Global Stability and Hopf-bifurcation Analysis of Biological Systems using Delayed Extended Rosenzweig-MacArthur Model

Abstract

This paper investigates the global asymptotic stability of a Delayed Extended Rosenzweig-MacArthur Model via Lyapunov-Krasovskii functionals. Frequency sweeping technique ensures stability switches as the delay parameter increases and passes the critical bifurcating threshold.The model exhibits a local Hopf-bifurcation from asymptotically stable oscillatory behaviors to unstable strange chaotic behaviors dependent of the delay parameter values. Hyper-chaotic fluctuations were observed for large delay values far away from the critical delay margin. Numerical simulations of experimental data obtained via non-dimensionalization have shown the applications of theoretical results in ecological population dynamics.

Spirals and Cycles of Biological Systems via Extended Rosenzweig-MacArthur Model with Ratio-dependent Functional Response

“…12, No. 9;2018 By Routh-Hurwitz conditions and Descartes rule of sign, the next proposition follows.…”

“…12, No. 9;2018 Behavior of the Model at Prey-free Equilibrium Point: Consider the ecological parameters; α = 0.0031, κ = 5.92, ξ = 0.50, σ = 1.50, µ = 0.42, η = 0.40, ε = 1.20, then system (10) has a prey-free equilibrium point in the absent of predators and super-predators interactions at E 1 (u * = 3.0630, y * = 0, v * = 0) with negative eigenvalues (−0.1484, −1.0954, −0.6753) satisfying equation 14. Thus, every trajectory of system (10) converges, and asymptotically stable to the fixed point E 1 as seen in the phase portrait of figure 2.…”

“…The model shows dynamical behaviors such as stability, limit cycles, hopf-bifurcation, persistence and global stability,periodic solutions, and stability dynamics with deviated arguments or delays. (see., Joshua, Akpan, & Madubueze 2016;Joshua, Akpan, Madubueze, Adebimpe, 2017, Joshua, & Akpan, 2018. In this article, system (4)is modified with a ratio-dependent functional response and its dynamical complexity is studied.…”