Abstract
We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse amongndifferent patches(n≥2). By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive equilibrium of this model is unique and globally asymptotically stable if it exists.
Highlights
Coupled systems on networks are used to describe a wide variety of physical, natural, and artificial complex dynamical systems, such as neural networks, biological systems, and the spread of infectious diseases in heterogeneous populations
We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse among n different patches (n ≥ 2)
We generalize the model of the n-patch predator-prey model of [12] to a three-species food chain model where prey species, mid-level predator species, and top predator species can disperse among n different patches (n ≥ 2)
Summary
Coupled systems on networks are used to describe a wide variety of physical, natural, and artificial complex dynamical systems, such as neural networks, biological systems, and the spread of infectious diseases in heterogeneous populations (see [1,2,3,4,5,6] and the references therein). In [10], Kuang and Takeuchi considered a predator-prey model in which preys disperse among two patches and proved the uniqueness and global stability of a positive equilibrium by constructing a Lyapunov function. A graph theoretic approach was proposed to construct Lyapunov functions for some general coupled systems of ordinary differential equations on networks, and the global stability was explored in [11, 12]. There may be more species in some habitats and they can construct a food chain; in this case it is more realistic to consider a multiple species predator-prey system Based on this fact, in this paper, we investigate the following three-species food chain model in a patchy environment:.
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