scholarly journals Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

2019 ◽  
Vol 17 (1) ◽  
pp. 1186-1202 ◽  
Author(s):  
Fengde Chen ◽  
Xinyu Guan ◽  
Xiaoyan Huang ◽  
Hang Deng
Keyword(s):  
Birth Rate ◽  
Allee Effect ◽  
Prey Species ◽  
Positive Equilibrium ◽  
Volterra Type ◽  
Predator Species ◽  
Dynamic Behaviors ◽  
Predator Prey ◽  
Density Dependent ◽  
Prey System

Abstract A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.

PERMANENCE IN A PERIODIC PREDATOR–PREY SYSTEM WITH PREY DISPERSAL AND PREDATOR DENSITY-INDEPENDENT

2006 ◽  
Vol 14 (04) ◽  
pp. 491-507 ◽  
Author(s):  
LONG ZHANG ◽  
ZHIDONG TENG
Keyword(s):  
Periodic Solution ◽  
Prey Species ◽  
Positive Periodic Solution ◽  
Analytic Method ◽  
Predator Density ◽  
Volterra Type ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Theoretical Results

In this paper, we study two-species predator–prey Lotka–Volterra-type dispersal system with periodic coefficients, in which the prey species can disperse among n-patches, but the predator species which is density-independent is confined to some patches and cannot disperse. By utilizing the analytic method, sufficient and realistic conditions on the boundedness, permanence, extinction, and the existence of positive periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations.

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.

Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator–Prey System with Beddington–DeAngelis Functional Response

2016 ◽  
Vol 26 (10) ◽  
pp. 1650165 ◽  
Author(s):  
Haiyin Li ◽  
Gang Meng ◽  
Zhikun She
Keyword(s):  
Hopf Bifurcation ◽  
Density Dependence ◽  
Functional Response ◽  
Positive Equilibrium ◽  
Stability Switches ◽  
Predator Prey ◽  
Geometric Criterion ◽  
Density Dependent ◽  
Prey System ◽  
The Stability

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator–prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.

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.

Spatiotemporal Patterns of a Predator–Prey System with an Allee Effect and Holling Type III Functional Response

2016 ◽  
Vol 26 (05) ◽  
pp. 1650088 ◽  
Author(s):  
Yuanyuan Li ◽  
Jinfeng Wang
Keyword(s):  
Allee Effect ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Rigorous Results ◽  
Predator Prey ◽  
Type Iii ◽  
Prey System ◽  
Response Subject ◽  
Steady State Solutions

A diffusive Gause type predator–prey system with Allee effect in prey growth and Holling type III response subject to Neumann boundary conditions is investigated. Existence of nonconstant positive steady state solutions is proved by Leray–Schauder degree theory and bifurcation theory. Global stability of the positive equilibrium of the system is also investigated. Moreover, bifurcations of spatially homogeneous and nonhomogeneous periodic solutions are analyzed. Our rigorous results justify some recent ecological observations.

scholarly journals Qualitative Analysis for a Reaction-Diffusion Predator-Prey Model with Disease in the Prey Species

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Meihong Qiao ◽  
Anping Liu ◽  
Urszula Foryś
Keyword(s):  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Disease In Predator ◽  
Disease Free

A diffusive predator-prey system with disease in predator species and no-flux boundary condition is considered. Sufficient conditions which ensure persistence of the system are obtained. Conditions of disease-free ecosystem are also studied. Furthermore, sufficient conditions for global asymptotic stability of the unique positive equilibrium and disease-free equilibrium of the system are derived using the approach of Lyapunov function.

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 The uneasy coexistence of predators and pathogens

2020 ◽  
Vol 43 (7) ◽  
Author(s):  
Andreas Eilersen ◽  
Kim Sneppen
Keyword(s):  
Host Species ◽  
Prey Species ◽  
Scaling Relations ◽  
Predator Species ◽  
Predator Prey ◽  
Mass Scaling ◽  
Prey System ◽  
Model Equations ◽  
Extinction Events ◽  
Parameter Values

Abstract. Disease and predation are both highly important in ecology, and pathogens with multiple host species have turned out to be common. Nonetheless, the interplay between multi-host epidemics and predation has received relatively little attention. Here, we analyse a model of a predator-prey system with disease in both prey and predator populations and determine reasonable parameter values using allometric mass scaling relations. Our analysis focuses on the possibility of extinction events rather than the linear stability of the model equations, and we derive approximate relations for the parameter values at which we expect these events to occur. We find that if the predator is a specialist, epidemics frequently drive the predator species to extinction. If the predator has an additional, immune prey species, predators will usually survive. Coexistence of predator and disease is impossible in the single-prey model. We conclude that for the prey species, carrying a pathogen can be an effective weapon against predators, and that being a generalist is a major advantage for a predator in the event of an epidemic affecting the prey or both species. Graphical abstract

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