Optimal predator control policy and weak Allee effect in a delayed prey–predator system

In this article, a system of delay differential equations to represent the predator–prey dynamics with weak Allee effect in the growth of predator population is discussed. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform p...

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Bibliographic Details
Published inNonlinear dynamics Vol. 90; no. 4; pp. 2929 - 2957
Main Author Biswas, Santanu
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.12.2017
Springer Nature B.V
Subjects
Automotive Engineering
Bistability
Chaos theory
Classical Mechanics
Computer simulation
Control
Delay
Differential equations
Dynamical Systems
Engineering
Gestation
Hopf bifurcation
Mathematical models
Mechanical Engineering
Original Paper
Parameters
Predator-prey simulation
Response time
Stability analysis
Time lag
Vibration
Chaos
Optimal predator control policy
Allee effect
Time delay
Bogdanov–Takens bifurcation
Hopf bifurcation
Saddle-node bifurcation
Bistability
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ISSN0924-090X
1573-269X
DOI10.1007/s11071-017-3854-x

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Summary:In this article, a system of delay differential equations to represent the predator–prey dynamics with weak Allee effect in the growth of predator population is discussed. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system are analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section, and maximum Lyapunov exponent. By constructing a suitable Lyapunov functional for the time-delayed model, global asymptotic stability analysis of the positive equilibrium points has been performed separately. It can be observed that the Allee parameter θ can destabilize the non-delay system, whereas θ and the attack rate of predator can stabilize the time-delayed model and can control the chaotic oscillations through period-halving bifurcation. The optimal predator control policy with Allee parameter ( θ ) as the control parameter is also discussed.
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ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-017-3854-x