall discuss these topics in detail. Throughout this paper, the basic assumption on b and c includes part of the range bc≤1, which contains the weak competition case (0<b,c<1) as well as b=c=1. At first, we make the following assumption.
(M)m1 belongs to C2+γ(¯Ω) for some γ satisfying 0<γ<1, and m2=¯m is a non-negative constant, m1≢m2. Moreover,
¯m=1|Ω|∫Ωm1dx>0. |
Assumption (M), biologically, implies that the total resources for two competitors are fixed at exactly the same level. Due to the assumption that m2 is a non-negative constant, it holds ∇m2=0, then the advective rate of species v (i.e., β) has no effect on the dynamics of system (1.3). Moreover, throughout this paper, we assume that d1,d2>0,d1≠d2 and α>0.
Besides the assumption (M), we define ω:=α/d1.
Then system (1.3) becomes
{ut=d1∇⋅(∇u−ωu∇m1)+u[m1(x)−u−bv],inΩ×R+,vt=d2Δv+v[¯m−cu−v],inΩ×R+,(∇u−ωu∇m1)⋅ν=∂νv=0,on∂Ω×R+,u(x,0)=u0,v(x,0)=v0,inΩ. | (1.4) |
We assume that the initial data u0,v0 of system (1.4) are non-negative and not identically zero, then by maximum principle [35], we can obtain u>0,v>0. Under the assumption (M), for all d1,d2>0 and α>0, there exist two semi-trivial steady states to system (1.4) (see [1,36]), denoted by (ˉud1,ω,m1,0) and (0,¯m) respectively, where ˉud1,ω,m1 is the unique positive solution ([37]) of
{d1∇⋅(∇ˉu−ωˉu∇m1)+ˉu(m1−ˉu)=0,inΩ,(∇ˉu−ωˉu∇m1)⋅ν=0,on∂Ω, | (1.5) |
and ¯m is the unique positive solution of
{d2Δˉv+ˉv(¯m−ˉv)=0,inΩ,∂νˉv=0,on∂Ω. | (1.6) |
To study the dynamics of system (1.4), we should study the stability of semi-trivial steady states (ˉu,0) and (0,¯m). The stability of (ˉud1,ω,m1,0) is determined by the principal eigenvalue of the following linear problem
{d2Δψ+(¯m−cˉud1,ω,m1)ψ+λ1ψ=0,inΩ,∂νψ=0,on∂Ω, | (1.7) |
which is denoted by λ1(d2,¯m−cˉud1,ω,m1). The stability of (0,¯m) is determined by the principal eigenvalue of the following linear problem
{d1∇⋅(∇ψ−ωψ∇m1)+ψ(m1−b¯m)+λ1ψ=0,inΩ,(∇ψ−ωψ∇m)⋅ν=0,on∂Ω, | (1.8) |
which is denoted by λ1(d1,ω,m1−b¯m).
Moreover, (ˉu,0) is linearly stable if λ1(d2,¯m−cˉu)>0 and it is unstable if λ1(d2,¯m−cˉu)<0. Similarly, (0,¯m) is linearly stable if λ1(d1,ω,m1−b¯m)>0 and it is unstable if λ1(d1,ω,m1−b¯m)<0.
To describe the first result, according to [12], for every d1>0,d2>0 and ω≤1/max¯Ωm1, we first introduce several notations:
Σu:={(d1,d2)∈Γ:(ˉud1,ω,m1,0) is linearly stable},
Σv:={(d1,d2)∈Γ:(0,¯m) is linearly stable},
Σ−:={(d1,d2)∈Γ: both (ˉud1,ω,m1,0) and (0,¯m) are linearly unstable},
where Γ:=R+×R+andR+=(0,∞).
