The discrete-time system has more complex and chaotic dynamical behaviors as compared to the continuous-time system. This paper extends a discrete Leslie–Gower predator-prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings.

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One of the most important goals of theoretical ecologists is to find a strategy for controlling the chaos in ecological models to maintain healthy ecosystems. We investigate the influence of fear and the supply of additional food to predators in a delayed tri‐trophic food chain model, where interference among the species is determined by Holling type‐II functional response. The proposed model incorporates the fact that the growth rates of prey and middle predators are decreased due to the fear of middle and special predators, respectively. The additional food is supplied to both predators by either nature or external agencies. Furthermore, the poisoned prey in the predator's environment is taken into account. In the proposed model, we use two time delays: one for the growth term of the middle predator and another for the gestation delay of the special predator. We determine the conditions for the existence of ecologically feasible equilibrium points and their local and global stability. In addition, we establish the conditions for the existence of Hopf bifurcation around interior equilibrium to seek periodic behavior in nondelayed and delayed models. Numerical investigations are performed to justify the proposed theoretical findings through phase portraits, time series of solutions, and bifurcation diagrams. We observe that the chaotic dynamics of a tri‐trophic food chain model can be controlled by the proper choice of the fear effect and by supplying additional food parameters. As a result, our findings provide more ecological insights into the dynamics of the delayed tri‐trophic food chain model with the fear effect and additional food supply.

One of the main objectives of theoretical ecologists involves finding mechanisms to control the chaos in ecological models to maintain positive densities of the species. Numerous researchers have suggested that, apart from the direct killing in the prey–predator relationship, there are some indirect effects, such as fear of predation. Induced fear can lead to slowing down the growth rate of the prey species, and this non-chemical strategy can be carried over to successive seasons or upcoming generations. In this work, we explore the impact of fear due to predation and its carry-over effect (COE) in a delayed tri-trophic food chain model, whereas the Holling type-II functional response is used to determine the interference among the species. The proposed model is an asymmetric interaction food chain model since the species in this model only kills other species. The growth rate of prey and middle predators is affected due to the respective fear of predation by middle and special predators. The non-delayed model considered in this paper generalizes the models developed by Hastings–Powell and Panday et al. The gestation delay in the special predator’s growth term is incorporated into the proposed model. We determined the essential conditions for the existence of ecologically feasible equilibrium points and their local and global stability. Furthermore, we developed the conditions for the occurrence of the Hopf bifurcation around an interior equilibrium to seek periodic behaviors of delayed and non-delayed models. Numerical examples were performed to justify the proposed theoretical findings and to show the impacts of fear and its COE parameters on the system dynamics through phase portraits, the time series of solutions, and bifurcation diagrams. We discovered that the chaotic behavior of the food chain model can be controlled by using the fear effect and its COE parameters. The dynamics of the delayed food chain model with the fear effect and its COEs are further explored in our findings. Our theoretical findings clearly provide a mechanism to protect and control species populations in ecological systems. It is also essential for developing optimized harvesting strategies in fisheries and pest management in agriculture.