Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

Abstract

Abstract. We study the existence of traveling wave solutions for a diffusive predatorprey system. The system considered in this paper is governed by a Sigmoidal response function which is more general than those studied previously. Our method is an improvement to the original method introduced in the work of Dunbar [2,3]. A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in [2,3].The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski's theorem which is much simpler than the extended version in Dunbar's work.

Dynamics of a predator–prey system with three species

“…When the spatiotemporal dynamics is concerned, since the pioneer work in [14,25], much attention has been paid to the traveling wave solutions of parabolic equations; we refer to Volpert et al [44] for some earlier results and a survey paper by Zhao [55] for some recent conclusions. If a system is of predator-prey type with two species, then several methods, including phase analysis, shooting methods, Conley index and fixed point theorem, have been applied to establish the existence of traveling wave solutions; we refer to some important results by Dunbar [9][10][11], Gardner and Smoller [16], Gardner and Jones [15], Chen et al [7], Huang et al [19], Huang and Zou [21], Huang [23], Hsu et al [18], Li and Li [26], Lin et al [30], Lin [28,29], Lin et al [32], Pan [37], Wang et al [45], Wang et al [47], Zhang et al [51].…”

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

“…However, many concrete models may not fit well with the frame established in Gardner's abstract approaches. Later, the authors of the works [1,12,15,18,19,20,23,26] improved the technique of Dunbar's approach to investigate the traveling wave solutions of system (1) with different types of functional responses. For examples, Huang [20] used a geometric approach to derive the existence of traveling waves for some classes of non-monotone reaction-diffusion systems.…”

Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections

“…However, for some non-cooperative systems, it is challenging to obtain the asymptotic spreading speed, especially for diffusive SIR models and virus models. Travelling wave solutions are an important tool which can be used to describe the spreading speed of population [6,15,7,14,22,56,59,16,54,43,25,17,18,57,3,41,49,62,30].…”

“…In [18], Huang made a further investigation for the model proposed in [17] to abandon the restriction condition on the diffusion coefficients. The Schauder's fixed point theorem is also widely applied to show the existence of travelling wave solutions connecting two steady states [15,7,22,56,59,54,43,25,17,18,57,3]. In [57], Zhang et al developed another method to show the existence of weak travelling wave solutions for a class of non-cooperative systems, which allows us to avoid the difficulties in studying the detailed final state (i.e., steady states, periodic solutions, etc.…”

“…In 1-D whole space-travelling wave solutions for model (2). In this section, we investigate the existence of travelling wave solutions when R 0 > 1 in an unbounded domain ( [36,15,7,22,56,59,54,43,25,17,18,57,3]). For convenience, we assume that D 0 = D 1 = D 2 = D. In such a special case, following the same derivation in [50], we can obtain the particular form for the kernel Γ(t, x): (2) is transformed into (omitting the hats on U for simplicity)…”