Define
{Lu:=infd1>0,0<ω≤1max¯Ωm1¯m¯ˉud1,ω,m1∈[0,+∞),Su:=supd1>0,0<ω≤1max¯Ωm1sup¯Ω¯mˉud1,ω,m1∈(0,+∞], | (1.9) |
and
Ξ:={(b,c)|b,c>0andbc≤1}∪{(b,c)|0<c≤1Su}∩{(b,c)|0<b≤1}. |
In particular, if (M) holds, then
0<Lu<¯m¯m1=1≤Su<∞. |
To characterize the set Σu in terms of c>0, for each c>0 and ω≤1/max¯Ωm1, we define
Ic:={d1:∫Ω(¯m−cˉud1,ω,m1)dx<0}:=I0c⋃I1c, | (1.10) |
where
{I0c:={d1:¯m−cˉud1,ω,m1≤(≢)on¯Ω},I1c:={d1∈Ic:sup¯Ω(¯m−cˉud1,ω,m1)>0}. | (1.11) |
In fact, we have the following equivalent descriptions:
{Σu:={(d1,d2)∈Γ:λ1(d2,¯m−cˉud1,ω,m1)>0},Σv:={(d1,d2)∈Γ:λ1(d1,ω,m1−b¯m)>0},Σ−:={(d1,d2)∈Γ:λ1(d2,¯m−cˉud1,ω,m1)<0andλ1(d1,ω,m1−b¯m)<0}. | (1.12) |
The first result is as follows.
Theorem 1.1. Suppose that (M) holds and suppose ω≤1/max¯Ωm1. Let Lu and Su be defined as in Eq (1.9), and let (b,c) be in Ξ. Then we have Σv=∅, and the following results hold:
(i)
Σu={∅,0<c≤Lu,{(d1,d2):d1∈Ic,d2>d∗2(d1)>0},Lu<c<Su,Γ,c≥Su, | (1.13) |
where d∗2(d1,ω) is defined as follows:
d∗2(d1)={0,d1∈I0c,1μ1(d2,¯m−cˉud1,ω,m1),d1∈I1c, |
Ic,I0c and I1c are defined in Eqs (1.10) and (1.11);
(ii) there exists at least one stable positive coexistence steady state for system (1.4) if (d1,d2)∈Σ−, where
Σ−={Γ,c≤Lu,∅,c≥Su,{(d1,d2):I1c,d2<1μ1(d2,¯m−cˉud1,ω,m1)},Lu<c<Su. | (1.14) |
Remark 1.1. In Theorem 1.1, we consider the case of ω≤1/max¯Ωm1. We divides the region of bc-plane discussed in Theorem 1.1 into three parts. Species v can invade the space occupied by the species u so that would coexist with species u when c≤Lu and 0<b≤1. When c takes on the intermediate value, that is Lu<c<Su, whether the two species will coexist or not is dependent on the values of random diffusion rates d1,d2 and the ratio ω. This result generalizes (Theorem 1.2 [22]) in which the authors discussed the case where the inter-specific competition coefficients b and c were 1. If c≥Su, then (ˉud1,ω,m1,0) is linearly stable. These results are similar to the non-advection case (Theorem 3.3 [12]). It means that in an advective environment, as long as the ratio of advection rate to random diffusion rate is appropriately small, advection contributes to dispersal much less than random diffusion, and a sufficient amount of diffusive movement can counterbalance the advection.
When ω≥1/min¯Ωm1, the following result holds.
Theorem 1.2. Suppose that (M) holds, and m1 is positive for all x in ¯Ω. If ω≥1/min¯Ωm1, then there exists at least one stable positive coexistence steady state for system (1.4) provided that (b,c)∈(0,1]×(0,1].
Remark 1.2. Theorem 1.2 implies that, when α/d1≤1/max¯Ωm1, for every (b,c)∈(0,1]×(0,1] and d2>0, species v can invade the space occupied by the species u, thereby the two species could coexist. This result generalizes (Theorem 1.5 [22]) for b=c=1 case. From a biological point of view, if the ratio of advection rate α to random diffusion rate d1 of species u is appropriately large, advection plays a dominant role in dispersal for species u. Species v adopts random diffusion strategy. Due to the advection, the species u concentrates at somewhere richer in resources, therefore, there leaves sufficient resources for species v to survival. Thereby, the two species coexist.
