Dynamics for the diffusive Leslie-Gower model with double free boundaries

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to\infty$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u,v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.


Introduction
Prey-predator systems (or consumer-resource systems) are basic differential equation models for describing the interactions between two species with a pair of positive-negative feedbacks. The classical Leslie-Gower prey-predator model is ( [18]) where a, b and µ are positive constants, u(t) and v(t) represent the population densities of prey and predator, respectively. In this model, the prey is assumed to grow in logistic patterns. It is known that this system has a globally asymptotically stable equilibrium a 1+b , a 1+b . The diffusive Leslie-Gower prey-predator model with homogeneous Neumann boundary condi-tions takes the form where Ω is a bounded domain of R N . The global stability of a 1+b , a 1+b for the problem (P) had been studied by many authors, see [5] for example.
In the problem (P), it is assumed that the habitats of prey and predator are the same and fixed, and no flux through the boundary. However, in some situations, predator and/or prey will have a tendency to emigrate from the boundary to obtain their new habitat, i.e., they will move outward along the unknown curve (free boundary) as time increases. The spreading and vanishing of multiple species is an important content in understanding ecological complexity. In order to study the spreading and vanishing phenomenon, many mathematical models have been established.
We assume that the prey distributes in the whole line R and the predator exists initially in a bounded interval and invades into the new environment from two sides. In such a situation the diffusive Leslie-Gower prey-predator model with double free boundaries can be written as v(t, x) ≡ 0, t ≥ 0, x ∈ (g(t), h(t)), g ′ (t) = −βv x (t, g(t)), t ≥ 0, h ′ (t) = −βv x (t, h(t)), t ≥ 0, where a, b, d, h 0 , µ and β are given positive constants. The initial functions u 0 (x), v 0 (x) satisfy (1.2) where p > 3, C b (R) is the space of continuous and bounded functions in R. The free boundary condition h ′ (t) = −βv x (t, h(t)) is the Stefen type, and the deduction can refer to [1] and [30].
Free boundary problems of the classical Lotka-Volterra type prey-predator models had been investigated systematically by many authors, please refer to [30] (with double free boundaries), [21,22,25] (with homogeneous Dirichlet (Neumann, Robin) boundary conditions at the left side and free boundary at the right side), and [34] (the prey distributes in the whole space R N , while the predator exists initially in a ball and invades into the new environment).
There were many related works for the classical Lotka-Volterra type competition models. Authors of [8,10,35] investigated a competition model in which the invasive species exists initially in a ball and invades into the new environment, while the resident species distributes in the whole space R N . In [13,29,35], two competition species are assumed to spread along the same free boundary at the right side and with homogeneous Dirichlet (Neumann, Robin) boundary conditions at the left side. Especially, the growth rates permit sign-changing in [35]. For the heterogeneous time-periodic environments, authors of [3] and [26] investigated the case with sign definite coefficients and the case with sign-changing growth rates, respectively.
The classical Lotka-Volterra type competition systems and prey-predator systems with different free boundaries had been studied in [14,27,28,33].
Without the predator in the environment (namely in the case v ≡ 0), (1.1) reduces to a free boundary problem for u considered in the pioneer work [7]. In this relatively simpler situation a spreading-vanishing dichotomy is known, and when spreading happens, the spreading speed has been determined through a semi-wave problem involving a single equation. More general results in this direction can be found in [2,4,6,9,16,23,24], where [2] concerns with a nonlocal reaction term, [4,24] considers time-periodic environment, [6] studies space-periodic environment, [9,16] investigates more general reaction terms. Particularly, in [23,24] the growth rates are allowed to change signs.
Free boundary problems of reaction diffusion equations and systems with advection had been studied by many authors, refer to [11,12,15,19,20,32,36] for example.
