Chaos in a seasonal food-chain model with migration and variable carrying capacity

Abstract

The carrying capacity’s functional dependence illustrates the reality that any species’ activities can enhance or diminish its carrying capacity. Migration is the need of many species to achieve better opportunities for survival. In a tri-trophic system, the middle predator often immigrates to consume its prey and often emigrates to secure themselves from predators. This work deals with formulating and investigating a mathematical model reflecting the aforementioned ecological aspects. We perform a detailed analysis to prove the boundedness of the solutions. Further, we examine the existence and stability of equilibrium points, followed by the bifurcation analysis. We explore various global and local bifurcations like Hopf, saddle-node, transcritical, and homoclinic for the critical parameters (measuring the impact of prey activities on the carrying capacity) and (measuring the migration rate of a predator). Higher values of generate unpredictability, which helps explain the enrichment paradox. The presence of a chaotic attractor and bi-stability of node-node type is demonstrated via numerical simulation. The migratory behavior of middle predators can control chaos in the system. Furthermore, we study the proposed model in the presence of seasonal fluctuations. Persistence of the non-autonomous system, existence, and global stability of periodic solutions are analyzed theoretically. The seasonality in brings the bi-stability between chaotic and periodic attractors, and seasonality in growth rate of the prey causes bi-stability between 2-periodic and 4-periodic attractors. Moreover, the bi-stability in the autonomous system shifts to the global stability of an equilibrium in the seasonal model due to the seasonality in . When birth and death rates are seasonal along with , the extinction of one or more populations is possible. The non-autonomous system also exhibits bursting oscillations when seasonality is present in the death rate. Our findings reveal that the population’s intense constructive and destructive actions can allow the basal prey to thrive while eradicating both predators.