Conformable Fractional Order Lotka–Volterra Predator–Prey Model: Discretization, Stability and Bifurcation

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
Fuat Gürcan ◽  
Güven Kaya ◽  
Senol Kartal
Keyword(s):  
Fractional Order ◽  
Discrete System ◽  
Positive Equilibrium ◽  
Dynamic Behaviors ◽  
Piecewise Constant ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
Piecewise Constant Approximation

Abstract The purpose of this study is to discuss dynamic behaviors of conformable fractional-order Lotka–Volterra predator–prey system. First of all, the piecewise constant approximation is used to obtain the discretize version of the model then, we investigate stability, existence, and direction of Neimark–Sacker bifurcation of the positive equilibrium point of the discrete system. It is observed that the discrete system shows much richer dynamic behaviors than its fractional-order form such as Neimark–Sacker bifurcation and chaos. Finally, numerical simulations are used to demonstrate the accuracy of analytical results.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

scholarly journals Dynamical analysis of a new fractional-order predator–prey system with Holling type-III functional

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lihua Dai ◽  
Junjie Wang ◽  
Yonggen Ni ◽  
Bin Xu
Keyword(s):  
Fractional Order ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Lyapunov Stability Theory ◽  
Dynamical Analysis ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a new fractional-order predator–prey model with Holling type-III functional response and stage structure. Based on the Lyapunov stability theory and by constructing a suitable Lyapunov functional, we obtain some sufficient conditions for the existence and uniqueness of positive solutions and the asymptotic stability of the positive equilibrium to the system. Finally, we give some numerical examples to illustrate the feasibility of our results.

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.

scholarly journals Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Jinlei Liu ◽  
Wencai Zhao
Keyword(s):  
Feedback Control ◽  
Deterministic Model ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Theoretical Derivation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Delays ◽  
Positive Equilibrium Point

In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

scholarly journals Dynamic Behaviors of a Discrete Two Species Predator-Prey System Incorporating Harvesting

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Ting Wu
Keyword(s):  
Equilibrium Point ◽  
Hamiltonian Function ◽  
Economic Benefits ◽  
Practical Significance ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
The Maximum Principle ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
Jury Criterion

A discrete two species predator-prey captured system is studied. Firstly, a sufficient condition of a positive equilibrium point for this system is obtained. Secondly, we observe that the two nonnegative equilibriums of the system are unstable through the eigenvalue discriminant method, and the positive equilibrium point is asymptotically stable by Jury criterion. Lastly, we obtain the optimal capture strategy of the system from the maximum principle by constructing a discrete Hamiltonian function. To show the feasibility of the main results, a suitable example together with its numerical simulations is illustrated in the last part of the paper. The example with certain practical significance might give an optimal scheme of the greatest economic benefits for the captors.

DYNAMICS OF A DIFFUSIVE PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT

2012 ◽  
Vol 05 (02) ◽  
pp. 1250023 ◽  
Author(s):  
YONGLI CAI ◽  
WEIMING WANG ◽  
JINFENG WANG
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Allee Effects ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Holling Ii Functional Response

In this paper, we investigate the dynamics of a diffusive predator–prey model with Holling-II functional response and the additive Allee effect in prey. We show the local and global asymptotical stability of the positive equilibrium, and give the conditions of the existence of the Hopf bifurcation. By carrying out global qualitative and bifurcation analysis, it is shown that the weak and strong Allee effects in prey can induce different dynamical behavior in the predator–prey model. Furthermore, we use some numerical simulations to illustrate the dynamics of the model. The results may be helpful for controlling and managing the predator–prey system.

scholarly journals Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

2020 ◽  
Vol 25 (2) ◽  
Author(s):  
Maruthai Selvaraj Surendar ◽  
Muniyagounder Sambath ◽  
Krishnan Balachandran
Keyword(s):  
Density Dependence ◽  
Periodic Solutions ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Spatially Homogeneous ◽  
Bifurcation And Stability ◽  
Theoretical Results

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out. 

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

TURING PATTERNS OF A PREDATOR–PREY MODEL WITH A MODIFIED LESLIE–GOWER TERM AND CROSS-DIFFUSIONS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250060 ◽  
Author(s):  
GUANG-PING HU ◽  
XIAO-LING LI
Keyword(s):  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Steady States

In this paper, a strongly coupled diffusive predator–prey system with a modified Leslie–Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

scholarly journals A class of discrete predator–prey interaction with bifurcation analysis and chaos control

2020 ◽  
Vol 15 ◽  
pp. 60
Author(s):  
Qamar Din ◽  
Nafeesa Saleem ◽  
Muhammad Sajjad Shabbir
Keyword(s):  
Chaos Control ◽  
Positive Equilibrium ◽  
Ecological Communities ◽  
Natural Ecosystems ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Boundary Equilibrium ◽  
Prey Interaction ◽  
Positive Equilibrium Point

The interaction between prey and predator is well-known within natural ecosystems. Due to their multifariousness and strong link population dynamics, predators contain distinct features of ecological communities. Keeping in view the Nicholson-Bailey framework for host-parasitoid interaction, a discrete-time predator–prey system is formulated and studied with implementation of type-II functional response and logistic prey growth in form of the Beverton-Holt map. Persistence of solutions and existence of equilibria are discussed. Moreover, stability analysis of equilibria is carried out for predator–prey model. With implementation of bifurcation theory of normal forms and center manifold theorem, it is proved that system undergoes transcritical bifurcation around its boundary equilibrium. On the other hand, if growth rate of consumers is taken as bifurcation parameter, then system undergoes Neimark-Sacker bifurcation around its positive equilibrium point. Methods of chaos control are introduced to avoid the populations from unpredictable behavior. Numerical simulation is provided to strengthen our theoretical discussion.

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