Dynamic analysis of a plankton–herbivore state-dependent impulsive model with action threshold depending on the density and its changing rate

Abstract A plankton-herbivore state-dependent impulsive model with nonlinear impulsive functions and action threshold including population density and rate of change is proposed. Since the use of action threshold makes the model have complex phase set and pulse set, we adopt the Poincaré map as a tool to study its complex dynamics. The Poincaré map is defined on the phase set and its properties in different situations are analyzed. Furthermore, the periodic solution of model are discussed, including the existence and stability conditions of the order-1 periodic solution and the existence of the order-k (k ≥ 2) periodic solutions. Compared with the fixed threshold in the existing literature, our results show that the use of action threshold is more practical, which is conducive to the sustainable development of population and makes people obtain more economic benefits. The analysis method used in this paper can study the complex dynamics of the model more comprehensively and deeply.

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Dynamical Behaviors for a Predator-Prey System with State-Dependent Impulsive Effects

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.130-134.385 ◽

2011 ◽

Vol 130-134 ◽

pp. 385-390

Author(s):

Ling Zhen Dong ◽

Lan Sun Chen

Keyword(s):

Dynamical System ◽

Periodic Solution ◽

Asymptotic Stability ◽

Impulsive System ◽

Impulsive Effects ◽

Predator Prey ◽

Dynamical Behaviors ◽

Prey System ◽

State Dependent ◽

Impulsive Model

With some theory about continuous and impulsive dynamical system, an impulsive model based on a special predator-prey system is considered. We assume that the impulsive effects occur when the density of the prey reaches a given value. For such a state-dependent impulsive system, the existence, uniqueness and orbital asymptotic stability of an order-1 periodic solution are discussed. Further, the existence of an order-2 periodic solution is also obtained, and persistence of the system is investigated.

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Modeling the Dynamics of a Single-Species Model with Pollution Treatment in a Polluted Environment

Discrete Dynamics in Nature and Society ◽

10.1155/2013/412409 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Bing Liu ◽

Shi Luan ◽

Yinghui Gao

Keyword(s):

Sufficient Conditions ◽

Single Species ◽

Fixed Time ◽

Polluted Environment ◽

State Dependent ◽

Pollution Treatment ◽

Impulsive Model ◽

The Given ◽

Theoretical Results ◽

Single Species Model

Without any question, environmental pollution is the main cause for the species extinction in recent times. In this paper, based on impulsive differential equation, the dynamics of a single-species model with impulsive pollution treatment at fixed time in a polluted environment is considered, in which we assume that the species is directly affected by the pollutants. Sufficient conditions for permanence and extinction of the species are given. The results show that the species is permanent when the impulsive period is less than some critical value, otherwise the species will be extinct. Although shortening the impulsive period can protect the species from extinction, it is expensive. To see how pollution treatment applications could be economical, we also establish a hybrid impulsive model involving periodic pollution treatment at fixed time with state-dependent pollution treatment applied when the pollution concentration reaches the given Environment Threshold (ET). It indicates that the hybrid method is the most effective method to protect the species from extinction. Numerical simulations confirm our theoretical results.

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Dynamical Behavior and Bifurcation Analysis of the SIR Model with Continuous Treatment and State-Dependent Impulsive Control

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501311 ◽

2019 ◽

Vol 29 (10) ◽

pp. 1950131 ◽

Cited By ~ 2

Author(s):

Qian Li ◽

Yanni Xiao

Keyword(s):

Periodic Solution ◽

Periodic Solutions ◽

Reproduction Number ◽

Continuous Treatment ◽

Positive Order ◽

Susceptible Population ◽

State Dependent ◽

Existence And Stability ◽

Impulsive Model ◽

Disease Free

In this study, we propose a state-dependent impulsive model describing the susceptible individuals-triggered interventions. We find that the model with susceptible individuals-guided impulsive interventions can exhibit very complex dynamical behaviors with rich biological meanings. We note that this formulated impulsive model has disease-free periodic solution, and we can investigate the threshold dynamics by defining the control reproduction number. We study the existence and stability of the disease-free periodic solution (DFPS) for [Formula: see text]. Our results show that, even if the basic reproduction number [Formula: see text], the DFPS can still be stable when the threshold level of susceptible population [Formula: see text], indicating that with a proper chosen [Formula: see text], the state-dependent impulsive strategy can effectively control the development of the infectious disease and eradicate the disease eventually. By employing the bifurcation theory, we investigate the bifurcation phenomenon near the DFPS with respect to some key parameters, and observe that a positive order-1 periodic solution can bifurcate from the DFPS via a transcritical bifurcation. By utilizing numerical simulation, we further explore the existence and stability of the positive order-[Formula: see text] periodic solutions, and found the feasibility of stable positive order-1, order-2 and order-3 periodic solutions, that imply the existence of chaos. In particular, we find that there can be three positive order-1 periodic solutions simultaneously, in which one is stable and the other two are unstable. Our finding indicates that the comprehensive strategy combining continuous treatment with state-dependent impulsive vaccination and isolation plays a crucial role in controlling the prevalence and further spread of the infectious diseases.

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The State-Dependent Impulsive Model with Action Threshold Depending on the Pest Density and Its Changing Rate

Complexity ◽

10.1155/2019/6509867 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Ihsan Ullah Khan ◽

Sanyi Tang ◽

Biao Tang

Keyword(s):

Periodic Solutions ◽

Integrated Control ◽

Dynamical Behavior ◽

Analytical Techniques ◽

The State ◽

Control Measures ◽

Lambert W Function ◽

Action Threshold ◽

Pest Density ◽

Changing Rate

Whether the integrated control measures are applied or not depends not only on the current density of pest population, but also on its current growth rate, and this undoubtedly brings challenges and new ideas to the state control measures that only rely on the pest density. To address this, utilizing the tactics of IPM, we constructed a Lotka-Volterra predator-prey system with action threshold depending on the pest density and its changing rate and examined its dynamical behavior. We present new criteria guaranteeing the existence, uniqueness, and global stability of periodic solutions. With the help of Lambert W function, the Poincaré map is constructed for the phase set, which can help us to provide the satisfactory conditions for the existence and stability of the semitrivial periodic solution and interior order-1 periodic solutions. Furthermore, the existence of order-2 and nonexistence of order-k(k≥3) periodic solutions are discussed. The idea of action threshold depending on the pest density and its changing rate is more general and can generate new remarkable directions as well compared with those represented in earlier studies. The analytical techniques developed in this paper can play a significant role in analyzing the impulsive models with complex phase set or impulsive set.

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