Traveling wave solutions in a diffusive producer-scrounger model

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed \begin{document}$c_{m}$\end{document} and a threshold value \begin{document}$R_{0}$\end{document} . By constructing a suitable invariant convex set $Γ$ and applying Schauder fixed point theorem, the existence for \begin{document}$c>c_{m}, R_{0}>1$\end{document} was established. Besides, a Lyapunov function is constructed subtly to explore the asymptotic behaviors of traveling wave solutions. The non-existences of traveling wave solutions for both \begin{document}$c 1$\end{document} and \begin{document}$R_{0}≤1, c > 0$\end{document} were obtained by two-sides Laplace transform and reduction method to absurdity.


1.
Introduction. Very recently, Cosner and Nevai [2] established a reaction-diffusion model to describe an ecological interaction of two species named Kleptoparasitism: ∂p ∂t = d 1 ∆p + (−b 1 − a 1 p) p + m(x) dp s+d , x ∈ Ω, ∂s ∂t = d 2 ∆s + (−b 2 − a 2 s) s + θm(x) ps s+d , x ∈ Ω. (1) In general, Kleptoparasitism is one kind of interspecies reaction that the scrounger species steals food from another species named producer. Here Ω is a bounded domain in R n (n ≥ 1) with smooth boundary; p(x, t), s(x, t) are the density of producer and scrounger at location x ∈ Ω and time t ≥ 0, respectively; d 1 , d 2 > 0 are diffusion rates and a i , b i > 0 (i = 1, 2) denote the density-independent and density-dependent death rates. Furthermore, m(x) represents the per-capita rate of resource discovery at location x for the producer; d > 0 represents the producer's ability to avoid the resource to be stolen by scrounger, and θ > 0 is the conversion rate. For more pertinence, the model (1) assumes that the scroungers are incapable of discovering food resource themselves, i.e., they just can steal food from the resource discovered by producers. The per-capita rate of resource discovery m(x) and the birth rate of producers are in proportion. In particular, the independent of location in m(x) means that the birth rate is the same everywhere in the domain. (2) Note that (2) is a mixed quasi-monotone system. Other familiar examples for mixed quasi-monotone systems are prey-predator models and epidemic models. The studies for traveling wave solutions of mixed quasi-monotone systems are developed during these thirty years. But it seems that the relative studies are very tricky indeed. In the 1980's, Dunbar [4]- [6] established the existence of traveling wave solutions for prey-predator models with Holling type-I and Holling type-II functional responses, using a shooting method based on Wazewski theorem, invariant manifold theorem and LaSalle theorem. His idea was used, developed and simplified in [10][11][12][13]18] for prey-predator models with more complicated functional responses. Another important technique for the study of traveling wave solutions of mixed quasi-monotone systems is Schauder fixed point theorem (e.g., see [1,7,9,19,21,22,25]). However, as emphasized in [19], the challenging and difficult task is to construct and verify a suitable invariant convex set Γ for using Schauder fixed point theorem.
Ducrot & Magal ( [3]), Li & Yang ( [14]), Li,Li & Yang ( [15]), Wang & Wu ( [19]), Xu ( [23]), Yang,Li,Li & Wang ( [24]) used some independent analytic techniques to construct an invariant convex set Γ for some SIR epidemic models. Their discussions to a large extent depend heavily on the forms of models, and some techniques maybe very challenging and different subtly. Actually, the producer and scrounger system (1) is different with the SIR models, the producer and scrounger in (1) both have their own dynamical function (−b 1 − a 1 p)p and (−b 2 − a 2 s)s, and the raise of producer (or scrounger) is embodied in the factor mp s+d (or θmsp s+d ). To the best of our knowledge, the model (1) is very new, and there is no research done for the spatio-temporal propagation and traveling wave solution on it now.
The purpose of the current paper is to study the existence and non-existence of traveling wave solutions for model (1). Our idea is mainly motivated by the works of above mentioned studies, and our concrete technique and method depend on the specific form of (1). Here we will firstly show that there exists a threshold constant R 0 > 0, the function of which is similar to the basic reproducing number of SIR epidemic model. Assume that R 0 > 1, one could find a constant c m > 0 such that it plays the function of minimal wave speed. In particular, we will construct four functions satisfying four inequalities and then a suitable invariant convex set Γ with the aid of these four functions as upper and lower bounds. We then show that a traveling wave solution of model (1) exists in Γ if c > c m by Schauder fixed point theorem. In order to discuss the asymptotic behaviors of traveling wave solutions, an important Lyapunov function is constructed and it's bounds are argued subtly. The non-existences of traveling wave solutions for both c < c m , R 0 > 1 and R 0 ≤ 1, c > 0 are discussed by reduction method to absurdity and two-sides Laplace transform.
The paper is organized as follow. In section 2, the existence and linear stability of constant equilibria are discussed in view of R 0 > 1 or R 0 < 1. In section 3, we construct an invariant convex set Γ, and apply Schauder fixed point theorem to establish the existence of traveling waves. We then introduce a Lyapunov function in order to discuss the asymptotic behaviors of traveling wave solutions. Section 4 focuses on the non-existence of traveling wave solutions. A concluding discussion is given in Section 5, and an appendix for proofs of some lemmas is at the last.
2. Linear stability of constant equilibria. In this section, the existence and the linear stability of constant equilibria of (2) are discussed. We first give a proposition about the existence of constant equilibria. Proposition 1. The following conclusions for the existence of constant equilibria of (2) hold.
Proof. The proofs of (1) and (2) are easy and omitted. The positive equilibrium exists if and only if the following algebraic system has a positive solution: Note that, the first curve in (3) decreases for s > 0 , while the second curve of (3) increases for s > 0 and comes to infinity as s → +∞. Then the two curves have an intersection point inside the first quadrant if and only if m−b1 a1 > b2d θm (R 0 > 1, see Figure 1). Furthermore, we know that the intersection point E + in the first quadrant is unique. In the following, we always assume m > b 1 , by a biological significance. This implies that (2) has at least two nonnegative equilibria. Now we begin to discuss the linear stability of the constant equilibria of (2). For any constant equilibrium (p * , s * ), the linearized system of (2) at (p * , s * ) is Substituting (p(x, t), s(x, t)) T = e λt+iσx (c 1 , c 2 ) T into (4) leads to the characteristic equation: where λ is a complex number, σ is a real number, i is the imaging unit, and (c 1 , c 2 ) T is a constant vector. Replacing (p * , s * ) by E 0 , E 1 , E + in (5), respectively, and applying the method of eigenvalue analysis, we can obtain the following proposition easily. The proof is simple and omitted here.
Proposition 2. Assume m > b 1 . The following conclusions for the linear stability of constant equilibria of (2) hold.
In what follows in this section, we always assume R 0 > 1, c > c m , and denote λ i := λ i (c) (i = 1, 2) for a given c > c m without causing confusion.
3.1. An invariant convex set Γ. In order to apply Schauder fixed point theorem to obtain a solution of (6), we have to construct a nonempty, closed and convex set Γ. The following four lemmas give four functions, which are used as the upper and lower bounds of Γ. In order for complete understanding of the construction of Γ, we will put the argument details into the Appendix.
is the positive root of a quadratic equation a 2 s 2 , and σ > p 1 is a sufficiently large number.
, and q > 1 is a sufficiently large number.
We could define a set Γ as follows: It is obvious that Γ is a nonempty, closed and convex set in C(R, R 2 ) and bounded with the maximum norm.