Figure 1 illustrates the region which is part of bc-plane considered in Theorems 1.1 and 1.2.
The rest of this paper is organized as follows. In Section 2, we present some preliminary results which will be used in verifying our results. Section 3 is devoted to establishing our main results. We give a short discussion in Section 4.
Since system (1.4) generates a monotone dynamical system, the potential population dynamics, to a large extent, can be determined by the qualitative properties of its steady states. For a monotone dynamical system, we have the following conclusions (see, e.g., (Proposition 9.1 and Theorem 9.2 [38]):
(I). If both semi-trivial steady states of a monotone dynamical system are unstable, then there is at least one locally stable coexistence steady state;
(II). If a monotone dynamical system has no coexistence steady state, then one of the semi-trivial steady state is unstable and the other is globally asymptotically stable.
Let μ1(h) denote the unique nonzero principal eigenvalue of
{Δφ+μh(x)φ=0,inΩ,∂νφ=0,on∂Ω, | (2.1) |
where h≢ constant and could change sign.
The following lemma collects some important properties of μ1(h). The proof can be found in [39,40,41].
Lemma 2.1. The problem (2.1) admits a nonzero principal eigenvalue μ1=μ1(h) if and only if ∫Ωhdx≠0 and h changes sign. More precisely, if h changes sign, then
(i)∫Ωhdx=0⇔0 is the only principal eigenvalue;
(ii)∫Ωhdx>0⇔μ1(h)<0;
(iii)∫Ωhdx<0⇔μ1(h)>0;
(iv)μ1(h1)>μ1(h2) if h1≤h2 a.e., and h1 and h2 both change sign;
(v)μ1(h) is continuous in h; more precisely, μ1(hl)→h in L∞(Ω).
In order to analyze the principal eigenvalue of problem (1.7), for convenience, we consider the following more general form of eigenvalue problem:
{dΔψ+h(x)ψ+λ1ψ=0,inΩ,∂νψ=0,on∂Ω. | (2.2) |
Denote the principal eigenvalue of problem (2.2) by λ1(d,h). By the variational approach, λ1(d,h) can be characterized by
λ1(d,h)=infψ∈H1(Ω)∖0∫Ωd|∇ψ|2dx−hψ2dx∫Ωψ2dx. |
The following lemma collects some important properties of λ1(d,h) as well as the connection between λ1(d,h) and μ1(h) (see [40]).
Lemma 2.2. The first eigenvalue λ1(d,h) of problem (2.2) depends smoothly on d>0 and continuously on h∈L∞(Ω). Moreover, it has the following properties:
(i)∫Ωhdx≥0⇒λ1(d,h)<0 for all d>0;
(ii)
∫Ωhdx>0⇒{λ1(d,h)<0,foralld<1μ1(h),λ1(d,h)=0,foralld=1μ1(h),λ1(d,h)>0,foralld>1μ1(h); | (2.3) |
(iii)λ1(d,h)<λ1(d,k) if h≥k and h≢k. In particular λ1(d,h)>0 if h≥(≢)0.
To study the stability of the semi-trivial steady state (0,¯m), it is needed to introduce the following linear eigenvalue problem
{d∇⋅(∇ψ−ωψ∇m)+h(x)ψ+λ1ψ=0,inΩ,(∇ψ−ωψ∇m)⋅ν=0,on∂Ω. | (2.4) |
Denote the principal eigenvalue of problem (2.4) by λ1(d,ω,h). Then λ1(d,ω,h) can be expressed by the following variational formula (see, e.g., [1])
λ1(d,ω,h)=infψ∈H1(Ω)∖0∫Ωdeωm|∇ψ|2dx−∫Ωeωmhψ2dx∫Ωeωmψ2dx, |
where h∈C(¯Ω) is not a constant. Then we obtain the following result [1].
