Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity

This paper is concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global attractivity of the semi-trivial solution. In addition, an attracting region was obtained by means of the method of upper and lower solutions.


1.
Introduction. The dynamical models in the form of reaction-diffusion equations have been extensively investigated in various natural sciences [33]. In order to reflect the real dynamical behaviors of models that depend on the past history of systems, it is reasonable to incorporate time delays into the systems [11,13,20,34]. Especially in mathematical biology, many models of population dynamics can be described by delayed reaction-diffusion equations [2,4,5,8,9,10,12,14,19,35,36]. Lam and Ni [18] investigated the interactions between diffusion and heterogeneity of the environment in the following classical diffusive Lotka-Volterra type model: where the function m(x) represents their common (spatially inhomogeneous) intrinsic growth rate or carrying capacity, the habitat Ω is a bounded region in R n (n ≥ 1) with smooth boundary ∂Ω. Lam and Ni [18] established the uniqueness and the global asymptotic stability of coexisting steady states under various circumstances, and also obtained a complete understanding of the change in dynamics when one of the interspecific competition coefficients is small.
In the natural world, there are many species that go through several stages during their lifetime, such as a single-species growth model with stage structure consisting of immature and mature stages is developed using a discrete time delay. For example, Aiello and Freedman [1] introduced the following population model in which α, β, γ, τ are all positive, u i (t) and u m (t) are the densities of immature (juvenile) and mature (adult) members, respectively. The αu m term in the first equation of (2) is the birth rate, assumed proportional to the number of adults and −γu i is juvenile mortality, the e −γs term corrects for juvenile mortality. In the second equation of (2), the βu 2 m term represents adult mortality. The reader will see that mature and immature mortality are treated differently: quadratic for mature and linear for immature. Aiello and Freedman [1] investigated the existence of a globally asymptotically stable positive equilibrium.
Predator-prey population with stage structure is very important in the models of multi-species populations interactions and has been studied widely (see, for example, [21,22,28,30]). In classical models of Lotka-Volterra type, such as system (1), it is assumed that all individuals have largely similar capabilities to hunt or reproduce [7]. However, for a number of animals, it seems reasonable to assume that the predator population feed on the mature prey because immature prey population are concealed in the mountain caves and are raised by their parents, and that the rate of predators attack at immature preys can be ignored (see, for example, [23]). Therefore, it is practical to introduce the stage structure into the competitive model. Furthermore, Liu et al. [23] considered the following two-species competitive population with stage structure: To find out how the stage structure affects the global behaviors of the competitive system (3), let delay τ 2 in (3) equal to zero, which means species 2 has only one stage, then we reduce system (3) to the following system Motivated by the works of Lam and Ni [18], and Liu et al. [23], in this paper, we are concerned with the following Lotka-Volterra competition type model with stage structure in heterogeneous environments: for all x ∈ Ω and t > 0, where u i (x, t) and u m (x, t) are the densities of immature and mature members of species 1 at time t and location x, respectively; v(x, t) is the population density of species 2 at time t and location x; d i > 0 (i = 1, 2) is the diffusion coefficient, τ is the time delay; Ω is a connected bounded open domain in R n (n ≥ 1) with smooth boundary ∂Ω, α(x) is the birth rate of species 1 at location x, γ is the death rate of the immature of species 1 and β is the mature death and overcrowding rate of species 1, as in the logistic equation; r(x) represents the spatially inhomogeneous carrying capacities for species 2 or intrinsic growth rates of species 2 at location x in Ω, which reflects the situation of the resources and thus may vary from point to point. Throughout this paper, we shall assume α(x) and r(x) satisfy the following hypothesis: (M): α(x), r(x) ∈ C (Ω) with ∈ (0, 1) are nonconstant, α(x) > 0 and r(x) > 0 on Ω;ᾱ = 1 |Ω| Ω α(x)dx,r = 1 |Ω| Ω r(x)dx. Note that u i don't appear the second and third equations of system (5), then we just need to investigate the following system: For simplicity, letū = βu m ,v = bv,ā = a/b,c = c/β, and drop the bars, then system (6) can be rewritten as In this paper, we consider system (7) with the following initial conditions and Neumann boundary conditions: The zero Neumann (no-flux) boundary condition means that no individual crosses the boundary of the habitat, ∂ ν = ν · ∇, where ν denotes the outward unit normal vector on ∂Ω. Moreover, the functions ϕ 1 (·, s) and ϕ 2 (·) are non-negative and not identically zero. By analyzing the principal eigenvalue, we shall show that the semi-trivial solution (θ d1,α,0 , 0) (respectively, (0, θ d2,r )) is globally asymptotically stable provided that τ < τ * (respectively, τ > τ * ). This result resembles that of the homogeneous system without diffusion, which has been investigated by Liu et al. [23], who have confirmed the negative effect of stage structure on the permanence and suggested that for a competitive community stage structure is also one of the important reasons that cause permanence and extinction. Moreover, this result is also consistent with result of the classical Lotka-Volterra competition model with just diffusion and heterogeneity of the environment, which has been considered in [15,16,18]. Note that system (7) possesses a mixed quasi-monotone property, then by the method of upper and lower solution, we can establish an attracting region of the solution, which implies that the system is uniformly persistent and permanent. The rest of the paper is organized as follows. In Section 2, some concepts are introduced for later use. Sections 3 is devoted to the stability of the steady state solution of system (7) by analyzing the sign of the principal eigenvalues of the linearized system of (7) at this steady state solution. In Section 4, using the upper and lower method, we investigate the asymptotic behavior of the solution of system (7).
For convenience, we introduce the following notations. Denote by L 2 (Ω, R + ) the Lebesgue space of integrable nonnegative functions defined on Ω,

