An investigation of the parameter space in a tri-trophic food chain model with refuge

Aside from humans, the instinct to run and hide from dangerous situations is ingrained in individuals of many other species. Taking refuge is a very common precautionary strategy among prey species to avoid direct contact with their predators. In this article, we investigate a three-species food chain model where the logistically growing basal prey seeks refuge to avoid the middle predator. To assess the impact of varying levels of refuge, we explore the system dynamics in the biparametric space of refuge and another important system parameter related to the half-saturation constant of the basal prey. The isospike diagrams along with the largest Lyapunov exponent diagrams show a stabilizing role of refuge. Apart from this, we observe an abundance of organized periodic structures embedded inside the chaotic zone of the biparametric space. The most notable ones include the shrimp-shaped structures and an “elliptic disc” shaped structure. The article also explores the coexistence between multiple pairs of coexisting attractors and one instance of spike-bubbling phenomena. The basins of attraction for the coexisting attractors also have a self-similar pattern. We also modify the present model by replacing the density-dependent prey refuge with constant proportion of prey refuge, and then explore the dynamics of the modified model within the same biparametric space and find similar complexity in its dynamics. We compare the effects of both types of refuge on the density variations of all three species when they show stable coexistence and observe that density-dependent refuge is more beneficial to the system from the standpoint of species conservation.