A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment

This paper is concerned with a prey-predator model with sign-changing intrinsic growth rate in heterogeneous time-periodic environment, where the prey species lives in the whole space but the predator species lives in a region enclosed by a free boundary. It is shown that the results for the case of the non-periodic environment remain true in time-periodic environment. In fact, we first establish a similar spreading-vanishing dichotomy, which implies that if the predator species could spread successfully, then the two species will coexist, and this is certainly for the situation that the predation is relatively weak. Furthermore, some criteria are also obtained for spreading and vanishing. At last, some rough estimates of the asymptotic spreading speed are given if spreading occurs.

1. Introduction. A variety of mathematical models obtaining from nature are based on the famous Lotka-Volterra predator-prey system. One of its modified versions has the following form: where B[u] = αu + β ∂u ∂n , α and β are non-negative constants and satisfy α + β > 0, n is the outward unit normal vector of the boundary ∂Ω. d 1 , d 2 , a, b, c, d, e and f are positive constants. In particular, when β = 0, the predator invades into the prey's living habitat. It is shown that whether the predator can invade successfully and coexist with the prey or not depends on the sign of the principal eigenvalue of the corresponding eigenvalue problem (4.96) − (4.98) in [2] no matter how small or large the predator's initial density of population is, the readers can refer to detailed argument of section 4.5 in [2]. Here, we emphasize that the sign of the principal eigenvalue depends on the parameters in system. However, the introduction of several bird species from Europe to North America in the 1900s was successful only after several attempts by changing the initial data. This reveals that the species will not invade unconditionally. Just like the problem raised by Lin [21], it is necessary to consider the impact of the free boundary on the dynamics of the species.

MENG ZHAO, WAN-TONG LI AND JIA-FENG CAO
In the natural world, the intrinsic growth rates of the predator and prey are not always positive constants as a result of the inhomogeneous distribution of the species. Meanwhile, due to seasonal variation and the cycle of day and night, their birth and death rates will present certain periodic characteristic. Particularly, when the environment becomes unfavourable to the growth of the species, the intrinsic growth rates may be negative in some periods and some areas. Hence, considering a predator-prey model with a free boundary and sign-changing coefficients in time-periodic environment has the greater theoretical and realistic significance. By the analysis above, we can use the corresponding periodic functions a(t, x), b(t, x), c(t, x), d(t, x), e(t, x) and f (t, x) to replace all constant coefficients a, b, c, d, e and f respectively, where the intrinsic growth rates a(t, x) and b(t, x) can change sign.
In the real world, the following phenomenon happens frequently: one kind of insect pests entering the favorable environment breeds wildly and expands its territory to a large area (almost all over the whole space). In order to control the pests, the most environment-friendly and effective strategy is to introduce one kind of natural enemies in this region. At the beginning, we put some natural enemy in some bounded area (initial habitat).
In general, the predator has a tendency to emigrate from the boundary to obtain its new habitat for improving its living environment, while the prey occupies the whole space, so it is reasonable to assume that the free boundary is caused only by the predator and the movement speed of the spreading front is proportional to the gradient of the population density of the predator at the front. In order to simplify the mathematics, in this paper we assume that the species can only invade further into the environment from the right end of the initial region and the left boundary is fixed at x = 0. Moreover, we take e(t, x), f (t, x), c(t, x) and d(t, x) to be positive constants, and by the suitable rescaling we may think that e(t, x) = f (t, x) = 1.
