Hopf bifurcation in a delayed prey-predator model with prey refuge involving fear effect

This work investigates a prey-predator model featuring a Holling-type II functional response,in which the fear effect of predation on the prey species,as well as prey refuge,are considered.Specifically,the model assumes that the growth rate of the prey population decreases as a result of the fear of predators.Moreover,the detection of the predator by the prey species is subject to a delay known as the fear response delay,which is incorporated into the model.The paper establishes the preliminary conditions for the solution of the delayed model,including positivity,boundedness and permanence.The paper discusses the existence and stability of equilibrium points in the model.In particular,the paper considers the discrete delay as a bifurcation parameter,demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter.The direction and stability of periodic solutions are determined using central manifold and normal form theory.Additionally,the global stability of the model is established at axial and positive equilibrium points.An extensive numerical simulation is presented to validate the analytical findings,including the continuation of the equilibrium branch for positive equilibrium points.

Codimension-2 bifurcation in a discrete predator-prey system with constant yield predator harvesting

This work investigates the bifurcation analysis in a discrete-time Leslie-Gower predatorprey model with constant yield predator harvesting.The stability analysis for the fixed points of the discretized model is shown briefy.In this study,the model undergoes codimension-1 bifurcation such as fold bifurcation(limit point),flip bifurcation(perioddoubling)and Neimark-Sacker bifurcation at a positive fixed point.Further,the model exhibits codimension-2 bifurcations,including Bogdanov-Takens bifurcation and generalized fip bifurcation at the fixed point.For each bifurcation,by using the critical normal form coefficient method,various critical states are calculated.To validate our analytical findings,the bifurcation curves of fixed points are drawn by using MATCONTM.The system exhibits interesting rich dynamics including limit cycles and chaos.Moreover,it has been shown that the predator harvesting may control the chaos in the system.

Hopf bifurcation and chaos in a Leslie–Gower prey–predator model with discrete delays

In this work,a Leslie–Gower prey-predator model with two discrete delays has been investigated.The positivity,boundedness and persistence of the delayed system have been discussed.The system exhibits the phenomenon of Hopf bifurcation with respect to both delays.The conditions for occurrence of Hopf bifurcation are obtained for different combinations of delays.It is shown that delay induces the complexity in the system and brings the periodic oscillations,quasi-periodic oscillations and chaos.The properties of periodic solution have been determined using central manifold and normal form theory.Further,the global stability of the system has been established for different cases of discrete delays.The numerical computation has also been performed to verify analytical results.

EFFECTS OF DELAY AND SEASONALITY ON TOXIN PRODUCING PHYTOPLANKTON-ZOOPLANKTON SYSTEM

A dynamical model for toxin producing phytoplankton and zooplankton has been formu- lated and analyzed. Due to gestation of prey, a discrete time delay is incorporated in the predator dynamics. The stability of the delay model is discussed and Hopf bifurcation to a periodic orbit is established. Stability and direction of bifurcating periodic orbits are investigated using normal form theory and center manifold arguments. Global existence of periodic orbits is also established. To substantiate analytical findings, numerical simu- lations are performed. The system shows rich dynamic behavior including chaos and limit cycles. The influence of seasonality in intrinsic growth parameter of the phytoplankton population is also investigated. Seasonality leads to complexity in the system.