Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.


1.
Introduction. The predator-prey models played a very important role in the theoretical studies of invasive species, resource management and environment protection. The Leslie-Gower [2,15] type models are that the density of predators follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. The importance of the Leslie-Gower model is highlighted by Collings [5], who argued that ratio-dependent Leslie model provides a way to avoid the biological control paradox wherein classical prey-dependent exploitation models generally do not allow for a prey equilibrium density that is both low and stable.
As it was said in [3], spatial features are now solidly established as essential considerations in ecology both in terms of theory and practice, and the mathematical challenges in advancing understanding of the role of space in ecology are substantial and mathematically seductive. Taking into account the spatial diffusion, system (2) in [21] can become as follows (also see system (1.1) in [22]) where 0 < b < 1 represents Allee effect threshold. β and µ are positive constants. d 1 and d 2 denote the diffusion coefficients corresponding to the prey u and the predator v with d 1 > 0 and d 2 ≥ 0. In system (1), the habitat is the whole real line, the predator follows a logistic dynamics with a varying carrying capacity proportional to the density of the prey, and the prey is subject to Allee effect. In recent years, traveling wave solutions of reaction-diffusion equations have attracted increasing interest (see e.g., [6,7,28,18,27,9,10,8]). It is well known that traveling wave is one kind of special solutions to the evolutionary systems, which has a fixed shape and translates at a constant speed c as time evolves. In addition, the minimum wave speed has a possible link to the population spreading speed. In fact, it is conjectured that the minimum wave speed is identical to the population spreading speed, which is proved to be true for some ecological models (see e.g., [16,17,11]). For the existence of the traveling wave solutions, a standard approach, such as the monotone iteration or comparison argument, has been established that is very efficient to deal with the traveling wave solutions for the monotone systems. However, for predator-prey systems, it is very difficulty to prove the existence of traveling wave solutions since predator-prey systems do not generate monotone semiflows. Recently, the analysis of the phase plane can be useful for the existence of traveling wave solutions for predator-prey systems, where original system becomes a four-dimensional ODE system, but the geometric structure in R 4 will be very complex (see e.g., [10,11]). The application of Schauder's fixed point theorem with the help of upper and lower solutions has also been proved to be quite successful. Although this method is very standard, the construction of suitable upper and lower solutions can be very challenge.
For system (1), Ni and Wang [22] analyzed the nonnegative constant steady state solutions and their stabilities, and investigated the stationary patterns induced by diffusions. To the best of our knowledge, few works have been done for the existence of traveling wave for system (1).
In this paper, we mainly establish the existence and nonexistence of traveling waves connecting predator-free steady state and coexistence steady state by constructing suitable upper and lower solutions. This kind of traveling wave can be called as wave of invasion (see [25]), which is of ecological interest since it corresponds to a situation where an environment is initially inhabited only by the prey species at its carrying capacity, and a small invasion of the predator drives the system to a stable coexistence steady state displacing an unstable predator-free steady state [12].
On the other hand, we will study the existence of small amplitude periodic traveling wave train solutions for our system based on the Hopf bifurcation theory. It is noted that these solutions are periodic in ξ = x + ct given by (3). As pointed out in [29], periodic traveling wave train solutions are spatio-temporal patterns which have periodic profile, maintain their shape and move at a constant speed.
Our work is organized as follows. Section 2 is devoted to the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by constructing the upper and lower solution method. By constructing the sequence of iterations, we obtain the asymptotic boundary conditions at +∞ (see (5)). Section 3 is concerned with the existence of small amplitude wave train solutions, taking into account two cases: the diffusion of both species and the diffusion of prey-only, by Hopf bifurcation theory.
2. Existence of traveling wave solution. In this section, we will discuss the existence of traveling wave solution for system (1) with the diffusion of two species, i.e., d 1 > 0 and d 2 > 0.
For the convenience of discussion, we introduce transformationsx = x √ d1 . Then, dropping the tildes, system (1) can be rewritten as It is clear that system (2) has the predator-free constant steady states (b, 0) and . As pointed out in [22], for positive constant steady states, the following results are obtained.