Lemma 2.3. Let λ1(d,ω,h) be the principal eigenvalue of problem (2.4). Suppose that h is a continuous function on ¯Ω and h is not a constant. Then we have the following result
h1≥h2⇒λ1(d,ω,h1)≤λ1(d,ω,h2), |
and the equality holds only if h1≡h2.
The proofs of our results depend heavily on the following result.
Lemma 2.4. Assume that m∈C2(¯Ω), m≢constant and ∫Ωmdx≥0 (m may change sign). Then there exists a unique positive solution to the steady state problem
{∇⋅(d∇ˉu−αˉu∇m)+ˉu(m−ˉu)=0,inΩ,(d∇ˉu−αˉu∇m)⋅ν=0,on∂Ω, | (2.5) |
which is denoted by ˉu. Moreover,
(i) if α/d≤1/max¯Ωm, then ∫Ωˉudx>∫Ωmdx;
(ii) if m>0 in ¯Ω, and α/d≥1/min¯Ωm, then ∫Ωˉudx<∫Ωmdx.
Proof. The existence and uniqueness of positive solution ˉu is well-known [36]. For the proof of (ⅰ) see Lemma 3.1 in [22]. For the proof of (ⅱ), see Lemma 4.1 in [24]. For completeness, we provide a proof of (ⅰ).
If α/d≤1/max¯Ωm, set
η=e−αdmˉu. |
Then η satisfies
{d∇⋅[eαdm∇η]+ˉu(m−ˉu)=0,inΩ,∂νη=0,on∂Ω. | (2.6) |
Let η(x∗)=max¯Ωη. Then we can always choose x∗∈Ω. If x∗∈∂Ω, then η(x)<η(x∗) for all x∈Ω. By the Hopf boundary lemma ([42]), ∂η∂ν(x∗)>0. This contradicts the boundary condition ∂νη=0 on ∂Ω, which implies that x∗ can be chosen in Ω. For such x∗∈Ω, it holds that Δη(x∗)≤0,∇η(x∗)=0. Thus, by the equation of η, we have ˉu(x∗)≤m(x∗). Therefore,
max¯Ωη=e−αdm(x∗)ˉu(x∗)≤e−αdm(x∗)m(x∗)≤dαe, | (2.7) |
where the last inequality follows from ye−αdy≤d/αe. It follows that η(x)≤d/αe for every x∈¯Ω. Then due to α/d≤1/max¯Ωm, we have
ˉu=eαdm(x)η(x)≤eαdmax¯Ωmmax¯Ωη≤dα,forallx∈¯Ω. | (2.8) |
Define a function f(y)=ye−αdy, y∈[0,d/α]. Since f′(y)>0 for y∈[0,d/α], f has an inverse function, denoted by s. It is obvious that s is defined in [0,d/αe], and its range is [0,d/α]. Hence, for each x∈¯Ω, f(ˉu(x)) and s(η(x)) is well-defined by Eqs (2.7) and (2.8).
Dividing system (2.6) by s(η) and integrating in Ω, we obtain
d∫Ωeαdms′(η)|∇η|2s2(η)dx+∫Ωˉus(η)(m−ˉu)dx=0, |
which can be written as
∫Ω(m−ˉu)dx=d∫Ωeαdms′(η)|∇η|2s2(η)dx+∫Ωˉu−s(η)s(η)(m−ˉu)dx. |
Next we prove the following inequality
[ˉu−s(η)](m−ˉu)≥0,inΩ. | (2.9) |
If ˉu(x)<m(x), we have f(ˉu(x))=ˉu(x)e−αdˉu(x)>ˉu(x)e−αdm(x)=η(x), then by the strictly monotone increasing property of s, it holds that ˉu(x)>s(η(x)), and then Eq (2.9) follows.
