Michaelis-Menten Type Prey Harvesting Research Articles

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Global Stability and Bifurcation Analysis in a Discrete-Time Two Predator-One Prey Model with Michaelis-Menten Type Prey Harvesting

This article studies a discrete-time Leslie-Gower two predator-one prey system with Michaelis-Menten type prey harvesting. Positivity and boundedness of the model solution are investigated. Existence and stability of fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we find out the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation can be developed. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is chosen, which are regulated by a hybrid control method. Examples are provided to illustrate our conclusions.

The Dynamics of a Bioeconomic Model with Michaelis–Menten Type Prey Harvesting

The Dynamics of a Bioeconomic Model with Michaelis–Menten Type Prey Harvesting

Stability and Bifurcation Analysis of a Predator-Prey Model with Michaelis-Menten Type Prey Harvesting

In this paper, we investigated stability and bifurcation behaviors of a predator-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interior equilibrium point were established. Some critical threshold conditions for transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation were explored analytically. Furthermore, It should be stressed that the fear factor could not only reduce the predator density, but also affect the prey growth rate. Finally, these theoretical results revealed that nonlinear Michaelis-Menten type prey harvesting has played an important role in the dynamic relationship, which also in turn proved the validity of theoretical derivation.

Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters

Abstract In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings.

Stability and bifurcation for a stochastic differential algebraic Holling-II predator–prey model with nonlinear harvesting and delay

In this paper, a stochastic delayed differential algebraic predator–prey model with Michaelis–Menten-type prey harvesting is proposed. Due to the influence of gestation delay and stochastic fluctuations, the proposed model displays a complex dynamics. Criteria on the local stability of the interior equilibrium are established, and the effect of gestation delay on the model dynamics is discussed. Taking the gestation delay and economic profit as bifurcation parameters, Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values, respectively. Moreover, the solution of the model will blow up in a limited time when delay [Formula: see text]. Then, we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method, which is the key to illustrate the effect of stochastic fluctuations. Finally, we demonstrate our theoretical results by numerical simulations.

Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting

In this paper, a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten type prey harvesting is investigated. It is shown that the model exhibits several bifurcations of codimension 1 viz. Neimark–Sacker bifurcation, transcritical bifurcation and flip bifurcation on varying one parameter. Bifurcation theory and center manifold theory are used to establish the conditions for the existence of these bifurcations. Moreover, existence of Bogdanov–Takens bifurcation of codimension 2 (i.e. two parameters must be varied for the occurrence of bifurcation) is exhibited. The non-degeneracy conditions are determined for occurrence of Bogdanov–Takens bifurcation. The extensive numerical simulation is performed to demonstrate the analytical findings. The system exhibits periodic solutions including flip bifurcation and Neimark–Sacker bifurcation followed by the wide range of dense chaos.

Spatiotemporal Dynamics Induced by Michaelis–Menten Type Prey Harvesting in a Diffusive Leslie–Gower Predator–Prey Model

This paper is devoted to study the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model with Michaelis-Menten type harvesting in the prey population. The existence and stability of possible non-negative constant equilibria are investigated. By regarding [Formula: see text] as a bifurcation parameter, the Hopf bifurcation from the positive constant equilibrium solution is investigated. The necessary and sufficient conditions of Turing instability are explicitly obtained. We show that at the critical value of the bifurcation parameter [Formula: see text] a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, the global steady state bifurcation from the positive constant equilibrium solution is investigated. In particular, the local steady state bifurcation from double zero eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that Michaelis–Menten type harvesting in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without harvesting.

Stability and Bifurcation Analysis in a Nonlinear Harvested Predator–Prey Model with Simplified Holling Type IV Functional Response

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

Periodic Solutions of a Nonautonomous Leslie-Gower Predator-Prey Model with Non-Linear Type Prey Harvesting on Time Scales

In this paper, we investigate the existence of periodic solutions of modified version of the Leslie-Gower predator-prey model with Holling-type II functional response in the presence of Michaelis-Menten type prey harvesting over a time scale. Sufficient conditions for the existence of periodic solutions are derived by using the continuation theorem of coincidence degree theory. The condition we obtain is easily verifiable and not much restricted.

Numerical Study of Predator-Prey Model with Beddington-DeAngelis Functional Response and Prey Harvesting

A modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and Michaelis-Menten type prey harvesting is studied. The equilibrium points of the system are investigated. To see the stability of each equilibrium point, we perform some numerical simulations. Our numerical simulations show that the extinction of prey or survival of both prey and predator are conditionally stable.

