The stability of internal equilibria in predator-prey models with breeding suppression

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

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Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

Abstract and Applied Analysis ◽

10.1155/2012/363051 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Shuang Guo ◽

Weihua Jiang

Keyword(s):

Hopf Bifurcation ◽

Center Manifold ◽

Three Dimensional ◽

Sufficient Conditions ◽

Predator Prey ◽

Center Manifold Theorem ◽

Predator Prey Model ◽

Normal Form Method ◽

Predator Prey Models ◽

The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

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The Stability of Gauss Model Having One-Prey and Two-Predators

Abstract and Applied Analysis ◽

10.1155/2012/219640 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

A. Farajzadeh ◽

M. H. Rahmani Doust ◽

F. Haghighifar ◽

D. Baleanu

Keyword(s):

Mathematical Biology ◽

Global Stability ◽

Equilibrium Points ◽

Predator Prey ◽

Local And Global Stability ◽

Stability Properties ◽

Interacting Species ◽

Predator Prey Models ◽

The Stability

The study of the dynamics of predator-prey interactions can be recognized as a major issue in mathematical biology. In the present paper, some Gauss predator-prey models in which three ecologically interacting species have been considered and the behavior of their solutions in the stability aspect have been investigated. The main aim of this paper is to consider the local and global stability properties of the equilibrium points for represented systems. Finally, stability of some examples of Gauss model with one prey and two predators is discussed.

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An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

International Journal of Differential Equations ◽

10.1155/2020/9176784 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Behrooz Basirat ◽

Hamid Reza Elahi

Keyword(s):

Time Lag ◽

Euler Polynomials ◽

Algebraic Equations ◽

Predator Prey ◽

Predator Prey Model ◽

Collocation Points ◽

Prey Model ◽

Collocation Approach ◽

Predator Prey Models ◽

The Stability

This paper deals with an approach to obtaining the numerical solution of the Lotka–Volterra predator-prey models with discrete delay using Euler polynomials connected with Bernoulli ones. By using the Euler polynomials connected with Bernoulli ones and collocation points, this method transforms the predator-prey model into a matrix equation. The main characteristic of this approach is that it reduces the predator-prey model to a system of algebraic equations, which greatly simplifies the problem. For these models, the explicit formula determining the stability and the direction is given. Numerical examples illustrate the reliability and efficiency of the proposed scheme.

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STABILITY ANALYSIS OF DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT DELAY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406014939 ◽

2006 ◽

Vol 16 (02) ◽

pp. 465-472 ◽

Cited By ~ 24

Author(s):

WEIHUA DENG ◽

YUJIANG WU ◽

CHANGPIN LI

Keyword(s):

Stability Analysis ◽

Differential Equations ◽

Finite Time ◽

Time Dependent ◽

Time Interval ◽

Finite Time Interval ◽

Predator Prey ◽

Gauss Type ◽

Predator Prey Models ◽

The Stability

In this Letter, we study the stability of differential equations with time-dependent delay. Several theorems are established for stability on a finite time interval, called "interval stability" for simplicity, and Liapunov stability. These theorems are applied to the generalized Gauss-type predator–prey models, and satisfactory results are obtained.

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Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics

Abstract and Applied Analysis ◽

10.1155/2014/935923 ◽

2014 ◽

Vol 2014 ◽

pp. 1-15

Author(s):

Mei Yan ◽

Yongfeng Li ◽

Zhongyi Xiang

Keyword(s):

Pest Control ◽

Control Method ◽

Impulsive Differential Equations ◽

Threshold Values ◽

Predator Prey ◽

Predator Prey Model ◽

Birth Pulse ◽

Predator Prey Models ◽

The Stability ◽

Chemical Pesticides

Normally, chemical pesticides kill not only pests but also their natural enemies. In order to better control the pests, two-time delayed stage-structured predator-prey models with birth pulse and pest control tactics are proposed and analyzed by using impulsive differential equations in present work. The stability threshold conditions for the mature prey-eradication periodic solutions of two models are derived, respectively. The effects of key parameters including killing efficiency rate, pulse period, the maximum birth effort per unit of time of natural enemy, and maturation time of prey on the threshold values are discussed in more detail. By comparing the two threshold values of mature prey-extinction, we provide the fact that the second control tactic is more effective than the first control method.

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Mathematical Analysis of a Fractional-Order Predator-Prey Network with Feedback Control Strategy

Computational Intelligence and Neuroscience ◽

10.1155/2021/9358881 ◽

2021 ◽

Vol 2021 ◽

pp. 1-17

Author(s):

Wei Zhang ◽

Yu Fei ◽

Zhouhong Li ◽

Chengdai Huang

Keyword(s):

Fractional Order ◽

Bifurcation Control ◽

Theoretical Derivation ◽

Predator Prey ◽

Stability Performance ◽

Proposed Model ◽

Predator Prey Models ◽

The Stability ◽

Feedback Control Strategy ◽

Theoretical Results

This paper examines the bifurcation control problem of a class of delayed fractional-order predator-prey models in accordance with an enhancing feedback controller. Firstly, the bifurcation points of the devised model are precisely figured out via theoretical derivation taking time delay as a bifurcation parameter. Secondly, a set comparative analysis on the influence of bifurcation control is numerically studied containing enhancing feedback, dislocated feedback, and eliminating feedback approaches. It can be seen that the stability performance of the proposed model can be immensely heightened by the enhancing feedback approach. At the end, a numerical example is given to illustrate the feasibility of the theoretical results.

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The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽

10.2298/fil1718811z ◽

2017 ◽

Vol 31 (18) ◽

pp. 5811-5825

Author(s):

Xinhong Zhang

Keyword(s):

Mortality Rate ◽

Stochastic Model ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Global Dynamics ◽

Type Ii ◽

Predator Prey ◽

Persistence In The Mean ◽

The Mean ◽

Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

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