This paper is organized as follows. In Section 2 we study the global existence, uniqueness, regularity and some estimates of (u, v, g, h). Section 3 is concerned with the long time behaviors of (u, v), and Section 4 deals with the criteria governing spreading and vanishing. Because the term v/u may be unbounded when u nears zero, it will bring some difficulties for the study.
At last we mention that for the free boundary problem of Holling-Tanner prey-predator model with double free boundaries the methods used here are valid and the corresponding results are still true.
2 Existence, uniqueness, regularities and estimates of global solu- Then g * ≤ 0 and h * ≥ 0. In order to facilitate the writing, we denote Λ = a, b, d, µ, β, α, p} with 0 < α < 1 − 3/p. For the given interval I ⊂ R + , we set , h(t)). and Moreover, for any given 0 < T < ∞ and 0 < α where the positive constant C depends only on T, Before giving the proof of Theorem 2.1, we first state a lemma which can be proved by the same way as that of [30, Lemma 3.1] and the details will be omitted.
on [g 1 (0), h 1 (0)], then we have Proof of Theorem 2.1. The proof will be divided into three steps.
Step 1: Local existence and uniqueness. The idea of this part comes from [28] and [30]. Let .
Next, we shall apply the contraction mapping theorem to show the existence and uniqueness result. Due to the choice of T , we see that, for (g, h) ∈ D 2 T × D 3 T , into (2.2). Then (2.2) is a Cauchy problem of w. The standard theory (cf. [17]) guarantees that the problem (2.2) admits a unique solution w ∈ C([0, T ] × R). As u 0 (x) > 0 in [−2h 0 , 2h 0 ], by use of the structure of D T and the continuity of w, we can find a T 2 > 0 depending on a, b, h 0 , g * , h * , Substituting this known function w(t, y) into (2.3) and taking advantage of the L p theory and Sobolev's imbedding theorem we have that the problem (2.3) admits a unique solution, denoted bỹ where C 2 depends on C 1 , β, h 0 , g * and h * . Now we define a mapping F : by the contraction mapping theorem, and (z, That is, (2.2)-(2.4) have a unique solution (w, z, g, h). By use of the L p and Schauder theories we can show for any given 0 < τ < T and L > 0 (cf. [24, Theorem 2.1]). Moreover, w, z > 0 by the maximum principle, and z y (t, −1) > 0, z y (t, −1) < 0 by the Hopf boundary lemma, the latter imply that . Therefore, the problem (1.1) admits a unique solution (u, v, g, h) and Step 2: Global existence. We extend the solution (u, v, g, h) of (1.1) to the maximal time interval [0, T 0 ) and show that T 0 = ∞.
Using (2.3), (2.4) and the Hopf boundary lemma we have that g Assume on the contrary that T 0 < ∞. Let (v,ḡ,h) be the unique solution of the following free boundary problem In view of Lemma 2.1, v ′ 0 (x) and max Let 0 < α < 1 − 3/p and Applying the L p theory to (2.3) and the embedding theorem we have and there exists a positive constant C 1 (T 0 ) depending only on Λ(T 0 ) and d such that Thus, by use of (2.4), g, h ∈ C 1+ α 2 ([0, T 0 ]) and , where C 2 (T 0 ) depends on Λ(T 0 ), C 1 (T 0 ) and β. Take advantage of the Schauder theory to (2.3) we and there exists a positive constant C 3 (T 0 ), which depends only on C 1 (T 0 ) and C 2 (T 0 ), but not on ε, such that Repeating the discussion of Step 1 we can find a positive constant T depending only on a, b, d, µ, β, α, p, A, C 2 (T 0 ) and δ(T 0 ) which was given by (2.5), such that the solution of (1.1) with initial time T 0 − T /2 can be extended uniquely to the time T 0 − T /2 + T . But this contradicts the definition of T 0 .
Step 3. The regularity and estimate (2.1) can be proved by the similar way to that of [31, Theorem 1.2] and the details are omitted here. The proof is finished.
Let (u, v, g, h) be the unique global solution of (1.1). As g ′ (t) < 0 and h ′ (t) > 0, we can define the limits lim At last, we shall give the uniform estimates of v and g ′ , h ′ when h ∞ − g ∞ < ∞. To this purpose, we first state a proposition. there exist T ε > T and l ε > max L, π 2 d/m , such that when the function w ∈ C 1,2 ((T, ∞) × (−l ε , l ε )) and satisfies w ≥ 0, and for t > T , w(t, ±l ε ) ≥ (≤) k if k > 0, while w(t, ±l ε ) ≥ (=) 0 if k = 0, we must have This implies This proposition can be proved by the same way as that of [ Proof. It is easy to see from the first equation of (1.1) that lim sup t→∞ max x∈R u(t, x) ≤ a. For any given ε > 0, there exists T ε ≫ 1 such that u(t, x) ≤ a + ε for all t ≥ T ε and x ∈ R. Then v satisfies Let w(t) be the unique positive solution of Then lim t→∞ w(t) = a + ε, and v(t, x) ≤ w(t) for all t ≥ T ε , x ∈ R by the comparison principle. Therefore, by the arbitrariness of ε.
Let T ε and l ε be given by Proposition 2.1 with d = θ = 1, m = a − b(a + ω), k = 0. It is clear that The proof is finished. The details are omitted here.