Global Hopf bifurcation of a delayed diffusive predator-prey model with Michaelis-Menten-type prey harvesting

This paper is concerned with a delayed diffusive predator–prey model with Michaelis–Menten-type prey harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, transcritical saddle-node, Hopf bifurcations, which can cause the behavior mutations of the system, such as the change of the stability, the change of the number of equilibria, and the emergence of the periodic orbits. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations near the positive steady state are given by the normal form theory and center manifold theorem. Meanwhile, the approximate expressions of periodic orbits are given. Furthermore, we derive the sufficient condition for global existence of the spatial homogeneous periodic orbits. Finally, some numerical simulations are carried out to support our analytical results.

Hopf bifurcation and global stability of a delayed predator–prey model with prey harvesting

In this paper, we consider a delayed prey–predator model with modified Leslie–Gower term and Michaelis–Menten type prey harvesting. Regarding time delay as a bifurcation parameter, and using the normal form and center manifold theorem of functional differential equations and partial differential equations, we obtain the existence, stability and direction of periodic solutions of functional differential equations and partial differential equations, respectively. Numerical simulations are carried out to depict our theoretical analysis.

Dynamics of a Diffusive Predator-Prey Model with Modified Leslie-Gower Term and Michaelis-Menten Type Prey Harvesting

A diffusive predator-prey model with modified Leslie-Gower term and Michaelis-Menten type prey harvesting subject to the homogeneous Neumann boundary condition is considered. We obtain the local and global stability of constant equilibria by eigenvalue analysis and iteration technique. Choosing some parameter concerning with harvesting as Hopf bifurcation parameter, we conclude the existence of periodic solutions near positive constant equilibrium. Using the normal form and center manifold theory, and numerical simulations, we demonstrate our theoretical results of stability and direction of periodic solutions. We also derive the non-existence and existence of non-constant positive steady states by energy method and degree theory.

Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator-prey model with time-delay and prey harvesting

From the saddle-node-Hopf bifurcation point of view, this paper considers a modified Leslie–Gower predator-prey model with time delay and the Michaelis–Menten type prey harvesting. Firstly, we discuss the stability of the equilibria, obtain the critical conditions for the saddle-node-Hopf bifurcation, and give the completion bifurcation set by calculating the universal unfoldings near the saddle-node-Hopf bifurcation point by using the normal form theory and center manifold theorem. Then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey-extinction and the coexistence equilibrium (or coexistence periodic or quasi-periodic solutions) are simultaneously stabilized. We also investigate the heteroclinic bifurcation, and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation. Finally, some numerical simulations are performed to support our analytic results.

Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting

In the present paper we discuss bifurcation analysis of a modified Leslie–Gower prey–predator model in the presence of nonlinear harvesting in prey. We give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources. The model displays a complex dynamics in the prey–predator plane. The permanence, stability and bifurcation (saddle–node bifurcation, transcritical, Hopf–Andronov and Bogdanov–Takens) of this model are discussed. We have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view. The local existence and stability of the limit cycle emerging through Hopf bifurcation is given. The emergence of homoclinic loops has been shown through simulation when the limit cycle arising though Hopf bifurcation collides with a saddle point. This work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed. Numerical simulations using MATLAB are carried out to demonstrate the results obtained.

Bifurcation Analysis and Control of Leslie–Gower Predator–Prey Model with Michaelis–Menten Type Prey-Harvesting

In this paper, a bi-dimensional system of ordinary differential equation prey–predator model with nonlinear prey-harvesting is analyzed. The model is a modified version of the well known Leslie–Gower type prey–predator model. The original Leslie–Gower type model has the unique interior equilibrium point which is globally asymptotically stable. We show here that the nonlinear harvesting in prey significantly modifies the dynamics of the system in comparison to the proportionate harvesting of prey. It has been observed that the system goes to extinction for a wide range of initial values. Moreover, the model can have two, one or no interior equilibrium point in the first quadrant where two interior equilibria collapse to one interior equilibrium point and then disappear through saddle-node bifurcation considering the rate of harvesting as bifurcation parameter. The local existence of limit cycle appearing through local Hopf bifurcation and its stability have also been investigated by computing first Lyapunov number. The conditions for existence of bionomic equilibria are analyzed. Pontryagin’s Maximum Principle is used to characterize the optimal singular control. Numerical examples are provided to validate our findings.