Abstract
In this paper, a kind of fractional-order predator–prey (FOPP) model with a constant prey refuge and feedback control is considered. By analyzing characteristic equations, we carry out detailed discussion with respect to stability of equilibrium points of the considered FOPP model. Besides, the effects of prey refuge and feedback control are also studied by numerical analysis. Our study reveals that prey refuge and feedback control can be used to adjust the biomass of prey species and predator species such that prey species and predator species finally reach a better state level.
Highlights
1 Introduction The investigation of the predator–prey model with prey refuge is an interesting and popular topic; here prey refuge can be reviewed as any strategy that decreases the risk of predation, including but not limited to burrows, prey aggregations, and heavy vegetation
Some eminent researchers [1,2,3,4,5,6] have studied the effects of prey refuge for integer-order predator–prey models and concluded that the prey refuge has a positive stabilizing effect on the predator–prey interaction, and prey individuals can be partially protected from predation
Ma [5] considered a kind of integer-order predator–prey system with a constant prey refuge, as follows:
Summary
The investigation of the predator–prey model with prey refuge is an interesting and popular topic; here prey refuge can be reviewed as any strategy that decreases the risk of predation, including but not limited to burrows, prey aggregations, and heavy vegetation. If 1 ≥ 0, we can derive from (8) that three eigenvalues λ1, λ2 and λ3 are negative, which imply that the equilibrium point E0 is locally asymptotically stable for all 0 < q < 1. We can see from the proof of Theorem 1 that even if the eigenvalues of the characteristic equation have positive real parts, the trivial equilibrium point E0 of fractional-order system (3) is still locally asymptotically stable when condition (H2) is satisfied. For the corresponding integer-order system, namely q = 1, if the eigenvalues of the characteristic equation have positive real parts, the trivial equilibrium point E0 is unstable. Theorem 2 The predator-extinction equilibrium point E1 of system (3) is locally asymptotically stable if either of the following criteria is satisfied:. D(P) denotes the discriminant of the cubic polynomial P(λ), as follows:
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