Dynamics of a Predator–Prey Model with Hunting Cooperation and Allee Effects in Predators

2020 ◽  
Vol 30 (14) ◽  
pp. 2050199
Author(s):  
Jun Zhang ◽  
Weinian Zhang
Keyword(s):  
Homoclinic Bifurcation ◽  
Allee Effects ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Analytical Parameter ◽  
Pitchfork Bifurcations ◽  
Ecosystem Diversity ◽  
Parameter Values

With both hunting cooperation and Allee effects in predators, a predator–prey system was modeled as a planar cubic differential system with three parameters. The known work numerically plots the horizontal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. Transitions among those cases with the rise of limit cycle and homoclinic loop were exhibited by numerical simulations. Although it is hard to obtain the explicit expression of coordinates, in this paper, we give the distribution of equilibria qualitatively, discuss all cases of coexisting equilibria, and obtain the Bogdanov–Takens bifurcation diagram to show analytical parameter conditions for those transitions. Our results give analytical conditions for not only the observed saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation but also the transcritical and pitchfork bifurcations at the predator-extinction equilibrium, which were not considered in the known work. Our analytic conditions provide a quantitative instruction to reduce the risk of predator extinction and promote the ecosystem diversity.

scholarly journals CODIMENSION TWO BIFURCATIONS IN A PREDATOR–PREY SYSTEM WITH GROUP DEFENSE

2001 ◽  
Vol 11 (08) ◽  
pp. 2123-2131 ◽  
Author(s):  
DONGMEI XIAO ◽  
SHIGUI RUAN
Keyword(s):  
Homoclinic Bifurcation ◽  
Qualitative Behavior ◽  
Homoclinic Loop ◽  
Saddle Node Bifurcation ◽  
Predator Prey ◽  
Codimension Two ◽  
Prey System ◽  
Group Defense ◽  
Nonmonotonic Functional Response ◽  
Parameter Values

In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

Stability and Bifurcation Analysis in a Nonlinear Harvested Predator–Prey Model with Simplified Holling Type IV Functional Response

2020 ◽  
Vol 30 (14) ◽  
pp. 2050205
Author(s):  
Zuchong Shang ◽  
Yuanhua Qiao ◽  
Lijuan Duan ◽  
Jun Miao
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Bifurcation Analysis ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
The Stability ◽  
Parameter Values

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

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.

MULTIPLE BIFURCATIONS IN A PREDATOR–PREY SYSTEM OF HOLLING AND LESLIE TYPE WITH CONSTANT-YIELD PREY HARVESTING

2013 ◽  
Vol 23 (10) ◽  
pp. 1350164 ◽  
Author(s):  
JICAI HUANG ◽  
YIJUN GONG ◽  
JING CHEN
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Phase Portraits ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Prey System ◽  
Multiplicity One ◽  
Multiple Focus ◽  
Parameter Values ◽  
Constant Yield

The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

scholarly journals Bifurcation, Chaos, and Pattern Formation for the Discrete Predator-Prey Reaction-Diffusion Model

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Lili Meng ◽  
Yutao Han ◽  
Zhiyi Lu ◽  
Guang Zhang
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Region ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Reaction Diffusion Model ◽  
Parameter Values ◽  
Bifurcation Chaos

In this paper, a discrete predator-prey system with the periodic boundary conditions will be considered. First, we get the conditions for producing Turing instability of the discrete predator-prey system according to the linear stability analysis. Then, we show that the discrete model has the flip bifurcation and Turing bifurcation under the critical parameter values. Finally, a series of numerical simulations are carried out in the Turing instability region of the discrete predator-prey model; some new Turing patterns such as striped, bar, and horizontal bar are observed.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

2008 ◽  
Vol 2008 ◽  
pp. 1-15 ◽  
Author(s):  
Can-Yun Huang ◽  
Min Zhao ◽  
Hai-Feng Huo
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Ii Functional Response

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

scholarly journals Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Caifeng Du
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

Cooperative hunting in a predator-prey system with Allee effects in the prey

2018 ◽  
Vol 31 (4) ◽  
pp. e12194 ◽  
Author(s):  
Sophia R.-J. Jang ◽  
Wenjing Zhang ◽  
Victoria Larriva
Keyword(s):  
Allee Effects ◽  
Predator Prey ◽  
Cooperative Hunting ◽  
Prey System
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