Long time behavior of (u, v)
This section concerns with the limits of (u(t, x), v(t, x)) as t → ∞.
Case 1: h ∞ − g ∞ < ∞. In this case we shall prove that For this purpose, we first give a proposition.
Proposition 3.1. Let d, C, µ and η 0 be positive constants, w ∈ W 1,2 p ((0, T ) × (0, η(t))) for some p > 1 and any T > 0, and Case 2: h ∞ − g ∞ = ∞. In this case we shall prove that uniformly in any compact subset of R. To do this we first show a proposition which alleges that Proof. Assume on the contrary that h ∞ < ∞. Then the condition h ∞ − g ∞ = ∞ implies g ∞ = −∞. There exists T ≫ 1 such that Similar to the proof of Theorem 2.2, we can find a constant σ = σ(T ) > 0 such that Let (w, η) be the unique global solution of Proof. This proof is similar to that of [30,Theorem 4.3]. For the completeness we shall give the details.  Chosen L ≫ 1 and 0 < ω, ε ≪ 1 such that a − b(a + ω) > 0. Let l ε be given by Proposition 2.1 with d = θ = 1, m = a − b(a + ω), k = 0. Using (3.4), we can choose a T 1 > 0 such that Then u satisfies In view of Proposition 2.1 we have lim inf By our assumption, a 1 > 0.
Given L ≫ 1 and 0 < ω, ε ≪ 1. Let l ε be determined by Proposition 2.1 with m = µ, θ = µ/(a 1 − ω) and k = 0. According to (3.5) and Similar to the above, By virtue of (3.6) we can find a T 3 > 0 such that Thus u satisfies The same as the above, where A is given by the above. Thanks to (3.7), there exists T 4 > 0 such that Consequently, v satisfies The same as the above, lim sup t→∞ v(t, x) ≤ā 1 uniformly on the compact subset of R.
Repeating the above procedure, we can find two sequences uniformly in the compact subset of R. Moreover, these sequences can be determined by the following iterative formulas: The direct calculation yields Using the inductive method we have the following expressions: Because of 0 < b < 1, one has This fact combined with (3.8) allows us to derive (3.3).

The criteria governing spreading and vanishing
We first give a necessary condition for vanishing.
uniformly in the compact subset of R (Theorem 3.1). We assume h ∞ − g ∞ > π d/µ to get a contradiction. For any given 0 < ε ≪ 1, there exists T ≫ 1 such that Let w be the unique solution of As ). This is a contradiction to (3.1), and hence (4.1) holds. The proof is complete.
From the above discussion we immediately obtain the following spreading-vanishing dichotomy and criteria for spreading and vanishing.

Discussion
In this paper we have examined the Leslie-Gower prey-predator model with double free boundaries x = g(t) and x = h(t) for the predator. We envision that the prey distributes in the whole space, while the predator initially occupy a finite region [−h 0 , h 0 ] and invades into the new environment. The dynamics of (1.1) exhibits a spreading-vanishing dichotomy: (i) When spreading happens, both prey and predator will stabilize at the unique positive equilibrium state a 1+b , a 1+b as t → ∞. This behavior is the same as that of the initial-boundary problem (P).
(ii) When vanishing occurs, the predator will spread within a bounded area and dies out in the long run, the prey will stabilize at a positive equilibrium state.
The criteria governing spreading and vanishing indicate that both spreading and vanishing are completely determined by the initial habitat of the predator and initial densities of the prey and predator, and the moving parameter/coefficient β of free boundaries.
These results tell us that in order to control the prey species (pest species) we should put predator species (natural enemies) at the initial state at least in one of three ways: (i) expand the initial habitat of predator, (ii) increase the moving parameter/coefficient of free boundaries, (iii) augment the initial density of the predator species.
These theoretical results may be helpful in the prediction and prevention of biological invasions.