Abstract
This paper is concerned with a fractional diffusion predator-prey system with Holling type-II functional response in a bounded domain with no flux boundary condition. The local and global stabilities are investigated and sufficient conditions of stabilities are obtained. The existence of the non-constant steady states is considered and the sufficient conditions of the existence for the non-constant steady states are also obtained. The results show that the predator and the prey can coexist under some suitable conditions with fractional diffusion.
Highlights
1 Introduction The population always move from the higher concentration to the lower concentration by spatial heterogeneity of species. This process is often regarded as Brownian motion, which is sometimes called Gaussian diffusion or normal diffusion, and this process can be described by employing the Laplacian operator
The movement of animals cannot often obey the rule of normal diffusion, but it obeys Lévy diffusion which is described by a fractional Laplacian operator
The recent research [ ] has showed that the diffusion of some animals represents Lévy-walk-like behaviors. Based on this fact, we shall consider ( . ) including Lévy diffusion instead of Gaussian diffusion. Thereinto, this diffusion has been successfully used in the epidemic model [ ] and quantum physics [, ]. (– )α is a corresponding infinitesimal generator which is a fractional Laplacian operator, and this fractional diffusion generated by (– )α describes a pure jump process and is an anomalous diffusion
Summary
The population always move from the higher concentration to the lower concentration by spatial heterogeneity of species. Many authors have established the existence of non-constant steady states in normal diffusion population models [ , – ]. The main objective of this paper is to investigate the existence of non-constant steady states which arises from the fractional diffusion system In Section , the existence of non-constant positive equilibrium is investigated. We shall prove that there exists a positive global solution for system For this end, we first give out the following lemma. For the existence of positive constant solution w∗, it is necessary to assume that a. The following theorem shows that the positive constant w∗ of
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