Preliminaries.
To begin our discussion, we first recall the solution of the following equation: where m(x) ∈ C (Ω) with ∈ (0, 1) is non-constant and satisfies m > 0 in a set of positive measure in Ω. From the proof of existence and uniqueness results of (9) (see [3]), we know that if the principal eigenvalue µ 1 (d, m) < 0, the unique solution of (9) is positive, denoted by θ d,m . The principal eigenvalue will be given in Definition 2.1. Dividing the equation of θ d,m in the first equation of (9) by θ d,m and integrating over Ω. we obtain that To characterize the principal eigenvalue (9), we need to introduce the following eigenvalue problem with indefinite weight ∆ϕ + λh(x)ϕ = 0 in Ω, where h is not a constant and could change sign in Ω. We say that λ 1 (h) is a principal eigenvalue if (11) has a positive solution (Note that 0 is always a principal eigenvalue). Proposition 2.1 in [15] says that the principal eigenvalue λ 1 (h) is nonzero if and only if h changes sign and Ω h = 0. Next, we collect some facts concerning the elliptic eigenvalue problem with an indefinite weight.
Definition 2.1. Given a positive constant d and a function h ∈ L ∞ (Ω), we define µ k (d, k) to be the kth eigenvalue (counting multiplicities) of In particular, we call µ 1 (d, h) the first eigenvalue of (12), and have the following variational characterization The following proposition collects some important properties of µ 1 (d, h) in connection with λ 1 (h) [15]. The more detailed proof can be found, for example, on page 95 of [3].
Proposition 1. The first eigenvalue µ 1 (d, h) of (12) depends smoothly on d > 0 and continuously on h ∈ L ∞ (Ω). Moreover, it has the following properties: with equality holds if and only if h 1 = h 2 a.e. in Ω. Assume in addition that h is nonconstant, then Next, let us consider the following eigenvalue problem For convenience, letλ(d, τ, h) be the principal eigenvalue of (14). In fact, we need to study the sign of the principal eigenvalueλ(d, τ, h) in order to investigate the stability of the solution of the model. Thanks to the proof of Theorem 2.2 in [31], we have the following result.
3. Global dynamics of a single species with age structure. This section is devoted to the global dynamics of the following single-species model with age structure. It will play an important role in the investigation of global dynamics of system (7), ) < 0, it follows from the discussion about (9) that system (16) has exactly one positive steady state solution, denoted by θ d,α,b . In what follows, we shall employ the approach proposed by Huang [17] to show that every solution u of (16) converges to θ d,α,b as t → ∞ regardless of initial values u(·, θ), θ ∈ [−τ, 0]. The phase space for system (16) is C = C([−τ, 0], C(Ω)). The linear operator d∆ generates an analytic and compact semi-group {T (t)} t≥0 on C(Ω). Moreover, the maximum principle implies that T (t) is strongly positive for t > 0, that is, It is obvious that F is locally Lipschitz continuous. Therefore, from [24], for each φ ∈ C, there exists a maximum t φ > 0 such that the initial value problem of the evolution equation has a unique solution In what follows, we present a series of lemmas to investigate the properties of the solution u φ (t). (17) is called a mild solution of (16); u ∈ C([t 1 − τ, t 2 ], C 0 (Ω)) with t 2 > t 1 is called a classical solution of (17) for t ∈ [t 1 , t 2 ) if ∂u/∂t, ∆u are continuous and u satisfies (17) It is easy to see that u φ (t) is Hölder continuous for t > 0.
× Ω. By a regularity theorem, we see that a mild solution of (16) is a classical solution for t ∈ [τ, t φ ) if t φ > τ . We now establish a uniform C 1 (Ω) estimate for a solution of (16) with initial values. Lemma 3.2. Let δ > 0, t 1 > δ + r and κ > 0 be fixed, then there are constants M = M (δ, t 1 , κ) and The proof of Lemma 3.2 is similar to that of Lemma 2.3 and Corollary 2.4 of [17] and hence is omitted. Moreover, the following result is a direct consequence of Lemma 3.2.
Let u φ (x, t) be the solution of (16) with the initial value φ ∈ C satisfying φ(·, t) ≥ 0. Next, we shall prove that lim t→∞ u φ (·, t) − θ d,α,b = 0. From the viewpoint of biology, only the nonnegative solution of (16) is of our interest. Moreover, if an initial function φ vanishes at θ = 0, that is, φ(·, 0) = 0, then it is easy to observe that u φ (·, t) = 0 for all t ≥ 0. Hence, from now on, let C + = {φ ∈ C : φ(0) > 0} and investigate the properties of the solution u φ of system (16) Proof. The proof of the statement u φ (·) 0 is similar to Lemma 3.1 of [17] and hence is omitted here. In what follows, we only prove the second conclusion of the lemma.
Since for each φ ∈ C + , the solution u φ (t) of (16) is nonnegative, bounded and exists for all t ≥ 0, we can define a semigroup {S(t)} t≥0 on C + as It follows from Corollary 1 and Lemma 3.3 that S(t) is compact for all t > τ . For The properties of the omega limit set ω φ can be found in [17].

