A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment

This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space . The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.


Introduction
It is an important issue how to understand the nature and spreading of an invasive species in mathematical ecology. In recent years, many mathematicians have established various invasion models and investigated them from the viewpoint of mathematical ecology, refer to [2,3,26,28,29,30] etc. For the most theoretical approaches, they are all based on or started with single species models. In order to realize the spreading mechanism of a new or invasive species, Du and Lin [13] deduced the following free boundary problem of the diffusive logistic equation where is the moving boundary to be determined, , , are given positive constants, 0 denotes the size of initial habitat, 0 is the ratio of expanding speed of the free boundary and population gradient at expanding front, and is an given positive initial function. They have obtained the spreading-vanishing dichotomy result.
Since then, the problems describing the spread by free boundary have been studied intensively, such as in [1,4,16,21,22,37]. As an example, Kaneko and Yamada in [24] studied the free boundary problem which condition 0 at 0 in 1 is replaced by 0. Du, Guo and Peng in [10], and Du and Liang in [12] considered the free boundary problem in the time periodic environment. Peng and Zhao in [31] studied the seasonal succession case. When the nonlinear term is replaced by a general function , this problem has been investigated by Du and Lou in [15] and Du, Matsuzawa and Zhou in [17]. For the population model, there are many authors who have studied the diffusive competition system with the free boundary, such as Du and Lin in [14], Guo and Wu in [19,20], Wang in [32,33,34], wang and Zhang in [35], wang and Zhao in [36], etc. For the initial value problem of the Leslie-Gower predatorprey model, A. Ducrot in [18] has studied some spreading properties of modified Leslie-Gower predator-prey reactiondiffusion system. In [6], we have considered the spreading speed properties for the Leslie-Gower predator-prey model with the fractional diffusion term ! ∈ 0, 1)). In [7], we have showed the existence and stability of the Leslie-Gower predator-prey model with nonlocal diffusion. Liu et al. in [27] obtained the asymptotic behavior of two species evolving in a domain with a free boundary in onedimensional environment.
In consideration of the environment heterogeneity, Du and Guo in [8,9] have studied the diffusive logistic model with a free boundary in heterogeneous environment, where the heterogeneous environment coefficients were required to have positive lower and upper bounds. They also obtained the corresponding spreading-vanishing dichotomy results. The predator-prey systems with heterogeneous environment have also been examined extensively, refer to [11,39]. For instance, Wang and Zhao in [38] gave the discussion of the competition model with free boundary in the higher dimensional and heterogeneous environment. Some epidemic models with free boundary have been considered by some authors, such as in [23].
Motivated by those results, we consider in this paper with the following Leslie-Gower predator-prey reaction-diffusion system with free boundary and radial symmetry where $ = $$ + 12 $ $ , $ ' = ' $$ + 12 $ ' $ (% = | |, ∈ ℝ , 4 ≥ 1), , , and ℎ are given positive constants, and the functions , , &, !, 5 ∈ 6 7 8 ([0, ∞)) for some ; ∈ (0, 1) satisfy for given positive constants < 2 ≤ < = . Here the sphere {% = ℎ( )} is the moving boundary to be determined. In this paper, we shall focus on the dynamical process of an invasive predator species with population density '( , | |) invading into the n−dimensional heterogeneous habitat of a native prey species with population density ( , | |). To do so, one shall consider that the prey population is initially uniformly well disturbed and undergoes the diffusion and growth in the entire space ℝ , while the predator population is initially introduced in some localized location, namely, ' (| |) occupies in a ball {% < ℎ }, and disperses through random diffusion over an expanding ball {% < ℎ( )}, whose boundary {% = ℎ( )} is the spreading front and satisfies the free boundary condition ℎ′( ) = − ' $ ( , ℎ( )), where is a given positive constant. Using such a framework, we are interested in deriving some information about the invasion of predator in the environment. Before stating our results, let us precise the assumptions on the initial data. We assume that the initial functions (%) and ' (%) satisfy The paper is organized as follows. In Section 2, we first state the existence, boundedness and uniqueness of the solution for the problem (2), as well as some comparison principle for the following proof. Section 3 is mainly devoted to the proof of the spreading-vanishing dichotomy result. In Section 4, some rough estimates for the spreading speed are obtained in the case that spreading of ' happens.

Some Preliminaries
In this section, we will give the existence, uniqueness and some estimates of the solution to the problem (2). Then, we show some comparison results, which will be used in the following sections.
Since the first inequality of 10) holds only in part of [0, ∞), the maximum principle cannot be used directly. We can prove that for any € > ℎ( * ), We need only to prove min It is a contradiction. Then, it is easy to get that we can use the strong maximum principle and the Hopf boundary lemma to obtain that "( , %) > 0 in (0, * ] × [0, ℎ( )], and " $ ( * , ℎ( * )) < 0. Then we deduce ℎ′( * ) < ℎ ‾ ′( * ). This contradicts with (9). This proves our claim that ℎ( ) < ℎ ‾ ( ) for all ∈ (0, L]. We omit the details of the proof which can be proved as the process with the above lemma.

The Spreading-Vanishing Dichotomy
In this section, we prove the spreading-vanishing dichotomy of the free boundary problem (2). By the estimate of ℎ′( ) in Theorem 2.1, it is easy to show that there exists ℎ D ∈ (0, ∞] such that lim →D ℎ( ) = ℎ D .

. (See [Du and Guo
We will give the following lemmas to obtain the asymptotic properties of solutions for the problem (2).
where x(| |) is the unique positive (radial) solution of the equation Proof. By the similar way to that of [34, Theorem 2.1], we can get some uniform estimates of , ', ℎ′). Then according to Lemma of [36], we can obtain that (14) holds. The details are omitted here.
We use a squeezing argument introduced in [16] to state that (12) holds. Considering the Dirichlet problem where ' > 0 is a given constant, and the boundary blow-up problem When ' is small enough and ‰ is sufficiently large, it is well known that these problems have positive radial solutions • OE ' and ] OE , respectively. As ' → 0 , and ‰ → ∞ , we can deduce from the comparison principle given in [16] From (16) and (17), we can obtain that (12) is true. This is a contradiction with (14) . Therefore, ℎ D ≤ ‰ * (1, !) holds.
Proof. The proof can be found in many papers, such as Theorem 3.5 of [38], Theorem 4.11 of [24], et al. For the convenience of the readers, we give the outline of the proof of this result.

Conclusion
We have examined the dynamical behavior of the population v(t,x) and u(t,x) with spreading front x = h(t) determined by (2), and also the dynamical behavior of the population u(t,x) and v(t,x). We have proved that for both problems, a spreading-vanishing dichotomy holds (Theorems 3.1, 3.3 and 3.4), and when spreading occurs the spreading fronts expand at a nearly constant speed for large time (Theorem 4.1 and Corollary 4.1.). These phenomena are in agreement with numerous documented observations for the spreading of species in ecology, but differ from the mathematical conclusions obtained from (2), which predicts successful spreading for all initial data.