Stability and Bifurcation Analysis in a Predator–Prey System with Constant Harvesting and Prey Group Defense

In this paper, we study a predator–prey system that the prey population gathers in herds to defend its predator and both are harvested by constant rate. The defensive strategy of the gathered prey makes the individuals at the border of the herd mostly suffer from the attacks of the predators. This behavior can be described by a modified Holling-type II functional response in mathematics. Notably, we consider harvesting under two cases: prey harvesting only and predator harvesting only. We investigate the existence of equilibria for both cases, and then find there exists the maximum sustainable yield for two cases to guarantee predator and prey to coexist. Moreover, both species can coexist under some conditions and initial values through investigation of stability of the interior equilibrium in the given system. These results demonstrate that, when hunting the prey or predator for economic interest, harvesting rate must be chosen at a suitable value (not merely less than the maximum sustainable yield) to maintain the coexistence of the predator and prey as well as ecological balance. Finally, we analyze the saddle-node bifurcation and Hopf bifurcation, and determine the direction of Hopf bifurcation by calculating the first Lyapunov number for both cases. In particular, Bogdanov–Takens bifurcation occurs only in the given system with predator harvesting.

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Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2260-y ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Xinxin Liu ◽

Qingdao Huang

Keyword(s):

Prey Species ◽

Equilibrium Points ◽

Maximum Sustainable Yield ◽

Predator Prey ◽

Prey System ◽

Existence And Stability ◽

Predator Model ◽

Model Equations ◽

The Given

AbstractA new way to study the harvested predator–prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system.

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BIFURCATIONS OF A HOLLING-TYPE II PREDATOR–PREY SYSTEM WITH CONSTANT RATE HARVESTING

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740902427x ◽

2009 ◽

Vol 19 (08) ◽

pp. 2499-2514 ◽

Cited By ~ 19

Author(s):

GUOJUN PENG ◽

YAOLIN JIANG ◽

CHANGPIN LI

Keyword(s):

Hopf Bifurcation ◽

Type Ii ◽

Saddle Node Bifurcation ◽

Predator Prey ◽

Prey System ◽

Bifurcation Phenomena ◽

Constant Rate ◽

Generalized Hopf Bifurcation ◽

Subcritical Hopf Bifurcation ◽

Dynamical Properties

The objective of this paper is to study the dynamical properties of a Holling-type II predator–prey system with constant rate harvesting. It is shown that the model has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the degenerate Bogdanov–Takens bifurcation of codimension 3, the supercritical and subcritical Hopf bifurcation, the generalized Hopf bifurcation. These results reveal far richer dynamics than that of the model with no harvesting.

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Mathematical study on a dynamical predator-prey model with constant prey harvesting and proportional harvesting in predator

10.21203/rs.3.rs-377501/v1 ◽

2021 ◽

Author(s):

Md Golam Mortuja ◽

Mithilesh Kumar Chaube ◽

Santosh Kumar

Keyword(s):

Handling Time ◽

Maximum Sustainable Yield ◽

Prey Population ◽

Sustainable Yield ◽

Mathematical Study ◽

Predator Prey ◽

Interior Equilibrium ◽

Predator Prey Model ◽

Prey Model ◽

Herd Behaviour

Abstract A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root func- tional response also has been incorporated in the system to describe the prey herd behaviour, assuming the average handling time is zero. The existence and the local stability of equilibria of the system have been discussed. It is examined that, two types of bifurcation occur in the system. The two types of bifurcations have been analyzed, and it has been found by analyzing the saddle-node bifurcation that, there is a maximum sustainable yield. It is ob- served that if harvesting rate is greater than the maximum sustainable yield, the prey population abolish from the system and then extinction of the preda- tor population happen. But if harvesting rate is lesser than the maximum sustainable yield, the extinction of the prey population can not be possible. By analyzing the Hopf bifurcation, it is obtained that, there exists an unstable limit cycle around the interior equilibrium point. Several numerical simulations are performed to check the results.

