Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion

. This paper is concerned with the diffusive Leslie–Gower prey–predator model with weak diffusion. Assuming that the diffusion rates of prey and predator are sufﬁciently small and the natural growth rate of prey is much greater than that of predators, the diffusive Leslie–Gower prey– predator model is a singularly perturbed problem. Using travelling wave transformation, we ﬁrstly transform our problem into a multiscale slow-fast system with two small parameters. We prove the existence of heteroclinic orbit, canard explosion phenomenon and relaxation oscillation cycle for the slow-fast system by applying the geometric singular perturbation theory. Thus, we get the existence of travelling waves and periodic solutions of the original reaction–diffusion model. Furthermore, we also give some numerical examples to illustrate our theoretical results.


Introduction
The abundant dynamical feature of interacting species is a hot issue in the research of ecological system. Based on laboratory experiments and observations, researchers have proposed many realistic biological models such as prey-predator model. Among these models, a typical one is the Leslie-Gower prey-predator model, which is firstly proposed in [19]. In the Leslie-Gower prey-predator model, the density of prey can influence the carrying capacity of predators such that the density of predators obeys a logistic Table 1. Parameters in model (2), where N represent the number of individual per unit area.

Parameter Interpretation Unit
K Carrying capacity of the prey N r 1 Natural growth rate of the prey time −1 r 2 Natural growth rate of the predator time −1 b 1 The maximum value of the per capita reduction rate of the prey time −1 b 2 The maximum value of the per capita reduction rate of the predator time −1 m 1 Environment protection to the prey N m 2 Environment protection to the predator N dynamics with a changing capacity proportional. Collings [7] highlights that the Leslie-Gower prey-predator model is sufficient in population dynamics because it can avoid the biological control paradox of classical prey-predator models-a prey density is low in a stable coexistence equilibrium. The modified Leslie-Gower predator-prey model is proposed as where U and V separately stand for the total numbers of preys and predators, all parameters are positive constants and their interpretations are described in Table 1.
The distributions of prey and predators are not homogeneous in real world. Thus, we introduce the diffusive terms into model (1), which describe the movement of preys and predators, and obtain the following modified Leslie-Gower reaction-diffusion model: where d 1 and d 2 are diffusion rate of the prey and predators, respectively. In recent years, many researchers have investigated the modified Leslie-Gower predator-prey model (1) and its reaction-diffusion case [1,2,14,15,25,26,29,33]. By using the perturbation methods, they derived some interesting conclusions such as the stability of equilibriums, bifurcations, travelling waves, periodic solutions and so on.
For mathematical simplicity, using the following scaling transformations and denotingm 1 andm 2 by m 1 and m 2 , we can get nondimensional model (2) as We study model (3) under assumption that the diffusion rates of predators and prey are similar and sufficiently small and prey grow much faster than predators. It is clear that this assumption is reasonable such as hares and coyotes. In this article, we mainly investigate the travelling waves and periodic solutions of model (3). Hence, using the travelling wave transformationξ = x − ct [9,10], we get where c stands for the velocity of travelling waves. Note that the solution of (4) with c = 0 represents standing waves, which is not our main focus in this article. Since system (4) is invariant under transformation (ξ, c) → (−ξ, −c), then we only need to consider the case c > 0. Furthermore, in this paper, we are keen on the travelling waves with c > 1, which means the velocity c is larger than the week diffusion rate d 1 and d 2 . Hence, by transform ξ =ξ/c, the equivalent system of (4) reads Under our assumption, we introduce following new notations: and we rewrite (5) as a singularly perturbed system where and δ are sufficiently small parameters, and ξ is called the slow variable. Furthermore, we assume that and δ satisfy 0 < δ 1.
For the fast variable ζ = ξ/ , we get Note that systems (6) and (7) are separately called the slow system and the fast system, and their dynamics are equivalent. The slow-fast system is an important component in biological models and investigated by many investigators, who obtained some new dynamics. For prey-predator model, Ambrosio et al. [3] and Zhang and Wang [32] separately studied the relaxation oscillation cycle of system (1) with one small parameter. In [20], Li and Zhu investigated the canard cycles and relaxation oscillation cycle of a slow-fast prey-predator system with Holling III and IV response functions. Furthermore, in [28], Shen investigated the dynamics of similar model with Holling IV response function. Atabaigi and Barati [4] also investigate the relaxation oscillation cycle and canard explosion phenomenon of a slow-fast Holling-Tanner model with Holling IV response function. Indeed, Zhang and Wang [31] studied the dynamics of a slow-fast Holling-Tanner model with Holling III response function. For high dimension model, Liu, Xiao and Yi [23] considered the relaxation oscillation cycle of a slow-fast prey-predator model with one prey and two competing predators, and Shen, Hsu and Yang [27] studied the dynamics of a slow-fast intraguild predation model. For reaction-diffusion cases, Ducrot, Liu and Magal [11] studied the large speed traveling waves for the diffusive Rosenzweig-MacArthur predator-prey model, and Cai, Ghazaryan and Manukian [5] studied the travelling waves for the diffusive Rosenzweig-MacArthur and Holling-Tanner models with two small parameters. Furthermore, the geometric singular perturbation theory is an useful analysis method in these paper.
Recently, the geometric singular perturbation theory has been a main method in the analysis of a slow-fast system and it contains many mathematical tools such as the Fenichel theory [12,18], the exchange lemma [21,22], the blow-up method [16,17] and the entryexit function [8,30] and so on. Whereas, the foundation of geometric singular perturbation theory is the Fenichel theory about locally invariant manifolds. Its main conclusion is that, when 0 < 1, there is a locally invariant slow manifold M in O( )-neighborhood of a normally hyperbolic sub-manifold of the critical manifold M 0 .
In this article, we will apply the methods and conclusions in geometric singular theory to analyse (6) and (7). We obtain the existence of heteroclinic orbits corresponding to travelling waves and canard cycles and relaxation oscillation cycle corresponding to periodic solutions of system (3). Furthermore, compared with the result for two-dimensional space in [3,32], we analyse the canard cycles and relaxation oscillation cycle of the fourdimensional slow-fast system (6) with two small parameters. Indeed, we also illustrate the canard explosion phenomenon which is the changing process from a small limit cycle in the singular Hopf bifurcation to the relaxation oscillation cycle.
The rest of this article is organized as follows. In Section 2, we reduce system (6) to the plane (u 1 , v 1 ). We analyse the existence of travelling waves in Section 3 and prove the existence of canard explosion phenomenon and relaxation oscillation cycle in Section 4. Finally, we give the conclusion in Section 5.

