Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive preypredator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.
In this paper,the Caputo fractional derivative is assumed to be the prey-predator model.In order to create Caputo fractional differential equations for the prey-predator model,a discretization process is first used.The fixed points of the model are categorized topologically.We identify requirements for the fixed points of the suggested prey-predator model's local asymptotic stability.We demonstrate analytically that,under specific parametric conditions,a fractional order prey-predator model supports both a Neimark-Sacker(NS)bifurcation and a Flip bifurcation.We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory.The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey-predator model.As the bifurcation parameter is increased,the system displays chaotic behavior.Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations,phase portraits,invariant closed cycles,and attractive chaotic sets in addition to validating analytical conclusions.The suggested prey-predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology,which will also visualize the chaotic state for various biological parameters.
In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the Allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.
In this paper, we will study a class of discrete Leslie-Gower prey-predator models, which is a discretization of the continuous model proposed by Leslie and Gower in 1960. First, we find all fixed points, use hyperbolic and non-hyperbolic conditions to give the types of fixed points, and then analyze the bifurcation properties of non-hyperbolic fixed points. The generating conditions of Flip bifurcation and Neimark-Sacker bifurcation at fixed points are studied. Finally, numerical simulations of Flip bifurcation and Neimark-Sacker bifurcation are given.
In biological pest control systems,several pests(including insects,mites,weeds,etc.)are controlled by biocontrol agents that rely primarily on predation.Following this biocontrol management ecology,we have created a three-tier prey-predator model with prey phase structure and predator gestation delay.Several studies have demonstrated that predators with Holling type-II functional responses sometimes consume immature prey.A study of the well-posedness and local bifurcation(such as saddle-node and transcritical)near the trivial and planer equilibrium points is carried out.Without any time lag,the prey development coeficient has a stabilizing impact,while increasing attack rate accelerates instability.Energy transformation rate and handling time are shown to cause multiple stability switches in the system.Numerical results demonstrate time delay is the key destabilizer that destroys stability.Our model can replicate more realistic events by including time-dependent factors and exploring the dynamic behavior of nonautonomous systems.In the presence of time delay,sufficient conditions of permanence and global attractivity of the nonautonomous system are derived.Finally,MATLAB simulations are performed to validate the analytical findings.
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.