Chaos to order — Effect of random predation in a Holling type IV tri-trophic food chain system with closure terms

Abstract

Complex dynamics of modified Hastings–Powell (HP) model (phytoplankton-zoo-plankton-fish) with Holling type IV functional response and density-dependent mortality (closure terms) for top predator species is investigated in this paper. Closure terms describe the mortality of top predator in plankton food chain models. Modified HP model with Holling type IV functional response gives rise to similar type of chaotic dynamics (inverted “teacup attractor”) as observed in original HP model with Holling type II functional response. It is observed that introduction of nonlinear closure terms eliminate chaos and system dynamics becomes stable. Observation of this paper support the “Steele–Henderson conjecture” that, nonlinear closure terms eliminate or reduces limit cycles and chaos in plankton food chain models. Chaotic or stable dynamics are numerically verified by Lyapunov exponents (LE) method and Sil’nikov eigenvalue analysis and also illustrated graphically by plotting bifurcation diagrams. It is assumed that mortality of fish population, caused by higher-order predators (which are not explicitly included in the model) is not constant, rather it exhibits random variation throughout the year. To incorporate the effect of random mortality of fish population, white noise term is introduced into the original deterministic model. It is observed that the corresponding stochastic model is stable in mean square when the intensity of noise is small.