Equilibriums and critical manifold
Under our assumption 0 < δ 1, it is clear that system (6) is a multiscale slowfast system. Thus, using the method in [5,13,24], we reduce the multiscale slow-fast system (6) twice by the geometric singular perturbation. The first reduction is respect to smaller parameter , and the second is respect to δ.
For = 0, the degenerate system of (6) reads and we have the critical manifold which is the set of equilibriums for the layer system of (7) as It is easy to verify each point of M =0 has eigenvalues −1 and −1/d besides the two zero eigenvalues. Hence, the critical manifold M =0 is normally hyperbolic and attracting based on the Fenichel theory [12,18]. Furthermore, the dynamics in critical manifold M =0 are defined by the limiting system which is a slow-fast system respect to small parameter 0 < δ 1. Based on the geometric singular perturbation theory [6,18], there is a two dimensional stable slow manifold M for > 0, which is viewed as a -order perturbation of the critical manifold M =0 , i.e., M = M =0 + O( ). Furthermore, the flows on M are govern by the δ-perturbation of system (8). Hence, we only need to investigate system (8).
In order to simplify our analysis processes, we make the following scale transformation: We still denote ξ by ξ. Hence, system (8) reads where η = (m 1 , m 2 , b). Note that the orbit direction of system (10) is reversed to that of system (8) because of the transformation (9) and It is easy to see that system (10) is a slow-fast system with respect to fast variable ξ. Hence, rescaling τ = δξ, we get the slow system of (10) as where τ is the slow variable. Similarly, let δ → 0 in system (10) and (11). We obtain the degenerate system which is defined on the critical manifold , and the layer system Clearly, the critical manifold M =0, δ=0 consists of four normally hyperbolic submanifolds Furthermore, it is clear that M 0a =0, δ=0 and M pa =0, δ=0 are attracting; M 0r =0, δ=0 and M pr =0, δ=0 are repelling; (0, m 1 ) is a turning point, and Through a straight calculation, we can get the following theorem about the existence and stability of equilibriums of system (10).
Theorem 1. For system (10), the following conclusions hold: is a stable node located in the right branch of function v = h 1 (u) for η ∈ D 1 and is an unstable node located in the left branch of function Here Note that the equilibriums in Theorem 1 are the spatially homogeneous equilibriums of model (3). In what follows, we will investigate the travelling waves about spatially homogeneous equilibrium A * and the periodic solutions of model (3). In other words, we will study the heteroclinic orbits about equilibrium A * , canard cycles and relaxation oscillation cycle of system (10).