If ˉu(x)≥m(x), we have f(ˉu(x))=ˉu(x)e−αdˉu(x)≤ˉu(x)e−αdm(x)=η(x). It follows that ˉu(x)≤s(η(x)), and Eq (2.9) holds either in this case.
Hence, we obtain ∫Ωˉudx>∫Ωmdx if α/d≤1/max¯Ωm.
Firstly, we prove the unstability of (0,¯m) for every d1,d2>0, and ω>0 by employing a method similar to that used in [22].
Lemma 3.1. Suppose that (M) holds. Then for every d1,d2>0 and ω>0, (0,¯m) is unstable provided b in (0,1] and c>0.
Proof. Let λ1(d1,ω,m1−b¯m) be the principal eigenvalue of problem (1.8) and let ψ>0 be the corresponding eigenfunction. Then
{d1∇⋅(∇ψ−ωψ∇m1)+ψ(m1−b¯m)+λ1ψ=0,inΩ,(∇ψ−ωψ∇m1)⋅ν=0,on∂Ω. | (3.1) |
Consider
{d1∇⋅(∇ψ−ωψ∇m1)+ψ(m1−¯m)+λ1ψ=0,inΩ,(∇ψ−ωψ∇m1)⋅ν=0,on∂Ω. | (3.2) |
Set ϕ=e−ωm1ψ. Then ϕ satisfies
{d1∇⋅(eωm1∇ψ)+eωm1(m1−¯m)ψ+λ1eωm1ψ=0,inΩ,∂νψ=0,on∂Ω. | (3.3) |
Dividing Eq (3.3) by ψ and integrating in Ω, we have
d1∫Ωeωm1|∇ψ|2ψ2dx+∫Ω(m1−¯m)eωm1dx+λ1∫Ωeωm1dx=0. | (3.4) |
Define
Ω+={x∈Ω|m1(x)≥¯m},Ω−={x∈Ω|m1(x)<¯m}, |
then Ω=Ω+⋃Ω−, and
∫Ω(m1−¯m)eωm1dx=∫Ω+(m1−¯m)eωm1dx+∫Ω−(m1−¯m)eωm1dx>∫Ω+(m1−¯m)eω¯mdx+∫Ω−(m1−¯m)eω¯mdx=eω¯m[∫Ω+(m1−¯m)dx+∫Ω−(m1−¯m)dx]=eω¯m∫Ω(m1−¯m)dx=0. | (3.5) |
It follows from Eqs (3.4) and (3.5) that λ1(d1,ω,m1−¯m)<0. In view of Lemma 2.3, we derive
λ1(d1,ω,m1−b¯m)≤λ1(d1,ω,m1−¯m)<0, |
which implies that (0,¯m) is linearly unstable.
Proof of Theorem 1.1. According to Lemma 3.1, it holds that Σv=∅. Next we give the proof of Eq (1.13) in two steps.
Step1.(d1,d2)∈Σu indicates that d1∈Ic.
From Eq (1.12), we know
Σu:={(d1,d2)∈Γ:λ1(d2,¯m−cˉud1,ω,m1)>0}. |
Suppose that d1∉Ic, where Ic is defined in Eq (1.10). Then ∫Ω(¯m−cˉud1,ω,m1)dx≥0. By Lemma 2.2 (ⅰ), λ1(d2,¯m−cˉud1,ω,m1)≤0 for all d2>0, i.e., (d1,d2)∉Σu. Hence (d1,d2)∈Σu implies that d1∈Ic.
Step2. Characterize the set Ic for all c>0 in detail.
If c≤Lu, by the definition of Lu in Eq (1.9), ∫Ω(¯m−cˉud1,ω,m1)dx≥0 for all d1>0. Hence Ic=∅ and Σu=∅. Moreover,
I1c≠∅ifandonlyifLu<c<Su. | (3.6) |
Indeed, if d′1∈I1c≠∅, then Lu<c and for some x0∈¯Ω, it follows that
c<¯mˉud′1,ω′,m1(x0)≤Su, |
where ω′=α/d′1. Hence I1c≠∅ gives rise to Lu<c<Su.
