A Leslie-Gower predator-prey model with a free boundary

In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the original model.


Introduction
A variety of models are used to describe the predator-prey interactions. The dynamical relationship between a predator and a prey has long been among the dominant topics in mathematical ecology due to its universal existence and importance. Recently, many works studied the predator-prey system with the Leslie-Gower scheme [1,3,8,9,11,17,20]. A typical Leslie-Gower predator-prey model is the following where N and P denote the population densities of the prey and predator populations respectively. The parameter r represents the intrinsic growth rate of the prey species and G stands for its carrying capacity. The parameter a is the growth rate for the predator and b (resp. c) is the maximum value which per capita reduction rate of N (resp. P ) can attain. G 1 denotes the extent to which environment provides protection to predator P . All parameters are assumed to be positive.
Our main objective is to understand the long time behavior of a Leslie-Gower predator-prey model via a free boundary. In this paper, we consider the following model: for all t > 0 and 0 < x < h(t), for all t > 0 and 0 < x < h(t), h ′ (t) = −µ(ux(t, h(t)) + ρυx(t, h(t))), for all t > 0, ux(t, 0) = υx(t, 0) = u(t, h(t)) = υ(t, h(t)) = 0, for all t > 0, u(0, x) = u 0 (x) and υ(0, x) = υ 0 (x), for all x ∈ [0, h 0 ], (4) with the positive parameters µ, ρ > 0. The initial data (u 0 , υ 0 ) satisfy From a biological point of view, model (4) describes how the two species evolve if they initially occupy the bounded region [0, h 0 ]. The homogeneous Neumann boundary condition at x = 0 indicates that the left boundary is fixed, with the population confined to move only to right of the boundary point x = 0. We assume that both species have a tendency to emigrate throught the right boundary point to obtain their new habitat: the free boundary x = h(t) represents the spreading front. Moreover, it is assumed that the expanding speed of the free boundary is proportional to the normalized population gradient at the free boundary. This is well-known as the Stefan condition.
Many previous works study free boundary problems in predator-prey models. We refer the reader, for instance, to [14,15,18,21] and references cited therein.
In this paper, we have been working under the following assumption (H1) : δα + δ < 1.
Organization of the paper. In Section 2, we use a contraction mapping argument to prove the local existence and uniqueness of the solution to (4), then make use of suitable estimates on the solution to show that it exists for all time t > 0. In Section 3, we derive several lemmas which will be used later. Section 4 is devoted to the long time behavior of (u, υ), proving a spreading-vanishing dichotomy and finally deriving criteria for spreading and vanishing. We estimate the spreading speed in Section 5 and then summarize through a brief discussion in Section 7.

Existence and uniqueness of solutions
In this section, we first state a result about the local existence and uniqueness of a solution to (4) in Lemma 2.1. Then we derive a priori estimates (Lemma 2.2) in order justify that the solution is defined for all time t > 0. The global existence of a solution to the system (4) is stated in Theorem 2.3.
Lemma 2.1. Assume that (u 0 , υ 0 ) satisfies the condition (5), then for any θ ∈ (0, 1), there is a T > 0 such that the problem (4) admits a unique solution (u(t, x), υ(t, x), h(t)), which satisfies The proof of Lemma 2.1 will be postponed to Section 6. Lemma 2.2. Let (u, υ, h(t)) be a solution of (4) for t ∈ [0, T ] for some T > 0. Then where The proof of Lemma 2.2 will be postponed to Section 6 as well.
Theorem 2.3. Assume that (u 0 , υ 0 ) satisfies the condition (5), then for any θ ∈ (0, 1), the problem (4) admits a unique solution (u(t, x), υ(t, x), h(t)), which satisfies On the proof of Theorem 2.3. We only give a brief sketch of the proof here since it is similar to those done in [5] and [6]: the global existence of the solution to problem (4) follows from the uniqueness of the local solution, Zorn's lemma and the uniform estimates of u, υ and h ′ (t) obtained in Lemma 2.2, above.

Known results from prior works
In this section, we recall from prior works some important results that will be used repeatedly in our arguments. We start with some results regarding the stationary state(s) of the model The stationary state will be determined via the eigenvalue problem    dφxx + aφ = σφ, 0 < x < L, as well as the spatial domain's size. The following lemma summarizes the result. (ii) If L > L * , then (9) has a minimal positive equilibrium φ, and all positive solutions to (9) approach φ in C([0, L]) as t → +∞.
For a detailed proof of (i) and (ii) one can refer to Proposition 3.1 and 3.2 of [2]. The result in (iii) is obtained through a simple computation and can be found in the proof of Corollary 3.1 in [22]. Now, we state a comparison principle that we will use in the proving the results of Section 4, below. This comparison principle is extracted from Lemma 4.1 and Lemma 4.2 of [13] with minor modifications. and Letū,ῡ ∈ C(Ω) ∩ C 1,2 (Ω) and u, υ ∈ C(Ω 1 ) ∩ C 1,2 (Ω 1 ). Assume that 0 <ū, u ≤ M 1 and 0 <ῡ, υ ≤ M 2 and that (ū,ῡ,h) satisfies Assume that the initial data of (11) satisfȳ and the initial data of (12) and (13) satisfy Then, the solution (u, υ, h) of (4) satisfies The proof of Lemma 3.2 is very similar to the proofs of Lemma 5.1 of [7], Lemma 4.1 and Lemma 4.2 of [13]. We hence omit the details here.
In order to discuss the spreading of the species, we will use Lemma A.2, Lemma A.3 of [19] and Proposition 8.1 of [16]. We restate these results here for the reader's convenience.
On the contrary, we will use the following lemma, which is Proposition 3.1 of [13], in order to discuss the vanishing case of the species.
, for some θ > 0. Assume that s(t) > 0 and ω(t, x) > 0 for all 0 ≤ t < ∞ and 0 < x < s(t). We further assume that for some constant M > 0. If the functions ω and s satisfy To discuss the asymptotic behaviors of u and υ in the vanishing case, we need the following lemma.
Lemma 3.6. Let (u, υ, h(t)) be the solution of (4) and recall that h∞ = lim t→+∞ h(t). If We skip the proof of the above lemma since it is similar to that of Theorem 4.1 in [16].
Furthermore, we need the following lemma which appears in [7] and [13] (page 893 and page 3388 respectively).
If s ≥ s min = 2 max{1, √ Dκ}, then problem of (18) admits a solution (U, V ) which satisfies the conditions The following lemma will be used to give a lower estimate of the "asymptotic spreading speed" (when spreading occurs). The notion of spreading and spreading speed will become more clear later on.
Before we state the needed lemma, let us first consider the following problem (which is relevant to the original problem (4). It will also initiate problem (22), the subject of We assume that (υ, h) is the unique solution of (20) and h(t) → +∞ as t → +∞. Setting Since lim t→+∞ h(t) = +∞, if h ′ (t) approaches a constant s * and ω(t, x) approaches a positive function V (x) as t → +∞, then V (x) must be a positive solution of (22) with s * = µρV ′ (0). We now state the lemma.