Travelling waves analysis
In this section, we investigate the heteroclinic orbits of system (10) On M p =0, δ=0 , system (12) is reduced to which allows assert that the value of u 1 decreases on segment (0, u M ) and (u * , 1) and increases on segment (u M , u * ). Hence, system (14) has a stable equilibrium u 1 = u * corresponding to A * (u * , v * ) and an unstable equilibrium With similar analysis, we can get the dynamics of (12) on M 0 =0, δ=0 . Hence, the dynamics of (10) and (11) with δ = 0 are shown in Fig. 2(a). Moreover, we have the following lemma about the heteroclinic orbits of positive equilibrium A * (u * , v * ).
Note that the picture of Lemma 1 is shown in Fig. 2(b). Furthermore, for system (8), we have the following remark.
Remark 1. Since the transformation (9), the flows of system (8) are reversed to that of system (10). Hence, the positive equilibrium A * (u * , v * ) of system (8) is unstable, and the heteroclinic orbits in Lemma 1 are also heteroclinic orbits of system (8) with oppositive direction.
Since M =0 is normally hyperbolic attracting manifold, then there exists a attracting slow manifold M near M =0 for sufficiently small > 0. Moreover, the heteroclinic orbits in Lemma 1 are transversal, then they persist on slow manifold M . Indeed, according to Remark 1, we have the following theorem for system (6).
Note that these heteroclinic orbits in Theorem 2 are corresponded to different travelling waves of model (3). The biological explanation is that, if η = (m 1 , m 2 , b) ∈ D 1 , the predators will be eventual extinction, and the prey will extinct or attach the carrying capacity. Furthermore, we give the following example to verify our conclusions.

Periodic solutions analysis
In this section, we mainly investigate the existence of periodic solutions for model (3) with sufficiently small parameters 0 < δ 1. This means that we are interested in the periodic orbits of the slow-fast system (6). However, the canard explosion is the most important periodic orbit phenomenon in a slow-fast system. In [16,17], this phenomenon is described as a small cycle arising in a singular Hopf bifurcation grows through a sequence canard cycles without head and canard cycles with head to a relaxation oscillation cycle. Canard cycles come from the perturbation of a singular slow-fast cycles, which consist of the attracting and repelling part of critical manifold M =0, δ=0 and the fast orbits of layer system (13). Hence, we will give some results about the canard cycles, canard explosion phenomenon and relaxation oscillation cycle in this section.
To begin with, we give a result about the exit-entry function of system (10).