Traveling wave solutions. Define an operator
are large enough such that for any ξ ∈ R, hold for any (P 1 , S 1 ), (P 2 , S 2 ) ∈ Γ satisfying (P 1 , S 1 ) ≤ (P 2 , S 2 ). The equation (6) can be rewritten as Obviously, the linear parts of (11) leads to two algebraic equations It is obvious that a fixed point of F is a solution of (6), and vice verse. Now we shall argue that F defined on Γ satisfies all the conditions in Schauder fixed point theorem. Since some proofs are standard, and we will ask the readers to refer [9,22] for some details.
From the continuity of the functions, for any ξ ∈ R, there hold Now, for any (P, S) ∈ Γ, we have which means F = (F 1 , F 2 ) maps Γ into Γ. The proof is complete.
3.3. Asymptotic behaviors of traveling wave solutions. In this subsection, we shall discuss the asymptotic behaviors of traveling wave solutions of system (2). In the following, we always assume that (P, S) ∈ Γ is a traveling wave solution of system (2) obtained in Theorem 3.9.
5. Concluding discussions. This article established the existence of traveling wave solutions connecting the nonzero boundary equilibrium E 1 to the positive equilibrium E + for the diffusive producer-scrounger model (2). It is showed that the existence of traveling wave is determined by a threshold value R 0 and a wave speed threshold c m . One may call that c m > 0 is the minimal wave speed although we cannot show the existence of traveling wave solution for c = c m at the moment. In detail, when R 0 > 1 and c > c m , the traveling waves do exist. But in the case R 0 > 1 and c < c m or the case R 0 ≤ 1, the traveling waves do not exist. However the dynamics in the case c = c m remains an open problem. It seems that R 0 plays a role like the basic reproducing number in SIR epidemic models.
In biology, the existence of traveling wave solutions connecting E 1 to E + implies a spatio-temporal co-existence state between the two species: producer and scrounger, while the scrounger species invade the habitat of producer species. In this sense, the producer species' high productivity and their weak ability to avoid food to be stolen, as well as the slow diffusion of the scrounger species are more suitable for their co-existence.
Appendix. In this appendix, we shall give all proofs of Lemma 3.2-3.5.