Bifurcation analysis in a predator–prey model for the effect of delay in prey

2016 ◽  
Vol 09 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Qiubao Wang
Keyword(s):  
Critical Point ◽  
Center Manifold ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Reduction Methods ◽  
Manifold Reduction ◽  
Theoretical Results

In this paper, we study dynamics in a predator–prey model with delay, in which predator can be infected, with particular attention focused on nonresonant double Hopf bifurcation. By using center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

scholarly journals Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jinbin Wang ◽  
Rui Zhang ◽  
Lifenq Ma
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Two Dimensional ◽  
Main Attention ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Physical Phenomena ◽  
Manifold Reduction ◽  
Theoretical Results

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Coexistence of Periodic Oscillations Induced by Predator Cannibalism in a Delayed Diffusive Predator–Prey Model

2019 ◽  
Vol 29 (07) ◽  
pp. 1950089
Author(s):  
Daifeng Duan ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Reduction ◽  
Periodic Oscillations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Cannibalism ◽  
Manifold Reduction

This paper is concerned with the effect of predator cannibalism in a delayed diffusive predator–prey system. We aim for the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to Hopf–Hopf bifurcation. An approach of center manifold reduction is applied to derive the normal form for such nonresonant Hopf–Hopf bifurcations. We find that the system exhibits very rich dynamics, including the coexistence of periodic and quasi-periodic oscillations. Numerically, we show that Hopf–Hopf bifurcation is induced if the strength of the predator cannibalism term belongs to an appropriate interval.

HOPF–FOLD BIFURCATION AND FLUCTUATION PHENOMENA IN A DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR–PREY MODEL

2013 ◽  
Vol 23 (09) ◽  
pp. 1350153
Author(s):  
SHUANG GUO ◽  
WEIHUA JIANG
Keyword(s):  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fold Bifurcation ◽  
Normal Form Method ◽  
Linearized System ◽  
Ratio Dependent ◽  
Simple Pair ◽  
Manifold Reduction

A class of delayed ratio-dependent Gause-type predator–prey model is considered. We study the eigenvalue problem for the linearized system at the coexisting equilibrium. For a critical case when the characteristic equation has a single zero root and a simple pair of pure imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. We analyzed the influence of the time delay on the Hopf–Fold bifurcation and showed the occurrence of quasi-periodic motion and bursting behavior. This phenomenon is in line with the seasonal variation law of the population.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

scholarly journals Simulation of Discrete Predator-Prey Model

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Adel A. Abed Al Wahab ◽  
Nihad Mahmoud Nasir ◽  
Adil I. Khalil
Keyword(s):  
Discrete Model ◽  
Initial Conditions ◽  
Current Model ◽  
Continuous Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Computerized Simulation ◽  
Theoretical Results ◽  
Over Time

It is well known that dynamical systems deal with situations in which the system transforms over time. In fact, undertaking a manual simulation of such systems is a difficult task due to the complexity of the computations. Therefore, a computerized simulation is frequently required for accurate results and fast execution time. Nowadays, computer programs have become an important tool to confirm the theoretical results obtained from the study of models. This paper aims to employ new MATLAB codes to examine a discrete predator–prey model using a difference equations system. The paper discusses the existences and stabilities of each possible fixed point appearing in the current model. Furthermore, numerical simulations fixed by a certain parameter to plot the diagrams are presented. Our results confirm that the systems sensitive to initial conditions are chaotic. Furthermore, the theoretical results as well as numerical examples illustrated that the discrete model exhibits complex behavior compared to a continuous model. The conclusion drawn is that the numerical simulation is an important tool to confirm theoretical results.

scholarly journals Paradox of Enrichment in a Stochastic Predator-Prey Model

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jawdat Alebraheem
Keyword(s):  
Initial Conditions ◽  
Random Noise ◽  
Unique Positive Solution ◽  
Predator Prey ◽  
Paradox Of Enrichment ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
Biological Phenomena ◽  
Theoretical Results

We propose a stochastic predator-prey model to study a novel idea that involves investigating random noises effects on the enrichment paradox phenomenon. Existence and stochastic boundedness of a unique positive solution with positive initial conditions are proved. The global asymptotic stability is studied to determine the occurrence of the enrichment paradox phenomenon. We show theoretically that intensive noises play an important role in the occurrence of the phenomenon, where increasing intensive noises lead to occurrence of the paradox of enrichment. We perform numerical simulations to verify and demonstrate the theoretical results. The new results in this study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.

scholarly journals A hybrid predator–prey model with general functional responses under seasonal succession alternating between Gompertz and logistic growth

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lei Hang ◽  
Long Zhang ◽  
Xiaowen Wang ◽  
Hongli Li ◽  
Zhidong Teng
Keyword(s):  
Global Attractivity ◽  
Seasonal Succession ◽  
Hybrid Population ◽  
Logistic Growth ◽  
Ultimate Boundedness ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

AbstractIn this paper, a hybrid predator–prey model with two general functional responses under seasonal succession is proposed. The model is composed of two subsystems: in the first one, the prey follows the Gompertz growth, and it turns to the logistic growth in the second subsystem since seasonal succession. The two processes are connected by impulsive perturbations. Some very general, weak criteria on the ultimate boundedness, permanence, existence, uniqueness and global attractivity of predator-free periodic solution are established. We find that the hybrid population model with seasonal succession has more survival possibilities of natural species than the usual population models. The theoretical results are illustrated by special examples and numerical simulations.

scholarly journals Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

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