Canard cycle and canard explosion
In order to find the canard cycles, we need to ensure that the generic fold point M (u M , v M ) satisfies the canard point condition, i.e., g(u M , v M , η) = 0, which means that the equilibrium of degenerate system (12)  Hence, we can construct a family of singular slow-fast cycles Γ c 0 (s) (see Fig. 6(a)) as where s ∈ (0, v M − m 1 ), and α(s), ω(s) are two roots of the equation h 1 (u 1 ) = v M − s, and a family of singular slow-fast cycles Γ ch 0 (s) (see Fig. 6(b)) as https://www.journals.vu.lt/nonlinear-analysis  (10) and (11) (u 1 , v 1 ). Then we rewrite system (10) as where Note that we can choose suitable value of b such that P 1 > 0.
In order to use the conclusions in [16,17], by a straightforward calculation, we can obtain the slow-fast normal form of (16) as where Clearly, λ = 0 implies that P 0 = 0, which is equal to b = (u M + m 2 )/v M . Hence, according to [16,17], we obtain Then the singular Hopf bifurcation curve λ H ( √ δ) and maximal canard curve λ c ( √ δ) of system (17) are and b c ( Hence, we have the following theorems about singular Hopf bifurcation from Theorem 3.1 in [17]. is shown in (19). So a Hopf bifurcation occurs when b passes through b H ( √ δ). It is supercritical ifÂ > 0 and subcritical ifÂ < 0.
Proof. Let η M be the parameters η = (m 1 , m 2 , b) satisfying bv M = u M + m 2 . Hence, we have Then the vertex point M (u M , v M ) is a nondegenerate canard point. Furthermore, system (17) is the normal form of system (10) in the neighborhood of the nondegenerate canard point M (u M , v M ). Thus, it is clear that the origin point is a nondegenerate canard point of system (17). According to Theorem 3.1 in [17], for suitable and λ, there is a equilibrium p e of system (17) near (0, 0), which satisfy p e → (0, 0) as ( , λ) → (0, 0). Furthermore, there exists a singular Hopf bifurcation curve λ H ( √ ) such that p e is stable for λ < λ H ( √ ) and unstable for λ > λ H ( √ ). Since dλ/db < 0, then equation . Hence, we complete the proof.
Remark 2. Theorem 4 shows that, for s ∈ (0, v M − m 1 ) and 0 < δ 1, there exists b = b(s, √ δ) such that the singular slow-fast cycles Γ c 0 (s) or Γ ch 0 (s) can be perturbed the canard cycles Γ c δ (s) or Γ ch δ (s) of system (16). Furthermore, the canard explosion occurs when the parameter b changes in a exponential small neighborhood of the maximal canard curve b c ( √ δ).
Let (ū * ,v * ) be the intersection point of v 1 =v * and v 1 = h 1 (u 1 ), wherev * defined by (15) in Lemma 2 when v 0 = v M . We denote a singular slow-fast cycle as which contains two slow segments and two fast segments; see Fig.8(b). Indeed, according to Theorem 3.2 in [3,32], we have the following theorem about relaxation oscillation cycle. Figure 8. (a) The dynamics of system (10) and (11) with δ = 0; (b) the relaxation oscillation cycle of system (10) with sufficiently small δ, where the pink curve is the singular slow-fast cycle Γ 0 , and the orange curve is the relaxation oscillation cycle Γ .
Remark 3. Due to transformation (9), the direction of canard cycles and relaxation oscillation cycle of system (8) is opposite to system (10).
Indeed, we give the following example to show the existence of relaxation oscillation cycle. Since the critical manifold M =0 is a normally hyperbolic attracting manifold, then for sufficiently small 0 < 1, there is a slow manifold M near M =0 such that canard cycles and relaxation oscillation cycle persist on it. Hence, we have the following theorem about periodic solution of (3). Theorem 6. There exist δ 0 = δ 0 (m 1 , m 2 , b) > 0 and 0 = 0 (m 1 , m 2 , b, δ) > 0 such that for all 0 < δ < δ 0 1 and 0 < < 0 1, the following conclusions hold: Note that the picture of a periodic solution of model (3) is given in Fig. 10, which corresponds to the relaxation oscillation cycle shown in Fig. 9.  Remark 4. The periodic solution of system (3) in Fig. 9 means that the predators and preys will coexit since the population of preys sometimes is very low.

Conclusion
In this paper, we investigate the diffusive Leslie-Gower prey-predator model (3) under the assumption that both prey and predator diffuse small and prey grows much faster than predator. Thus, this prey-predator model is a singular problem with two small parameters. Using the travelling waves transformation, model (3) is transformed to a multiscale slowfast system (6) with 0 < δ 1. Applying the geometric singular perturbation theory, we prove the existence of heteroclinic orbits of system (6) for η ∈ D 1 , which correspond to the travelling waves of model (3). Indeed, we also prove the existence of relaxation oscillation cycle of system (10) for η ∈ D 2 and the canard explosion phenomenon of system (10) when M (u M , v M ) is a canard point. These periodic cycles imply the existence of periodic solutions of model (3). We also give some numerical examples to show illustration of our theoretical results. Our results have important theoretical significance for the biological control of pests and the conservation of biodiversity. Meanwhile, our method used in this paper can be applied into other similar models.