Abstract
A Lotka–Volterra predator–prey system incorporating fear effect of the prey and the predator has other food resource is proposed and studied in this paper. It is shown that the trivial equilibrium and the predator free equilibrium are both unstable, and depending on some inequalities, the system may have a globally asymptotically stable prey free equilibrium or positive equilibrium. Our study shows the fear effect is one of the most important factors that lead to the extinction of the prey species. Such a finding is quite different from the known result. Numeric simulations are carried out to show the feasibility of the main results.
Highlights
1 Introduction The aim of this paper is to investigate the dynamic behaviors of the following Lotka– Volterra predator–prey system incorporating fear effect of the prey and the predator having other food resource: du =
– du – au2 – puv, dt 1 + kv d1v2, where u and v are the density of prey species and the predator species at time t, respectively. r0 is the birth rate of the prey species, d is the death rate of the prey species, a is the intraspecific competition of the prey species, m is the intrinsic grow rate of the predator species; p denotes the strength of interspecific between prey and predator; c is the conversion efficiency of ingested prey into new predators; d1 is the intraspecific competition of the predator species; k is the level of fear, which is due to anti-predator behaviors of the prey
The aim of this paper is to investigate the dynamic behaviors of the system (1.1), and to find the influence of the fear effect on the prey species
Summary
Holds, i.e., the positive equilibrium is locally asymptotically stable as long as it exists. Is unstable since one of the eigenvalues is positive It follows from (2.3) that the Jacobian matrix of the system (1.1) about the predator free equilibrium. Both eigenvalues of J(E3(u∗, v∗)) have negative real parts, E3(u∗, v∗) is locally asymptotically stable. This ends the proof of Theorem 2.2. Remark 2.2 Theorem A shows that under some suitable assumption, the trivial equilibrium E0 and the predator free equilibrium E1 of system (1.3) are globally asymptotically stable, Theorem 2.2 shows that E0 and E1 of system (1.1) are unstable. System (1.3) has no prey free equilibrium, while under some assumption, system (1.1) admits a prey free equilibrium, which is locally asymptotically stable
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