On the other hand, if Lu<c<Su, then there exists some d″1>0(ω″=α/d″1) and y0∈¯Ω such that
¯m−cˉud″1,ω″,m1(y0)>0and∫Ω(¯m−cˉud″1,ω″,m1)dx<0, |
i.e., d″1∈I1c≠∅. This finishes the proof of Eq (3.6).
We now claim that Ic admits the following decomposition:
{Ic=∅,ifc≤Lu,Ic=(I0c∪I1c)⊂R+,ifLu<c<Su,Ic=I0c=R+,ifc≥Su. | (3.7) |
To finish the proof of Eq (3.7), it suffices to show that if Su<+∞ and c≥Su, then Ic=I0c=R+. By the definition of Su in Eq (1.9), we deduce that
¯m−cˉud1,ω,m1≤0,inΩ. |
Hence to show that Ic=I0c=R+, it suffices to show that ¯m−cˉud1,ω,m1≢0 for c≥Su. Since ¯m is a constant, this is obviously true if c>Su. This finishes the proof of Eq (3.7).
Therefore
c≥Su⇒Ic=I0c⇒¯m−cˉud1,ω,m1≤≢0⇒λ1(d2,¯m−cˉud1,ω,m1)>0, | (3.8) |
i.e., Σu=Γ.
Now assume that Lu<c<Su and d1∈Ic. If d1∈I0c, by Lemma 2.2 (ⅲ), λ1(d2,¯m−cˉud1,ω,m1)>0, then for any d1∈I0c, (ˉud1,ω,m1,0) is linearly stable. If d1∈I1c, then λ1(d2,¯m−cˉud1,ω,m1)>0 for all d2>1/μ1(d2,¯m−cˉud1,ω,m1) by Lemma 2.2 (ⅱ). Thus after defining
d∗2(d1)={0,d1∈I0c,1μ1(d2,¯m−cˉud1,ω,m1),d1∈I1c, |
we obtain that, when Lu<c<Su, (d1,d2)∈Σu if and only if d1∈I1c,d2>1/μ1(d2,¯m−cˉud1,ω,m1) or d1∈I0c,d2>0.
Finally, we prove Eq (1.14).
If c≤Lu, then ∫Ω(¯m−cˉud1,ω,m1)dx≥0 for all d1>0. Hence by Lemma 2.2 (i), λ1(d2,¯m−cˉud1,ω,m1)≤0 for all (d1,d2)∈Γ and λ1(d2,¯m−cˉud1,ω,m1)=0 if and only if c=Lu and ¯m≡Luˉud1,ω,m1. Due to the fact that ¯m is a constant, it is easy to see that the equality ¯m≡Luˉud1,ω,m1 is not true, which implies that for all (d1,d2)∈Γ,λ1(d2,¯m−cˉud1,ω,m1)<0, i.e., (ˉud1,ω,m1,0) is unstable. If c≥Su, by Eq (3.8), λ1(d2,¯m−cˉud1,ω,m1)>0, then for all (d1,d2)∈Γ, (ˉud1,ω,m1,0) is linearly stable. If Lu<c<Su, for d1∈I0c, λ1(d2,¯m−cˉud1,ω,m1)>0 for all d2>0. For d1∈I1c, λ1(d2,¯m−cˉud1,ω,m1)<0 for all d2<1/μ1(d2,¯m−cˉud1,ω,m1). Then
Σ−={Γ,c≤Lu,∅,c≥Su,{(d1,d2):d1∈I1c,d2<1μ1(d2,¯m−cˉud1,ω,m1)},Lu<c<Su. |
By the theory of monotone dynamical system, we derive that system (1.4) has at least one coexistence steady state for Σ−. This finishes the proof of Theorem 1.1.