Remark 1.
We can see that if β >β, then system (2) has no coexistence steady state. This implies that when the capturing rate β is large, system (2) cannot achieve a state of coexistence of both species.
In the following, we always assume β <β. According to [22], constant steady state (u * , u * ) are unstable for all d, µ > 0. So we only focus on the existence of traveling wave solutions for system (2) connecting the predator-free constant steady state (1, 0) and the coexistence steady state (u * , u * ). Here c > 0 is the wave speed. Then the traveling wave solution satisfies the following form with the boundary conditions Here, subscripts denote differentiation with respect to the corresponding variable. First, we introduce the following function spaces and give the definition of the upper and lower solutions.
Definition 2.1. A pair of functions (ũ,ṽ) and (û,v) in X 0 are coupled upper and lower solutions of system (4) ifũ ≥û,ṽ ≥v, and there exists a finite set D = {D j ∈ R : j = 1, 2, · · · , n} such thatũ,û,ṽ,v ∈ C 2 (R\D), and (ũ,ṽ), (û,v) satisfy the following inequalities Define the functions , we easily obtain that is nondecreasing in u and nonincreasing in v, and H 2 (u, v) is nondecreasing in u and v. So system (4) is equivalent to the following equation For (u(ξ), v(ξ)) ∈ X 0 , we consider an operator P = (P 1 , P 2 ) : X 0 → X defined as follows Clearly, a fixed point of P is a solution of system (4). Similar to Lemma 3.2 of [13] and Lemma 3.2 of [19], we can prove the existence of the fixed point by using the Schauder's fixed point theorem and obtain the following result (also see [29,4]).
Lemma 2.2. Let c > 0. Suppose that system (4) has a pair of upper and lower solutions (ũ,ṽ) and (û,v) in X 0 satisfying Then system (4) has a positive solution In what follows, we focus on constructing the upper and lower solutions (ũ(ξ), v(ξ)) and (û(ξ),v(ξ)). However, there is a negative power nonlinearity in the predator equation of system (4). Thus, we need a positive lower solution for the prey density.
For constructing the proper upper and lower solutions, we first assume that β <β . Note thatβ <β. It is easy to obtain that if β <β holds, then u * and u * exist, and Note that r 2 < ηr 1 < c 2 . It is clear that dη 2 r 2 1 − cηr 1 + µ < 0. It shows that q > 1.
It is clear that Thus, lim ξ→−∞ (u(ξ), v(ξ)) = (1, 0). Next, we will show that lim ξ→+∞ (u(ξ), v(ξ)) = (u * , u * ). Denote Obviously, it is obtained that By the second equation of system (2), we obtain that By Lemma 2.4 and Lemma 2.5 in [29] (also see Corollary 1 and Proposition 2.1 in In the following, similar to the methods in [29], we will obtain the relation of inequality betweenũ,ṽ,û andv. Note that λ 1 < 0 < λ 2 . For sufficiently small ε > 0, we can choose and there exists M 1 and ξ 1 > M 1 + N such that u(ξ 1 ) > u + − ε and for ξ > M 1 , Thus, we can obtain that By the arbitrariness of ε, we can obtain that Similarly, we have that Moreover Note that v + ≥ v − > 0. From (10), it follows that Furthermore, according to b+1 2 < u − ≤ u + < 1, it is obtained that Denote u + 1 = 1 and u − 1 = b+1 2 . Construct sequences {u + k } k=1,2,··· and {u − k } k=1,2,··· , where Obviously, u + k is decreasing in k and u − k is increasing in k. For all integer k, it holds that u + ≤ u + k and u − ≥ u − k . Moreover, it is easy to obtain that for all integer k, In fact, if it is false, then for some integer k 0 , at least one of the following inequalities holds u + k0 < u * , u − k0 > u * . First, assume that for some integer k 0 , u + k0 < u * . By the first equation of (11), we can obtain that Then, by the second equation of (11), we have that Similarly, the case for u − k0 > u * can be treated. Next, for all integer k, it is seen that . Under the condition ρ < 1, i.e., we can obtain that lim It follows from (12) and (13) that Note that u + ≥ u − . Thus, u + = u − = u * . Therefore, this completes the proof of existence of a traveling wave solution connecting (1, 0) and (u * , u * ). Similar to the proof of Theorem 2.6 in [4] (also see [29]), it is obtained that for 0 < c < 2 √ dµ, system (2) has no positive traveling wave solution connecting (1, 0) and (u * , u * ).