According to the above argument, we shall use the following free boundary problem with no-flux condition on the fixed boundary to describe the above phenomenon, where d 1 , d 2 , µ, h 0 are given positive constants, x = h(t) stands for the free boundary to be determined, which grows at a speed proportional to the gradient of the population density of the predator at the front: h (t) = −µu x (t, h(t)). The ecological background and derivation of this free boundary condition can refer to [1]. We mainly want to realize the dynamics of predator, prey and the free boundary. Throughout this paper, we assume that the functions a(t, x) and b(t, x) satisfy the following condition: for some ν ∈ (0, 1), and are T -periodic in time t for some T > 0 and positive somewhere in [0, T ] × [0, ∞); and the initial functions u 0 (x) and v 0 (x) satisfy: We remark that there are a series of researches associated with (1) recently. When the functions a, b are positive constants, Wang and Zhao [34] studied problem (1) for double free boundaries. In 2014, Wang [25] investigated some free boundary problems for that the predator and prey share the common region, which was extended by Wang and Zhang [30]. For the higher dimensions and radially symmetric case, problem (1) was studied by Zhao and Wang [36] for the heterogeneous environment. In [36], the authors showed that many results of [25,34] continue to hold in this more general and ecologically realistic situation. In [34,25,30,36], the complete description about spreading-vanishing dichotomy was given, namely, as time t → ∞, the species either successfully establishes itself in the new environment (called spreading), or fails to establish and dies out eventually (called vanishing). Moreover, criteria for spreading and vanishing, long time behaviour of (u, v) and asymptotic spreading speed of the spreading front when spreading happens had also been obtained. For the time-periodic case, Chen et al. [4] and Wang [32] studied the diffusive competition model in time-periodic environment. A large number of related works but for the diffusive Lotka-Volterra type competition model have been studied intensively, and we refer the readers to [8,14,33,15,35,23,24] and some of the references cited therein.
In the absence of v, problem (1) reduces to the following diffusive logistic model: which has been studied by Wang [28]. In the special case that the function a has positive lower and upper bounds, problem (3) was studied by Du et al. [6], in which the authors discussed the higher demension and radially symmetric case. Chen et al. [3] also have considered a diffusive logistic problem with a free boundary in time-periodic environment, including favorable habitat and unfavorable habitat.
If the function a in problem (3) is independent of time t and changes sign, this problem was considered by Lei et al. [20], Wang [26] and Zhou and Xiao [37]. In particular, when a is a positive constant, Du and Lin [7] investigated problem (3) for the first time, which has been extended to the higher dimensional and radially symmetric case by Du and Guo [5].
Besides, Gu et al. [11] studied initially the long time behavior of solutions of a diffusion-advection logistic model with double free boundaries in one dimensional space when the influence of advection is small. Further, Gu et al. [12] considered the rightward and leftward asymptotic spreading speeds when spreading happens. Later, Gu et al. [13] investigated the long time behavior of solutions of Fisher-KPP equation with advection and free boundaries. For the general nonlinear advectiondiffusion equations, Kaneko and Matsuzawa [17] discussed the spreading speeds of the fronts and sharp asymptotic profiles of solutions in free boundary problems in detail.
The rest of this paper is organized as follows. The global existence, uniqueness and estimates of solution are given in Section 2. To establish the criteria for spreading and vanishing, in Section 3 we provide some basic results. Section 4 is devoted to the long time behavior of (u, v) and we get a spreading-vanishing dichotomy. The criteria for spreading and vanishing will be given in Section 5. In Section 6, we give the rough estimation of asymptotic spreading speed. We will give some discussions in Section 7.
2. Existence and uniqueness. In this section, we will give some fundamental results on solutions of problem (1) under (H ). We firstly prove the following local existence and uniqueness result by the contradiction mapping theorem.
Step 1. Transformation of the problem (1). Let α(t, y) = a(t, h(t)y), β(t, y) = b(t, h(t)y), and Step 2. Existence of the solution (w, z, h) to (5) and (6). For 0 < τ 1, we set Then Z τ is a bounded and closed convex set of C( ∞ τ ), and for any given z ∈ Z τ , we have For the given z ∈ Z τ , similarly to the argument of [7, Theorem 2.1], by using the contraction mapping theorem we can prove that, when 0 < τ 1, the problem (5) has a unique solution (w(t, y), h(t)), which continuously depends on z and satisfies where C 1 depends only on α, c, d 1 , µ, h 0 , ν, p, u 0 W 2 p and 1 + v 0 L ∞ . For such a (w(t, y), h(t)) determined uniquely by the above, we put w(t, y) zero extension to [0, τ ]×[1, ∞) and consider the problem (6). By an analogous argument as above, we can prove that, when 0 < τ 1, the problem (6) has a unique solution z(t, y) which depends continuously on (w, h), and thus continuously depends on z. Moreover, there exists a positive constant and z y ∈ L ∞ ( ∞ τ ). We can also verify that z ∈ C ν 2 ,ν ( ∞ τ ) and, upon using (8), Define a map F : As mentioned above, we see that F is continuous in Z τ , and z ∈ Z τ is a fixed point of F if and only if (w, z, h) solves the problem (5) and (6). Estimation (9) indicates that F is compact.