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The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource

Mathematics ◽

10.3390/math9080865 ◽

2021 ◽

Vol 9 (8) ◽

pp. 865

Author(s):

Jialin Chen ◽

Xiaqing He ◽

Fengde Chen

Keyword(s):

Discrete Time ◽

Food Resource ◽

Sufficient Conditions ◽

Iteration Scheme ◽

Predator Species ◽

Predator Prey ◽

Interior Equilibrium ◽

Fear Effect ◽

Prey System ◽

Trivial Equilibrium

A discrete-time predator–prey system incorporating fear effect of the prey with the predator has other food resource is proposed in this paper. The trivial equilibrium and the predator free equilibrium are both unstable. A set of sufficient conditions for the global attractivity of prey free equilibrium and interior equilibrium are established by using iteration scheme and the comparison principle of difference equations. Our study shows that due to the fear of predation, the prey species will be driven to extinction while the predator species tends to be stable since it has other food resource, i.e., the prey free equilibrium may be globally stable under some suitable conditions. Numeric simulations are provided to illustrate the feasibility of the main results.

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Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

Advances in Difference Equations ◽

10.1186/s13662-020-02879-4 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Heping Jiang ◽

Huiping Fang ◽

Yongfeng Wu

Keyword(s):

Hopf Bifurcation ◽

Neumann Boundary ◽

Herd Behavior ◽

Predator Prey ◽

Predator Prey Model ◽

Dynamical Behaviors ◽

Prey System ◽

Prey Model ◽

Homogeneous Neumann Boundary Condition ◽

The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

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Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽

10.1155/2020/4285294 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Kankan Sarkar ◽

Subhas Khajanchi ◽

Prakash Chandra Mali ◽

Juan J. Nieto

Keyword(s):

Growth Rate ◽

Hopf Bifurcation ◽

Dynamical Behavior ◽

Prey Population ◽

Uniform Persistence ◽

Functional Responses ◽

Normal Form Theory ◽

Predator Prey ◽

Center Manifold Theorem ◽

Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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Community Effects of Fishing

Fish Ecology, Evolution, and Exploitation ◽

10.23943/princeton/9780691192956.003.0012 ◽

2019 ◽

pp. 200-214

Author(s):

Ken H. Andersen

Keyword(s):

Trophic Cascades ◽

Maximum Sustainable Yield ◽

Sustainable Yield ◽

Community Effects ◽

Predator Prey ◽

Community Model ◽

Trade Offs ◽

Large Predators

This chapter uses the community model to repeat many of the classic impact calculations of a single stock on the entire community. Here, a focus is the appearance of trophic cascades initiated by the removal of large predators. When a component of an ecosystem is perturbed, the effects are not isolated to the component itself but cascade through the ecosystem. Perturbations are mainly propagated through the predator–prey interactions. The chapter also considers the trade-offs between a forage fishery and a consumer fishery, and the extension of the maximum sustainable yield (MSY) concept to the community, before finally returning to the single-stock aspects.

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Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

Open Mathematics ◽

10.1515/math-2019-0014 ◽

2019 ◽

Vol 17 (1) ◽

pp. 141-159 ◽

Cited By ~ 9

Author(s):

Zaowang Xiao ◽

Zhong Li ◽

Zhenliang Zhu ◽

Fengde Chen

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Sufficient Conditions ◽

Stage Structure ◽

Prey Refuge ◽

Predator Species ◽

Predator Prey ◽

Predator Prey Model ◽

Prey System ◽

Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

Abstract and Applied Analysis ◽

10.1155/2013/168340 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Shaoli Wang ◽

Zhihao Ge

Keyword(s):

Hopf Bifurcation ◽

Logistic Growth ◽

Positive Equilibrium ◽

Prey Refuge ◽

Normal Form Theory ◽

Center Manifold Reduction ◽

Predator Prey ◽

Prey System ◽

Form Theory ◽

The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

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