Stability and Bifurcation of a Prey-Predator System with Additional Food and Two Discrete Delays

Kumar, Ankit; Dubey, B.

doi:10.32604/cmes.2021.013206

Cited by 3 publications

(2 citation statements)

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“…The series of prey-predator models are paid widespread attention because they reflect the ecological phenomenon existing in the real world generally. The dynamic behavior of those models are investigated in-depth and a large number of valuable results have been obtained in the past few decades [1][2][3][4]. To keep the ecological balance, it is necessary to increase the survival rate of the prey in some ecosystems.…”

Section: Introductionmentioning

confidence: 99%

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

2022

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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.

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Section: Introductionmentioning

confidence: 99%

“…The existence conditions of the coexistence equilibrium point of the proposed system are deduced. (2) The conditions of emergence of Hopf bifurcation for the generalized system are determined. In other words, the critical value of delay that the system switches from asymptotical stability to symmetric periodic oscillation is deduced.…”

Section: Introductionmentioning

confidence: 99%

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

2022

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Local and global dynamics of a prey–predator system with fear, Allee effect, and variable attack rate

Kumar,

K P

2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Fear prompts prey to adopt risk-averse behaviors, such as reduced foraging activity, increased vigilance, and avoidance of areas with high predator presence, which affects its reproduction. In a real scenario, a population requires a minimum density to avoid extinction, known as an Allee threshold. In light of these biological factors, we propose a predator–prey model with (i) a fear effect in a prey population, (ii) an Allee effect in a predator population, and (iii) a non-constant attack rate that modifies the functional response. We ensured the non-negativity and boundedness of the solutions and examined the local and global stability status for each existing steady state solutions. We investigated some deep dynamical properties of the system by varying different parameters, such as cost of fear in prey and strength of the Allee effect in predators and their mortality rate. In codimension one bifurcations, we observed saddle node, Hopf, homoclinic, and coalescence of two limit cycles. Additionally, codimension two bifurcations were observed, including Bautin and Bogdanov Takens bifurcations. To provide a clearer understanding of these bifurcations, we conducted biparametric analysis involving the fear and Allee parameters, as well as the fear parameter and predator mortality rate. Our investigation shows that cost of fear and strength of Allee strongly influences the survival status of the predator. Furthermore, bistability and tristability reveal that the survival and extinction of predator are dependent on the initial population level. Numerical simulations and graphical illustrations are provided to support and validate our theoretical findings.

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Stability and bifurcation in a predator-prey system with effect of fear and additional food

K P,

Kumar

2024

MATH

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<abstract><p>In the present study, we propose and analyze a three-dimensional prey-predator model. The prey grows logistically in the absence of the predator and their relationship follows the Crowley-Martin type functional response. In this paper, we examine the impact of supply of the additional food to the predators and the influence of fear in the prey population. Since the predator depends partially on the provided other resources, we incorporate a novel parameter, the degree of dependence, which basically demonstrates how dependent the predator is on the prey population. We investigate the steady-state solutions, and their local and global behavior, which contributes to understanding the long-term dynamics of the interaction. We have shown that the degree of dependence and the cost of fear both can cause periodic orbits to appear in the system via a Hopf-bifurcation. Our findings show that with the newly introduced parameter, we can control the oscillations from the system, which helps to balance the ecosystem. The direction and stability have also been investigated using the center manifold theorem and normal form theory. Last, we perform an extensive numerical simulation to validate our theoretical findings. Our main goal of this work is to maintain the ecological balance in the presence of fear effect and additional food for predators.</p></abstract>

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Stability and Bifurcation of a Prey-Predator System with Additional Food and Two Discrete Delays

Cited by 3 publications

References 40 publications

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

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

Local and global dynamics of a prey–predator system with fear, Allee effect, and variable attack rate

Stability and bifurcation in a predator-prey system with effect of fear and additional food

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