Recall the fact z(0, y) = z 0 (y). Using the mean value theorem and (9), we have Therefore, if we take 0 < τ 1, then F maps Z τ into itself. Hence, F has at least one fixed point z ∈ Z τ , namely, (5) and (6) has at least one solution (w, z, h). Moreover, from the above discussion, we see that (w, z, h) satisfies Step 3. Existence and uniqueness of the solution (w, z, h) to (1). Define ), . It is easy to see that (4) holds.
In the following, we prove the uniqueness. Let (u i , v i , h i ), with i = 1, 2, be two solutions of (1), which are defined for t ∈ [0, τ ] and 0 < τ 1. We can think of that Then (w i , h i ) solves (5) with z = z i and satisfies (7). and Remembering the facts w 2 W 1,2 We can apply the standard L p theory to (10) and then use the Sobolev's imbedding theorem to derive Now we estimate . Combining this with (12), we get Therefore, by use of (11), Recall W (0, y) = 0, H(0) = H (0) = 0. Taking advantage of the mean value theorem and (13) and (14), it follows that provided 0 < τ 1.
. For any (t, x) ∈ k τ , detailed derivation yields . Taking into account (15), we have that . The uniqueness is obtained and the proof is finished.
The global existence of the solution to (1) is guaranteed by the following estimates.
To prove h (t) > 0 for 0 < t < τ 0 , we use the transformation to straighten the free boundary x = h(t). A series of detailed calculations assert
It remains to show that h (t) ≤ M 3 for all t ∈ (0, τ 0 ) with some M 3 independent of τ 0 . To this aim, we define and construct an auxiliary function We will choose M so that u(t, x) ≥ u(t, x) holds over Ω M . Direct calculations show that, for (t, x) ∈ Ω M , It follows that . It is obvious that On the other hand, for Therefore, Applying the maximum principle to u − u over Ω M gives that u(t, x) ≥ u(t, x) for (t, x) ∈ Ω M , which implies that According to the above lemmas, we have the following result.
Theorem 2.4. The problem (1) has a unique global solution (u, v, h) in time and where and Proof. 3. Preliminaries. Assume that (u, v, h) is the unique solution of (1) obtained in Section 2. We need the following comparison principle for later applications.
If h(0) ≥ h 0 and u 0 ( Proof. The proof is similar to that of [8, Lemma 2.6], we omit the details here.
Similarly, we have the following result.
Next, we will state some known results about the principal eigenvalue, which play an important role in later sections.
For any given l > 0, let λ 1 (l, d, c(t, x)) be the principal eigenvalue of the following T -periodic eigenvalue problem  Then we state the following condition: (Cr ) The functions a(t, x), b(t, x) ∈ C r (T ) for some −2 < r ≤ 0. That is, there uniformly in [0, T ].
We can see that if the function a ∈ C r (T ) for some −2 < r ≤ 0, then a satisfies the condition (A1 ).
When the condition (Cr ) holds, we define for any L > 0, where V is the unique positive solution of the following T-periodic boundary value problem This shows that if the invasive species u cannot spread successfully, it will die out in the long run.

Proof. Applying Theorem 2.4 and [25, Proposition 3.1], we have that
We divide the proof of (25) into two parts, and show that, respectively, lim sup Arguing as Step 2 in the proof of [32, Theorem 3.2], we can get (27). Next, we prove (28). For any given l > 0, let w(t, x) be the unique positive solution of By the comparison principle, v(t, x) ≤ w(t, x) for all (t, x) ∈ (0, ∞) × [0, l]. Using the argument of [28, Lemma 3.2], we can prove that where V l (t, x) is the unique positive solution of the following initial-boundary value problem If h ∞ = ∞, then we have uniformly in [0, T ] × [0, L] for any L > 0, where U , V , U and V will be given in the proof.

