Abstract
This paper is concerned with the dynamics of a predator–prey system with three species. When the domain is bounded, the global stability of positive steady state is established by contracting rectangles. When the domain is mathbb{R}, we study the traveling wave solutions implying that one predator and one prey invade the habitat of another prey. More precisely, the existence of traveling wave solutions is proved by combining upper and lower solutions with a fixed point theorem, and the asymptotic behavior of traveling wave solutions is obtained by the idea of contracting rectangle. Moreover, we show the nonexistence of traveling wave solutions by applying the theory of the asymptotic spreading.
Highlights
In population dynamics, predator–prey systems have been widely studied due to their importance as well as plentiful dynamical behaviors
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, 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]
Summary
Predator–prey systems have been widely studied due to their importance as well as plentiful dynamical behaviors. We study the global stability of the positive equilibrium with the help of contracting rectangles [42] when the domain is bounded, the existence as well as noexistence of traveling wave solutions when x ∈ R. Positive solutions of (4.1)–(4.2) describe the following biological process: at any fixed location x ∈ R, there was only one prey a long time ago (t → –∞ such that x + ct → –∞), and the predator and two preys will coexist after a long-term species interaction (t → +∞ such that x + ct → +∞). We will establish the existence of a nontrivial positive solution of (4.3) by combining Schauder’s fixed point theorem with the method of upper and lower solutions (for quasimonotone systems, we refer to [20, 34, 46, 49]).
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