Hopf Bifurcation and Control of a Fractional-Order Delay Stage Structure Prey-Predator Model with Two Fear Effects and Prey Refuge

Abstract

A generalized delay stage structure prey-predator model with fear effect and prey refuge is considered in this paper via introducing fractional-order and fear effect induced by immature predators. Hopf bifurcation and control of this system are investigated though regarding the delay as the parameter. Firstly, by using the method of linearization and Laplace transform, the roots of the characteristic equation of the linearized system of the original system are discussed, and the sufficient conditions for the system exhibits an unstable state of symmetrical periodic oscillation (Hopf bifurcation) are explored. Secondly, a linear delay feedback controller is added to the system to increase the stability domain successfully. Thirdly, numerical simulations are performed to validate the theoretical analysis, and the various impacts on the dynamical behavior of the system occurring by fear effects, prey refuge, and each fractional-order are illustrated, respectively. Furthermore, the influence of feedback gain on the bifurcation critical point is analyzed. Finally, an analysis based on the results and in-depth research about this system under the biological background is stated in the conclusion.

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

“…Given the practical implications of ecosystems, we focus on the coexisting equilibrium. If 1 + aβ − β > 0, then there is unique positive equilibrium E * (ρ * , θ * ) of system (6), where…”

“…(i) If (H4) and (H8) are satisfed, then the trajectories of system (6) converge to the equilibrium point E * for all τ ≥ 0. (ii) If either (H5) or (H6), (H7), and (H8) are satisfed, the trajectories of system (6) tend to E * for τ < τ 0 and is far from E * for τ > τ 0 . System (iii) A Hopf bifurcation is produced at E * when τ � τ 0 and a periodic Oscillation appears.…”

“…We supply several numerical examples for system (3) to testify to the main results in Sections 3 and 4. In numerical simulations, τ 0 ′ of system (3) corresponds to τ 0 of system (6).…”

“…Recently, many populations have been on the verge of extinction due to destructive human exploitation, pollution, and hunting that have had a huge impact on nature and species. Te studies of ecological competition dynamic models have been pushed to the hot spot [6][7][8][9]. Predator-prey systems are an essential component to revealing the competition mechanism, growth law, and control strategy among populations, so it is important to establish a scientifc predator-prey model.…”

Dynamics of a Fractional-Order Prey-Predator Model with Fear Effect and Harvesting

“…Zhang et al [3], Cresswell [4], Xu [5] showed that the fear that consider by the predator is greater effect on the prey species than the direct killing. Lan et al [6] interpreted that the fear effect may occur from not only adult predators but also from juvenile one. Fractional derivatives and integrals have provided a better tool to understanding some biological models, especially differential equations models have been interested due to the memory effect which exists in most biological systems [7,8].…”