scholarly journals Dynamics of a Predator-Prey Model with Fear Effect and Time Delay

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Junli Liu ◽  
Pan Lv ◽  
Bairu Liu ◽  
Tailei Zhang
Keyword(s):  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Delay Dependent ◽  
The Impact ◽  
Dependent Factor

In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. The dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. The explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. Thus, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

Codimension Bifurcation Analysis of a Modified Leslie–Gower Predator–Prey Model with Two Delays

2018 ◽  
Vol 28 (05) ◽  
pp. 1850060 ◽  
Author(s):  
Jianfeng Jiao ◽  
Ruiqi Wang ◽  
Hongcui Chang ◽  
Xia Liu
Keyword(s):  
Time Delays ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
Two Delays ◽  
Delay Differential

The Bogdanov–Takens (B–T) and triple-zero bifurcations of a modified Leslie–Gower predator–prey model with two time delays are studied in this paper. By generalizing and using the normal form theory and center manifold theorem for delay differential equations, the normal forms of the B–T and triple-zero bifurcations of the model at its interior equilibria are obtained. In addition, some numerical simulations are presented to illustrate our main results.

scholarly journals Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Manifold Theory ◽  
Delay Dependent

A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

scholarly journals Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

2018 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
Jai Prakash Tripathi ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Interaction Model ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Predator Species ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model

AbstractIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

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

Complexity ◽  
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.

scholarly journals Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xinhong Zhang ◽  
Qing Yang
Keyword(s):  
Functional Response ◽  
Dynamical Behavior ◽  
Uniform Boundedness ◽  
Jump Diffusion ◽  
Necessary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
A New Technique ◽  
The One

<p style='text-indent:20px;'>In this paper, we consider a stochastic predator-prey model with general functional response, which is perturbed by nonlinear Lévy jumps. Firstly, We show that this model has a unique global positive solution with uniform boundedness of <inline-formula><tex-math id="M1">\begin{document}$ \theta\in(0,1] $\end{document}</tex-math></inline-formula>-th moment. Secondly, we obtain the threshold for extinction and exponential ergodicity of the one-dimensional Logistic system with nonlinear perturbations. Then based on the results of Logistic system, we introduce a new technique to study the ergodic stationary distribution for the stochastic predator-prey model with general functional response and nonlinear jump-diffusion, and derive the sufficient and almost necessary condition for extinction and ergodicity.</p>

Bifurcation Analysis and Spatiotemporal Patterns in a Diffusive Predator–Prey Model

2014 ◽  
Vol 24 (06) ◽  
pp. 1450081 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu ◽  
Yuepeng Wang
Keyword(s):  
Pattern Formation ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Target Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Impact

In this paper, we consider a species predator–prey model given a reaction–diffusion system. It incorporates the Holling type II functional response and a quadratic intra-predator interaction term. We focus on the qualitative analysis, bifurcation mechanisms and pattern formation. We present the results of numerical experiments in two space dimensions and illustrate the impact of the diffusion on the Turing pattern formation. For this diffusion system, we also observe non-Turing structures such as spiral wave, target pattern and spatiotemporal chaos resulting from the time evolution of these structures.

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