This article investigates a fractional-order Predator-prey model incorporating prey refuge and anti-Predator behaviour on Predator species. For our proposed model, we prove the existence, uniqueness, non-negativity and boundedness of solutions. Further, all biologically possible equilibrium points and their stability analysis for the proposed system are carried out with the linearization process. Moreover, by using an appropriate Lyapunov function, the global stability of the co-existence equilibrium point is studied. Finally, we provide numerical simulations to demonstrate how the theoretical approach  is  consistent.

A discrete Predator-prey model with harvesting effects on both Predator and prey is examined to reveal its chaotic dynamics. The model's existence and local stability analysis are investigated. It is demonstrated that the model experiences period-doubling bifurcation and Neimark-Sacker bifurcation by using bifurcation theory. Moreover, numerical examples are used to demonstrate the consistency of analytical conclusions as well as the model's complexity owing to harvesting effects. It is shown that changing the harvesting parameters affects not only the number of fixed points in the model, but also the occurrence of different bifurcations.

In this paper, we have proposed a study on controllability and optimal harvesting of a prey Predator model and mathematical non linear formation of the equation equilibrium point of Routh harvest stability analysis. The problem of determining the optimal harvest policy is solved by invoking Pontryagin′s maximum principle dynamic optimization of the harvest policy is studied by taking the combined harvest effect as a dynamics variable.

In ecological systems, Predator-prey contact is seen as something that happens naturally. How does the density of prey populations affect Predators? This is a naturally occurring issue in ecosystems. Even though it plays a little role in population dynamics, Predators in most ecological models lower prey numbers by direct killing.  Research on vertebrates has shown that Predator aversion may impact prey population dynamics and reproductive rates. There has been new research on mathematical models of Predator-prey systems that include a range of Predator-functional responses that include the fear effect. Researchers in this research failed to account for the impact of fear on prey mortality rates. In light of the above, our study focuses on analyzing a Predator-prey system that incorporates the cost of perceived fear into reproductive processes using a Holling type-IV functional response. The scheme also includes intraspecific competition within the Predators and a gestation delay to make the interactions more realistic and natural. The increase of the Predator population is constrained by a high Predator-to-prey density ratio by this extra intraspecific competition term. These dynamic model's fundamental aspects such as non-negative, boundedness of solutions, and viability of equilibria are investigated, and adequate conditions are discovered. Both the local and global stability of the system are obtained with sufficient conditions on its functionals and parameters. This study makes a major impact in that it creates a novel technique to quantify some important, regulating system resilience parameters, it also studies the presence of Hopf bifurcation when the time lag parameters exceed the critical values by looking at the related characteristic equation. Furthermore, we addressed how time delay factors reaching thresholds cause the Hopf bifurcation. Numerous numerical examples are used to validate all of these theoretical inferences, and simulations are given to help visualize the examples.

In this paper, a new stage structured prey-Predator model with linear functional response is proposed and studied. The stages for prey have been considered. The proposed mathematical model consists of three nonlinear ordinary di erential equations to describe the interaction among juvenile prey, adult prey and Predator populations. The model is analyzed by using linear stability analysis to obtain the conditions for which our model exhibits stability around the possible equilibrium points. Besides this a rigorous global stability analysis has been performed for our proposed model by using Li and Muldowney approach (geometric approach). Global stability conditions for the proposed model are described in the form of theorem. This is not a case study, hence the real parameters are not available for this model. However, model may be simulated by using hypothetical set of parameters. Investigation of real parameters for the proposed model is an open problem.

In this paper, we investigate the dynamical complexities of a prey Predator model prey refuge providing additional food to Predator. We determine dynamical behaviors of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Hopf bifurcation conditions are derived analytically. We especially approximate a family of limit cycles emanating from a Hopf point. The analytical results are in well agreement with the numerical simulation results. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Hopf, and Bogdanov-Takens bifurcations.

In this paper, we study the bifurcation of nontrivial steady state solutions for a cross-diffusion prey-Predator model with homogeneous Neumann boundary conditions. The existence of positive steady state solutions near a bifurcation point is proved using a crossing curve bifurcation theorem. We consider a situation where the transversality condition is not satisfied. Unlike the case in saddle-node bifurcation, the solution set is a pair of transversally intersecting curves.