Remark 2.
When the relationship between the predator and prey is relatively weak, namely, c and d is small enough, it is easy to see that (31) holds. Proof.
Step 1. The construction of U , V , U and V .
Since b ∈ C r (T ), it follows from [28,Theorem 4.2] that the problem admits a unique positive solution V (t, x) ∈ C r (T ), and It follows that a + cV ∈ C r (T ) since a ∈ C r (T ). Applying [28, Theorem 4.2] again, the problem has a unique positive solution U (t, x) ∈ C r (T ), and has a unique positive solution V (t, x) ∈ C r (T ), and It follows that a + cV ∈ C r (T ). Applying [28,Theorem 4.2] again, the problem Step 2. Arguing as the step 1 in the proof of [32, Theorem 3.2], we can get the second inequality of (4.11).
For any given L > 0, let w(t, x) be the unique positive solution of On the other hand, by the comparison principle, Step 3. For any ε > 0, denote a ε (t, x) = a(t, x) + c[V (t, x) + ε(1 + x) r ]. It follows from the condition (Cr ) that uniformly in [0, T ]. Hence a ε (t, x) ∈ C r (T ). For such fixed ε, there exists l * such that λ 1 (l, d 1 , a ε ) < 0 for all l > l * . For any fixed ε > 0 and l > l * , capitalize on the second limit of (33) and h ∞ = ∞, there exists τ 1 such that Consider the following auxiliary T -periodic boundary value problem Since λ 1 (l, d 1 , a ε ) < 0, it is well known (see, for example, [2, Corollary 3.4]) that the above problem admits a unique positive solution, denoted by Z ε l (t, x). Let U ε l be the unique positive solution of the following initial-boundary value problem where M > 1 is sufficiently large such that M Z ε l (τ, x) > u(τ, x) in [0, l]. Obviously, the function By the comparison principle, we have Using the argument of [28, Lemma 3.2], we can prove that and we have known that where Z ε is the unique positive solution of T -periodic boundary value problem The existence and uniqueness of Z ε is guaranteed by [28,Theorem 4.2]. It follows that lim sup Note that a ε (t, x) → a(t, x) + cV (t, x) as ε → 0, by the continuous dependence of solution with respect to parameter, we have that In the same way, we can show that and for any L > 0.
By Theorem 4.1 and Theorem 4.2, we immediately have that the following spreading-vanishing dichotomy holds under the conditions (Cr ) and (31). 5. Criteria for spreading and vanishing. Since a(t, x) and V (t, x) satisfy the condition (Cr ), then a+cV satisfies the condition (A1 ). Making use of Proposition 2, it yields that Σ d1 = {l > 0 : λ 1 (l, d 1 , a + cV ) = 0} = ∅. By the monotonicity of λ 1 (l, d 1 , a + cV ) in l, we see that Σ d1 contains at most one element. We write it as h * , namely, λ 1 (h * , d 1 , a + cV ) = 0. Now we give a necessary condition of vanishing.
Proof. We assume h ∞ > h * to get a contradiction. If h ∞ > h * , then λ 1 (h ∞ , d 1 , a + cV ) < 0. By the continuity of λ 1 (h ∞ , d 1 , a + cV ), there exists ε > 0 such that λ 1 (h ∞ , d 1 , a + cV − ε) < 0. In view of (25) and the fact that Let w(t, x) be the unique solution of where Z(t, x) is the unique positive solution of the following T -periodic boundary value problem Since This is in contradiction to Theorem 3.1.
In the following, for the parameter h 0 satisfying h 0 < h * and (u 0 , v 0 ) fixed, we discuss the effect of the coefficient µ on the spreading and vanishing for the species u.
Similarly to the argument of Lemma 5.2, we can get the following corollary.
Proof. This proof can be done by a similar process in [31, Lemma 3.2] with minor modification.
Similarly to the argument of Lemma 5.3, we can get the following corollary.
Proof. The proof is similar to that of [25,Theorem 5.2]. We give the details below for completeness. Since (Cr ) holds, Σ d1 = ∅. The conclusion (i) follows from Lemma 5.1. Next, we prove the conclusion (ii).