Dynamical analysis of a delayed tri-trophic food chain consisting of prey, an intermediate, and a top predator is investigated in this paper. The additive Allee effect is introduced in the prey population, and it is assumed that there is a time lag due to the gestation effect in the intermediate predator. The interference among the prey and the intermediate predator is according to Holling type II, while the interaction between the intermediate and top predators follows the Crowley–Martin functional response. The local stability and bifurcation analysis of the proposed model at the interior equilibrium point are studied. Numerical simulations are provided to ensure the mathematical results.

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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.

In this article, we propose the Holling type‐II predator‐prey model involving cannibalism and gestation delay in predators. We study the existence of all possible equilibrium points of the proposed model. We give the condition for the local stability and Hopf bifurcation analysis for the nondelayed model. Next, we also establish the local stability and Hopf bifurcation analysis for the corresponding delayed model. Finally, we discuss how cannibalism and delay play an important role in stabilizing and destabilizing the proposed system both theoretically and numerically.

Fear and prey refuges are two significant topics in the ecological community because they are closely associated with the connectivity of natural resources. The effect of fear on prey populations and prey refuges (proportional to both the prey and predator) is investigated in the nonlinear-type predator-harvested Leslie–Gower model. This type of prey refuge is much more sensible and realistic than the constant prey refuge model. Because there is less research on the dynamics of this type of prey refuge, the current study has been considered to strengthen the existing literature. The number and stability properties of all positive equilibria are examined. Since the calculations for the determinant and trace of the Jacobian matrix are quite complicated at these equilibria, the stability of certain positive equilibria is evaluated using a numerical simulation process. Sotomayor’s theorem is used to derive a precise mathematical confirmation of the appearance of saddle-node bifurcation and transcritical bifurcation. Furthermore, numerical simulations are provided to visually demonstrate the dynamics of the system and the stability of the limit cycle is discussed with the help of the first Lyapunov number. We perform some sensitivity investigations on our model solutions in relation to three key model parameters: the fear impact, prey refuges, and harvesting. Our findings could facilitate some biological understanding of the interactions between predators and prey.

This study presents an extended dissipative analysis of fractional order fuzzy networked control system with uncertain parameters. First, we designed the network-based fuzzy controller for the considered model. Second, a novel Lyapunov-Krasovskii functional (LKF) approach, inequality techniques, and some sufficient conditions are established, which make the proposed system quadratically stable under the extended dissipative criteria. Subsequently, the resultant conditions are expressed with respect to linear matrix inequalities (LMIs). Meanwhile, the corresponding controller gains are designed under the larger sampling interval. Finally, two numerical examples are presented to illustrate the viability of the obtained criteria.