scholarly journals Dynamical Behaviors of a Stage-Structured Predator-Prey Model with Harvesting Effort and Impulsive Diffusion

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Lingzhi Huang ◽  
Zhichun Yang
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Comparison Theorem ◽  
Discrete Dynamical System ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model ◽  
Impulsive Diffusion

We consider a delayed predator-prey model with harvesting effort and impulsive diffusion between two patches. By the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map, we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system. Furthermore, the permanence of the system is derived. The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

Almost periodic solution of a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference

2014 ◽  
Vol 07 (03) ◽  
pp. 1450028 ◽  
Author(s):  
Shengbin Yu ◽  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Almost Periodic Solution ◽  
Type Ii ◽  
Almost Periodic ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Globally Attractive

In this paper, we consider a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones.

scholarly journals Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients

BIOMATH ◽  
2017 ◽  
Vol 6 (1) ◽  
pp. 1707107
Author(s):  
Moussaoui Ali ◽  
M. A. Aziz Alaoui ◽  
R. Yafia
Keyword(s):  
Periodic Solution ◽  
System Modeling ◽  
Sufficient Conditions ◽  
Environmental Parameters ◽  
Positive Periodic Solution ◽  
Auxiliary Function ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Constant Coefficients

In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T-periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results.

scholarly journals Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Feedback Controls

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Kerong Zhang ◽  
Jianli Li ◽  
Aiwen Yu
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Feedback Controls ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Globally Attractive

We consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows:x˙t=xta1t-btxt-ctyt/αt+βtxt+γtyt-e1tut,u˙t=-d1tut+p1txt-τ,y˙t=yta2t-rtyt/xt+kt-e2tνt, andν˙(t)=-d2(t)ν(t)+p2(t)y(t-τ). Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.

scholarly journals Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zhimin Zhang
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Almost Periodic Solution ◽  
Globally Attractive

We consider a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones. One example is presented to verify our main results.

AN IMPULSIVE PREDATOR-PREY SYSTEM WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE AND TIME DELAY

2008 ◽  
Vol 01 (01) ◽  
pp. 1-17 ◽  
Author(s):  
HONG ZHANG ◽  
PAUL GEORGESCU ◽  
LANSUN CHEN
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Time Delay ◽  
Functional Response ◽  
Discrete Dynamical System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Crucial Effect ◽  
Two Stages

This paper deals with an impulsive predator-prey model with Beddington–DeAngelis functional response and time delay, in which the evolution of the predators takes them through two stages, juvenile and mature. It is assumed that only mature predators are able to hunt for prey and reproduce and the time delay is understood as being the time spent by the juvenile predators from birth to maturity. It is first seen that the dynamics of the model can be completely determined through the use of a reduced system consisting of the equations for prey and mature predators, respectively. Using the discrete dynamical system determined by the stroboscopic map, one first determines the mature predator-free periodic solution of the reduced system. By means of comparison techniques, one then deduces sufficient criteria for the global stability of the mature predator-free periodic solution and for the permanence of the reduced system, which yield similar properties for the initial system. As a result, it is observed that time delay and pulses have a crucial effect upon the dynamics of our model.

Flip bifurcation and Neimark–Sacker bifurcation in a discrete predator–prey model with harvesting

2019 ◽  
Vol 13 (01) ◽  
pp. 1950093
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
Equilibrium Point ◽  
Discrete Dynamical System ◽  
Singular System ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

In this paper, a difference-algebraic predator–prey model is proposed, and its complex dynamical behaviors are analyzed. The model is a discrete singular system, which is obtained by using Euler scheme to discretize a differential-algebraic predator–prey model with harvesting that we establish. Firstly, the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory. Further, by applying the new normal form of difference-algebraic equations, center manifold theory and bifurcation theory, the Flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point are studied, where the step size is treated as the variable bifurcation parameter. Lastly, with the help of Matlab software, some numerical simulations are performed not only to validate our theoretical results, but also to show the abundant dynamical behaviors, such as period-doubling bifurcations, period 2, 4, 8, and 16 orbits, invariant closed curve, and chaotic sets. In particular, the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.

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 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.

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