SHULING YAN AND SHANGJIANG GUO
The proof of Proposition 3 can be found in [17] and hence is omitted. From Proposition 2 we have Let χ > 0 be a constant and define (18) is not true, that is, σ > 0, then either σ = β * − 1 or σ = 1 − β * . Firstly, suppose that σ = β * − 1 then it follows from the compactness of ω φ and Ω × [−τ, 0] that one of the following two cases is true.

4.2.
Global stability. Next, let us discuss the global stability of the steady state solutions.
Remark 4. The quantity ζ γτ is said to be the degree of the stage structure for the species. Theorem 4.5 implies that in the stage-structured competitive system, stage structure brings negative effects on permanence of one species as well as contribution its extinction. And such effect can be estimated by e ζ . In view of this point, the effect of stage-structure on the Lotka-Volterra competition model with diffusion and heterogenous of the resources is similar to its on the Lotka-Volterra competition model without diffusion and heterogenous; see, for example [23]. More precisely, the increase of species degree of stage structure can not only drive it into elimination but also ensure its competitor permanent.

5.
Asymptotic behavior and global attractor. The purpose of this section is to investigate the asymptotic behavior of the solution of system (7). We first rewrite the system (7) as in Ω, where . Note that f 1 (·, u, u τ , v) is monotone non-decreasing in u τ and monotone non-increasing in v, f 2 (·, u, u τ , v) is monotone non-increasing in u. It follows that possesses a mixed quasi-monotone property in some subset Ξ of R 2 (see [26] for more details). Moreover, this property leads to the following definition.
Theorem 5.2. LetŨ andÛ be a pair of coupled upper and lower solutions of (37), then system (37) has a unique solution U * in Û ,Ũ . Moreover, there exist sequences {Ū (m) } and {U (m) } converging monotonically from above and below, respectively, to U * as m → ∞.
To investigate the asymptotic behavior of the solution to (37), where the reaction function is mixed quasi-monotonic, we first consider the following elliptic system corresponding to (37): in Ω, The definition of upper and lower solutions of system (38) can be found in Definition 4.1 in [26], and it says that any pair of coupled upper and lower solutionsŨ s and U s of (38) are also upper and lower solutions of (37). Hence by Theorem 5.2, the sector Û s ,Ũ s is an invariant set of system (37). Unlike the usual scalar boundary problem, the upper and lower solutions are coupled. Our construction the lower solutions of system (38) is based on the positive steady state of the following problem where m i (x) is given continuous nonnegative functions on Ω. Recall from the problem (9), we know that system (39) has a unique positive solution θ di,mi if µ(d i , m i ) < 0, Next, we seek a pair of the formŨ = (θ d1,α,0 , θ d2,r ) andÛ = (θ d1,m1 , θ d2,m2 ) satisfying in Ω, and in Ω.
Thus, when τ ∈ (τ * , τ * ), it is possible that the solution of system (37) enters an attracting region, moreover, if lim t→∞ u(·, t) =ū = u = u * and lim t→∞ v(·, t) = v = v = v * , then system (7) may have a co-existence steady state solution. This observation is consistent with the result the weak competition Lotka-Volterra system which has been discussed in [16].