On the Bifurcation Pattern and Normal Form in a Modified Predator–Prey Nonlinear System

2007 ◽  
Vol 2 (3) ◽  
pp. 267-273 ◽  
Author(s):  
Dibakar Ghosh ◽  
A. Roy Chowdhury
Keyword(s):  
New York ◽  
Normal Form ◽  
Bifurcation Theory ◽  
Positive Influence ◽  
Detailed Structure ◽  
Predator Prey ◽  
Second Stage ◽  
Prey System ◽  
Stability Structure ◽  
Almost All

Detailed bifurcation pattern and stability structure is studied in a modified predator–prey system, with nonmonotonic response function. It is observed that almost all the parameters of the system have a positive influence as far as bifurcation is concerned. The analysis is done with the help of the package MATCONT. In the second stage of the analysis the detailed structure of the normal form is obtained after the corresponding position of Hopf bifurcation and Bogdanov–Takens bifurcation are identified with the help of a modified approach recently proposed by Kuznetsov (1995, Elements of Bifurcation Theory, Springer, New York, Chap. 8). It is important to note that the positions of Hopf and Bogdanov–Taken bifurcation as obtained from the analytic studies in this approach coincides exactly with those obtained from MATCONT.

scholarly journals Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhenjiang Yao ◽  
Bingnan Tang
Keyword(s):  
Fractional Order ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Predator Prey ◽  
Mixed Time Delays ◽  
Predator Prey Model ◽  
Prey System ◽  
Change Of Variable ◽  
Set Up ◽  
The Stability

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.

DYNAMIC BEHAVIOR ANALYSIS OF A DIFFERENTIAL-ALGEBRAIC PREDATOR–PREY SYSTEM WITH PREY HARVESTING

2013 ◽  
Vol 21 (03) ◽  
pp. 1350022 ◽  
Author(s):  
WEI LIU ◽  
YUXIAN CHEN ◽  
CHAOJIN FU
Keyword(s):  
System Theory ◽  
Hopf Bifurcation ◽  
Algebraic System ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Formal Series ◽  
Series Expansions ◽  
Predator Prey ◽  
Prey System ◽  
Differential Algebraic System

This paper studies a differential-algebraic predator–prey system with prey harvesting, which consists of two differential equations and an algebraic equation. By using the differential-algebraic system theory, bifurcation theory and formal series expansions, we investigate the Hopf bifurcation and center stability of the differential-algebraic predator–prey system. Some sufficient conditions on these issues are obtained. In addition, numerical simulations illustrate the effectiveness of our results and their biological implications are discussed.

scholarly journals Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Changtong Li ◽  
Hao Sun ◽  
Yuzhen Wang
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Bifurcation Theory ◽  
A Priori Estimates ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
Steady State Solutions

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

scholarly journals Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Spatial Memory ◽  
Reaction Diffusion ◽  
Diffusion Term ◽  
Predator Prey ◽  
Spatially Inhomogeneous ◽  
Prey System ◽  
Mode 2 ◽  
The Stability

Abstract In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.

Dynamics of a Modified Predator-Prey System to allow for a Functional Response and Time Delay

2016 ◽  
Vol 6 (4) ◽  
pp. 384-399 ◽  
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
System Theory ◽  
Algebraic System ◽  
Bifurcation Theory ◽  
Dynamical Behaviour ◽  
Centre Manifold ◽  
Predator Prey ◽  
Prey System ◽  
Differential Algebraic System ◽  
The Stability ◽  
Manifold Theory

AbstractA modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

2021 ◽  
Author(s):  
Haixia Li ◽  
Wenbin Yang ◽  
Meihua Wei ◽  
Aili Wang
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
A Priori Bound ◽  
Steady State Solutions

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

scholarly journals Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Xianning Liu ◽  
Lansun Chen
Keyword(s):  
Periodic Solution ◽  
Bifurcation Theory ◽  
Positive Periodic Solution ◽  
Impulsive Effect ◽  
Impulsive Systems ◽  
Predator Prey ◽  
Prey System ◽  
Theory Application ◽  
Explicit Bounds ◽  
Impulsive Perturbations

A general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtained via the comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence of a positive periodic solution is also studied by the bifurcation theory. Application is given to a Lotka-Volterra predator-prey system with periodic impulsive immigration of the predator. It is shown that the results are quite different from the corresponding system without impulsive immigration, where extinction of the prey can never be achieved. The prey will be extinct or permanent independent of whether the system without impulsive effect immigration is permanent or not. The model and its results suggest an approach of pest control which proves more effective than the classical one.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

2019 ◽  
Vol 29 (11) ◽  
pp. 1950145 ◽  
Author(s):  
Yu-Xia Wang ◽  
Wan-Tong Li
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Spatial Patterns ◽  
Bifurcation Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Prey System

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect

2019 ◽  
Vol 12 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Seval Işık
Keyword(s):  
Discrete Time ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Small Neighborhood ◽  
Invariant Circle ◽  
Predator Prey ◽  
Discrete Dynamic System ◽  
Local Asymptotic Stability ◽  
Prey System ◽  
Theoretical Results

This paper deals with a discrete-time predator–prey system which is subject to an Allee effect on prey. We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system. Moreover, using bifurcation theory, it is shown that the system undergoes Neimark–Sacker bifurcation in a small neighborhood of the unique positive fixed point and an invariant circle will appear. Then the direction of bifurcation is given. Furthermore, numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages. Thus, the main theoretical results are supported with numerical simulations.

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