Dynamics and Bifurcation Analysis of a Filippov Predator–Prey Ecosystem in a Seasonally Fluctuating Environment

2019 ◽  
Vol 29 (02) ◽  
pp. 1950020 ◽  
Author(s):  
Wenjie Qin ◽  
Xuewen Tan ◽  
Xiaotao Shi ◽  
Junhua Chen ◽  
Xinzhi Liu
Keyword(s):  
Pest Control ◽  
Control Strategy ◽  
Sliding Mode ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Periodic Forcing ◽  
Optimal Control Strategy ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fluctuating Environment

Mathematical models can assist to design and understand control strategies for limited resources in Integrated Pest Management (IPM). This paper studies the dynamical behavior of a Filippov predator–prey model with periodic forcing. Firstly, bifurcation analyses are carried out to show that the Filippov predator–prey ecosystem may have very complex dynamics, i.e. the system may have periodic, quasi-periodic, chaotic solutions, as well as period doubling bifurcations. Meanwhile, the model is analyzed theoretically and numerically to understand how resource limitation and periodic forcing affect pest population outbreaks, the intersection between the initial densities (pest and natural enemy populations) and pest control has been discussed. Furthermore, the sliding surface, sliding mode dynamics, the existence and stability of sliding periodic solution of the proposed model and its application in IPM strategy are investigated. Our results show that several hidden factors can adversely affect our control strategy in limited resource and fluctuating environment. Thus, choosing a proper threshold value ET may play a decisive role in pest control, which confirms that IPM is the optimal control strategy.

CHAOTIC BEHAVIOR OF A PERIODICALLY FORCED PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE AND IMPULSIVE PERTURBATIONS

2006 ◽  
Vol 09 (03) ◽  
pp. 209-222 ◽  
Author(s):  
SHUWEN ZHANG ◽  
DEJUN TAN ◽  
LANSUN CHEN
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Periodic Variation ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Perturbations

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

scholarly journals Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Changtong Li ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Pest Control ◽  
Impulsive Control ◽  
Period Doubling ◽  
Dynamical Behaviour ◽  
Period Doubling Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model

Abstract An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

DYNAMIC ANALYSIS OF A HOLLING I PREDATOR-PREY SYSTEM WITH MUTUAL INTERFERENCE CONCERNING PEST CONTROL

2005 ◽  
Vol 13 (01) ◽  
pp. 45-58 ◽  
Author(s):  
YUJUAN ZHANG ◽  
ZHILONG XU ◽  
BING LIU ◽  
LANSUN CHEN
Keyword(s):  
Biological Control ◽  
Periodic Solutions ◽  
Pest Control ◽  
Chemical Control ◽  
Numerical Solutions ◽  
Dynamical Behavior ◽  
Continuous System ◽  
Mutual Interference ◽  
Predator Prey ◽  
Predator Prey Model

A Holling I predator-prey model with mutual interference concerning pest control is proposed and analyzed. The prey and predator are considered to be a pest and a natural enemy, respectively. The model is forced by the addition of periodic impulsive terms representing predator import (biological control) and pesticide application (chemical control) at different fixed moments. By using Floquet theory and small amplitude perturbations, we show the existence and stability of pest-free periodic solutions. Further, we prove that when the stability of pest-free periodic solutions is lost, the system is permanent by using analytic methods of differential equation theory. Numerical solutions are also given, which show that when stability of pest-free periodic solutions is lost, more exotic behavior can occur, such as quasi-periodic oscillation or chaos. We investigate the effect of impulsive perturbations on the unforced continuous system, and find that the forced system has a different dynamical behavior with a different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system. Finally, we compare the validity of the combination of biological control and chemical control with classical methods and conclude that the synthetical strategy is more effective than classical methods if we take effective chemical control.

An Imprecise Eco-Epidemic Model with Pesticide in Relevance to Agricultural Pest Control

2018 ◽  
Vol 13 (03) ◽  
pp. 109-131
Author(s):  
Anjana Das ◽  
M. Pal
Keyword(s):  
Pest Control ◽  
Dynamical Behavior ◽  
Predator Population ◽  
Agricultural Pest ◽  
Stability Criteria ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Existence And Stability ◽  
Imprecise Parameters

In this paper, we have proposed and analyzed an agricultural pest control system. For this purpose, an eco-epidemiological type predator–prey model has been proposed with the consideration of a sound predator population and two classes of pest populations namely susceptible pest and infected pest. Further to consider uncertainty, we modify our model and transform it into a fuzzy system with incorporation of imprecise parameters. The dynamical behavior of the proposed model has been investigated by examining the existence and stability criteria of all feasible equilibria. An optimal control problem is formed by considering the pesticide control as the control parameter and then the problem is solved both theoretically and numerically with the help of some computer simulation works.

scholarly journals Generalized Predator-Prey Model with Nonlinear Impulsive Control Strategy

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenjie Qin ◽  
Guangyao Tang ◽  
Sanyi Tang
Keyword(s):  
Pest Control ◽  
Impulsive Control ◽  
Dynamical Behavior ◽  
Limited Resource ◽  
Control Measures ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Pest Outbreaks ◽  
Wide Range

A generalized predator-prey model concerning integrated pest management and nonlinear impulsive control measures is proposed and analyzed. The main purpose is to understand how resource limitation affects the successful pest control and pest outbreaks. The threshold conditions for the stability of the pest-free periodic solution are given firstly. Once the threshold value exceeds a critical level, both pest and its natural enemy populations can oscillate periodically. Secondly, in order to address how the limited resources affect the pest control, as an example the Holling II functional response function is chosen. The numerical results show that predator-prey model with limited resource has complex dynamical behavior. In addition, it is confirmed that the model has the coexistence of pests and natural enemies for a wide range of parameters.

DYNAMIC COMPLEXITIES IN A LOTKA–VOLTERRA PREDATOR–PREY MODEL CONCERNING IMPULSIVE CONTROL STRATEGY

2005 ◽  
Vol 15 (02) ◽  
pp. 517-531 ◽  
Author(s):  
BING LIU ◽  
YUJUAN ZHANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Control Strategy ◽  
Impulsive Control ◽  
Period Doubling ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pest Eradication ◽  
Impulsive Control Strategy

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

scholarly journals The Dynamic Complexity of a Holling Type-IV Predator-Prey System with Stage Structure and Double Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Yakui Xue ◽  
Xiafeng Duan
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Dynamical Behavior ◽  
Stage Structure ◽  
Point Of View ◽  
Maturation Time ◽  
Dynamic Complexity ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv

We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

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>

Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

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