The spreading-vanishing dichotomy
We have seen in Lemma 2.2 that h ′ (t) > 0 for all t > 0. This allows us to define This will allow us to define the notions of spreading and vanishing as follows.

The Spreading Case
The following theorem shows that h∞ = +∞ is sufficient for a successful spreading: is the solution of (4). If h∞ = +∞, then we have Proof. We will divide the proof of this theorem into two steps.
Step 2. Indeed, we can continue the above strategy to obtain the following sequences, whose monotonicity is a straightforward conclusion Since the constant sequences {ū i } and {ῡ i } are monotone non-increasing and bounded from below, and the sequences {u i } and {υ i } are monotone non-decreasing, and are bounded from above, the limits of these sequences exist. Let us denote their limits, as i → +∞, byū,ῡ, u and υ respectively. We then havē u = 1 − δυ, u = 1 − δῡ,ῡ =ū + α and υ = u + α.
From hypothesis (H1), we can easily conclude thatū = u = u * and this implies that The proof of Theorem 4.2 is now complete.

Sharp criteria for spreading and vanishing
In this section, we derive some criteria governing the spreading and vanishing for the free-boundary problem (4).
Proof. The proof of Lemma 4.4 is essentially the same as that of Theorem 5.1 in [13]. By Theorem 4.3, we know that if h∞ < ∞, then In the following, we assume that h∞ > π 2 min 1, D κ to get the contradiction.
First, as h∞ > π 2 , there exists ε > 0 such that h∞ > π 2 x) be the solution of the following problem: By the comparison principle, we have u(t, x) ≤ u(t, x), for all t > T and 0 < x < h(T ). Since Secondly, as h∞ > π 2 D κ , there exists T > 0 such that h(T ) > π 2 D κ and u(t, x) > 0, for all t > T and 0 < x < h(T ). Let υ(t, x) be the solution of the following equation By the comparison principle, we have υ(t, x) ≤ υ(t, x), for all t > T and 0 < x < h(T ). Since which is a contradiction to Theorem 4.3.
Finally, since h ′ (t) > 0 for all t > 0, then together with the above arguments we can see that h∞ = +∞ when h 0 ≥ π 2 min 1, D κ .
Then by a calculation, we obtain from (42) that

Proof of existence and uniqueness
This section is devoted to prove the results about local existence and uniqueness of the solution to the main problem (4).
Proof of Lemma 2.1. The main idea is adapted from [4]. Let ζ ∈ C 3 ([0, ∞)) such that , for all y. Define Note that, as long as We then compute .
As mentioned above, we will construct a contraction mapping from X T into X T in order to prove the existence of a local solution. We begin this construction now. As 0 ≤ t ≤ T, the coefficients A, B and C are bounded and A 2 is between two positive constants.
G : = [DA 2 (h 1 (t), y(t)) − DA 2 (h 2 (t), y(t))]V 1yy + [(DB(h 1 (t), y(t)) − DB(h 2 (t), y(t))) − (C(h 1 (t), y(t)) − C(h 2 (t), y(t)))]V 1y + G(U 1 , V 1 ) − G(U 2 , V 2 ). Again, using standard L p estimates and the Sobolev embedding theorem, we have Ū C 1+θ 2 ,1+θ (R) , where the constants C 4 , C 5 , and C 6 > 0 depend on A, B, C and C i , for i = 1, 2, 3. We also have Ū C(R) + V C(R) + h′ 1 −h ′ As a first choice, we pick M = max (ii) A spreading-vanishing dichotomy can be established by using Lemma 4.4 and the critical length for the habitat can be characterize by h * , in the sense that the two species will spread successfully if h∞ > h * , while the two species will vanish eventually if h∞ ≤ h * . If the size of initial habitat h 0 is not less than h * , or h 0 is less than h * , but µ ≥μ or 0 < D ≤ D * , then the two species will spread successfully. While if the size of initial habitat is less than h * and µ ≤ µ or D * < D ≤ κ, then the two species will disappear eventually.
(iii) Finally, Theorem 5.1 reveals that the spreading speed (if exists) is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem induced from the original model.