Proof of Theorem 1.2. If (M) holds, m1>0 in ¯Ω, and ω≥1/min¯Ωm1, then by Lemma 2.4, the unique positive solution ˉud1,ω,m1 of Eq (1.9) satisfies ∫Ωˉud1,ω,m1dx<∫Ωm1dx.
Indeed, by the theory of monotone dynamical system, we only need to prove that (ˉud1,ω,m1,0) is unstable. It is sufficient to show that the principal eigenvalue λ1 of the problem
{d2Δψ+(¯m−cˉud1,ω,m1)ψ+λ1ψ=0,inΩ,∂νψ=0,on∂Ω, | (3.9) |
is negative.
Consider the following principal eigenvalue problem
{d2Δψ+(¯m−ˉud1,ω,m1)ψ+λ1ψ=0,inΩ,∂νψ=0,on∂Ω. | (3.10) |
Dividing system (3.10) by ψ, integrating in Ω, we obtain
∫Ω(ˉud1,ω,m1−¯m)dx−d2∫Ω|∇ψ|2ψ2dx=∫Ωλ1dx. |
Since ω≥1/min¯Ωm1(x), it holds that ∫Ω(ˉud1,ω,m1−¯m)dx=∫Ω(ˉud1,ω,m1−m1)dx<0. Then λ1(d2,¯m−ˉu)<0. From Lemma 2.2 (ⅲ)
λ1(d2,¯m−cˉud1,ω,m1)≤λ1(d2,¯m−ˉu)<0, |
for c∈(0,1].
Then (ˉud1,ω,m1,0) is unstable. From the proof of Lemma 3.1, we know that (0,¯m) is unstable for (b,c)∈(0,1]×(0,1] and by the theory of monotone system, system (1.4) has at least one stable positive coexistence steady state.
We study the dynamics of a Lotka-Volterra competition-diffusion-advection model. The two competing species u and v adopt the dispersal strategies of a combination of random dispersal and biased movement upward along the resource gradient ∇mi (i = 1, 2) (the species are very smart and they will move toward regions richer in resources). In this paper, it is assumed that the two species have the same level of total resources but the species v adopts the homogeneous distribution and its competitor, the species u, adopts heterogenous distribution. This means that ∇m2≡0 which implies that the species v only experiences random dispersal but has no biased movement.
Based on the above assumptions, we mainly study the effect of the inter-specific competition coefficients on the dynamics of system (1.4). From Lemma 3.1, (0,¯m) is unstable for each one of the following conditions: d1,d2>0, α≥0, and 0<b≤1,0<c. This phenomenon indicates that the homogeneous distribution is disadvantageous for v, therefore the species u can always invade the species v. To study the coexistence regime for system (1.4), we discuss two cases. For the case of α/d1≥1/min¯Ωm1, the species u and v will coexist for every d2>0 and (b,c)∈(0,1]×(0,1], see Theorem 1.1. From a biologic perspective, this case can be explained noting that the species u is characterized by a relatively strong advection, and then it leaves sufficient habitat for v to evolve. This result generalizes that in [22] for b=c=1. For the case of α/d1≤1/max¯Ωm1, whether u and v will coexist depends on the values of d1,d2 and c. In order to discuss this case, we define Su,Lu and Ic. We find that if c≤Lu, for all d1>0 and d2>0 the two species coexist. When c takes on the intermediate values, that is, Lu<c<Su, whether the two species will coexist or not is dependent on the values of d1,d2 and ω. If c≥Su, species v can not displace species v. These results are obtained in Theorem 1.2 which is more complex than Theorem 1.3 in [22].
For α/d1 in the interval (1/max¯Ωm1,1/min¯Ωm1), the dynamics of system (1.4) becomes more complicated. Moreover, for a general form of system (1.3), the dynamical behavior is known to be very limited and this case is far away from a complete understanding.
The authors express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript.
This work was partially supported by NNSF of China (No. 11971185).
All authors declare no conflicts of interest in this paper.
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