Note thatβ <β. Summarizing the above discussion, we obtain the following result.

Wave train solutions induced by Hopf bifurcation.
In this section, we will study the existence of small amplitude periodic solutions of system (4) based on Hopf bifurcation theory, which is equivalent to wave train solutions of system (2).
3.1. Predator and prey diffuse. Denote It is obvious that Note that du * dβ < 0 and u * = b+1 2 if and only if Thus, we can obtain that α < 0 if 0 < β <β, α = 0 if β =β, and α > 0 if β < β <β. Furthermore, denote f (z) = zf 1 (z), where It is clear that f 1 (u * ) = 0 and f 1 (u * ) > 0. Thus, Under the condition α > 0, we can obtain that α 2 βu * < α. Thus, there exists µ such that α 2 βu * < µ < α holds. First, we transform system (4) into first order ordinary differential equations. Let u (ξ) = U (ξ) and v (ξ) = V (ξ). Then system (4) becomes as follows It is clear that (u * , 0, u * , 0) is an equilibrium point of (14). The Jaccobian matrix of the linearization of (14) at (u * , 0, u * , 0) is given by Thus, the corresponding characteristic equation is To obtain the existence of periodic solutions induced by Hopf bifurcation, we seek for a pair of purely imaginary roots of (16). First, we assume that α 2 βu * < µ < α. Let λ = iω (ω > 0) be a root of (16). Next, substitute iω into (16). Then, separating the real and imaginary parts, we can obtain that Note that α − µ > 0. Thus, it follows from (17) that By µ > α 2 βu * , one can obtain that (18) has a unique root d = d H > −1. Note that It is easy to see that d H is strictly increasing on c, and there exists a unique In the following, we will analyze the transversity conditions with d = d H (c) for c > c * . From (16), it follows that Then, Re dλ < 0. Hence, based on the Hopf bifurcation theorem [20], it is obtained that system (4) possesses the small amplitude periodic solutions with period This periodic solution corresponds to a small amplitude traveling wave train solution of system (2).
Theorem 3.1. Assume that β <β and α 2 βu * < µ < α. Then there exists a unique c * > 0 such that for c > c * , as the parameter d crosses the Hopf bifurcation points d = d H (c), system (2) has a family of wave train solutions with period 3.2. Sedentary predator and diffusing prey. As we know, in the animal world, most predators are fast-moving for food, e.g, tigers have to run faster to catch a running deer. However, there are other cases in ecosystems where predators move slower than the prey (see e.g., [24,23]). For example, the pacman frog which has a wide mouth that enables it to swallow prey that crosses its path uses a 'sit-and-wait' strategy. The web-builders spider also applies the same strategy to prey upon small insects. Thus it is a relevant case to consider. In this subsection, we will assume that the predator moves very slowly relative to the prey. This is reasonable that we take d = 0. Let u (ξ) = U (ξ). Then system (4) becomes as follows It is clear that (u * , 0, u * ) is an equilibrium point of (19). The Jaccobian matrix of the linearization of (19) at (u * , 0, u * ) is given by Thus, the characteristic equation of (19) is We will search for a pair of pure imaginary roots of (21). Substituting λ = iω(ω > 0) into (21) and gathering terms, we can obtain ω 2 = α − µ and ω 2 = µ(βu * −α) c 2 −µ . Thus, under the condition α > µ, if then (21) has a pair of pure imaginary roots. Regarding λ = λ(c) as a function of c and differentiating (21), we obtain dλ dc = − ∂F 2 /∂c ∂F 2 /∂λ = (µ + c 2 )λ 2 + µ(α − βu * ) 3c 2 λ 2 + 2c(µ − c 2 )λ + c 2 (α − µ) .