Chaos in a nonautonomous eco-epidemiological model with delay

• Published in: Applied Mathematical Modelling Vol. 79; pp. 865 - 880
• Main Authors: Samanta, Sudip, Tiwari, Pankaj Kumar, Alzahrani, Abdullah K., Alshomrani, Ali Saleh
• Format: Journal Article
• Language: English
• Published: England Elsevier Inc 01.03.2020; Elsevier BV
• Subjects: Asymptotic methods; Chaos; Chaos theory; Computer simulation; Dynamical systems; Eco-epidemiology; Economic models; Epidemiology; Global stability; Hopf bifurcation; Incubation delay; Liapunov exponents; Limit cycle oscillations; Mathematical models; Nonlinear dynamics; Positive periodic solution; Predator-prey simulation; Quasi-Periodic Oscillations; Seasonal forcing; Stability; Time lag; Positive periodic solution; Chaos; Seasonal forcing; Incubation delay; Eco-epidemiology; Global stability
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2019.11.006

•We propose and analyse a delayed nonautonomous predator-prey model with disease in prey.•Derived the sufficient conditions for global asymptotic stability of the positive periodic solutions.•Autonomous system develops only limit cycle oscillations through a Hopf-bifurcation for increasing the values of delay.•Corresponding nonautonomous system shows chaotic dynamics for increasing the delay parameter.•We draw Poincare map and maximum Lyapunov exponent to identify the chaotic behaviour of the system. In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov’s functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system’s behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.