Complex Behavior in a Fish Algae Consumption Model with Impulsive Control Strategy

Yang, Jin; Zhao, Min

doi:10.1155/2011/163541

Cited by 10 publications

(7 citation statements)

References 25 publications

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“…The solution of system – is a piecewise continuous function

boldZ : {boldR}_{+} \times {boldR}_{+}^{4}

, with the properties that Z ( t ) is continuous on ( n T ,( n + 1) T ], n ∈ N and

boldZ (n T^{+}) = \lim_{t \to n T^{+}} boldZ (T)

exists. The smoothness properties of f , recall , assure the global existence and uniqueness of the solution of the system – .Definition The system – is said to be uniformly persistent , if there is a ν > 0(independent of the initial conditions) such that each and every solution Z ( t ) of system – satisfies the following conditions:

\underset{t \to \infty}{liminf} X (t) \geq ν, \underset{t \to \infty}{liminf} Y (t) \geq ν, \underset{t \to \infty}{liminf} V (t) \geq ν, \underset{t \to \infty}{liminf} A (t) \geq ν .

Definition The system – is said to be permanent , if there exists a compact region

{normalΘ}_{0} \subset int {boldR}_{+}^{4}

such that each and every solution Z ( t ) of – enters and remains in the region Θ 0 .…”

Section: The System With Impulsesmentioning

confidence: 99%

“…Definition 5. 3 The system (6)-(12) is said to be permanent [38], if there exists a compact region ‚ 0 int R 4 C such that each and every solution Z.t/ of (6)-(12) enters and remains in the region ‚ 0 .…”

Section: Definition 52mentioning

confidence: 99%

“…[38]) Let Z.t/ be a solution of the system(6)-(12) with Z.0 C / 0. Then Z i .t/ 0, i D 1, : : : , 4 for all t 0.…”

mentioning

confidence: 99%

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Effects of awareness program for controlling mosaic disease in Jatropha curcas plantations

Basir

Venturino

Roy

2016

Math Methods in App Sciences

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The shrinkage of fossil fuel resources motivates many countries to search alternative energy sources. Jatropha curcas is a small drought‐resistant shrub from whose seeds a high grade fuel biodiesel can be produced. It is cultivated in many tropical countries including India. However, the plant is affected by the mosaic virus (Begomovirus) through infected white‐flies (Bemisia tabaci) which causes mosaic disease. Disease control is an important factor to obtain healthy crop but in agricultural practice, farming awareness is equally important. Here, we propose a mathematical model for media campaigns for raising awareness among people to protect this plant in small plots and control disease. In order to archive high crop yield, we consider the awareness campaign to be arranged in impulsive way to make people aware from infected white‐flies to protect Jatropha plants from mosaic virus. The study reveals that the spread of mosaic disease can be contained or even eradicated by the awareness campaigns. To attain an effective eradication, awareness campaign should be implemented at sufficiently short time intervals. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The solution of system – is a piecewise continuous function

boldZ : {boldR}_{+} \times {boldR}_{+}^{4}

, with the properties that Z ( t ) is continuous on ( n T ,( n + 1) T ], n ∈ N and

boldZ (n T^{+}) = \lim_{t \to n T^{+}} boldZ (T)

\underset{t \to \infty}{liminf} X (t) \geq ν, \underset{t \to \infty}{liminf} Y (t) \geq ν, \underset{t \to \infty}{liminf} V (t) \geq ν, \underset{t \to \infty}{liminf} A (t) \geq ν .

Definition The system – is said to be permanent , if there exists a compact region

{normalΘ}_{0} \subset int {boldR}_{+}^{4}

such that each and every solution Z ( t ) of – enters and remains in the region Θ 0 .…”

Section: The System With Impulsesmentioning

confidence: 99%

Section: Definition 52mentioning

confidence: 99%

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Effects of awareness program for controlling mosaic disease in Jatropha curcas plantations

Basir

Venturino

Roy

2016

Math Methods in App Sciences

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“…The biologist can use them to study the relationship between species in different domains [6][7][8][9][10]. Yang and Zhao [11] have established a fish-algae consumption model to explore how to apply the complex dynamics between fish-algae populations to expound the mechanism of algae blooms; these results will be helpful in controlling algae bloom. González-Olivares and Rojas-Palma [12] have established a Gause type predator-prey model with Allee effects and considered three standard functional responses, respectively.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Behavior Analysis in a Predator-Prey Model

Wang

Zhao

et al. 2016

Discrete Dynamics in Nature and Society

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A predator-prey model is studied mathematically and numerically. The aim is to explore how some key factors influence dynamic evolutionary mechanism of steady conversion and bifurcation behavior in predator-prey model. The theoretical works have been pursuing the investigation of the existence and stability of the equilibria, as well as the occurrence of bifurcation behaviors (transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation), which can deduce a standard parameter controlled relationship and in turn provide a theoretical basis for the numerical simulation. Numerical analysis ensures reliability of the theoretical results and illustrates that three stable equilibria will arise simultaneously in the model. It testifies the existence of Bogdanov-Takens bifurcation, too. It should also be stressed that the dynamic evolutionary mechanism of steady conversion and bifurcation behavior mainly depend on a specific key parameter. In a word, all these results are expected to be of use in the study of the dynamic complexity of ecosystems.

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“…This theory is very important since complex problems can be modeled by systems with impulse conditions. The reader may find some important results and applications in [1,2,3], [15], [17] and [22,23,24,25], for instance.…”

mentioning

confidence: 99%

Stability and forward attractors for non-autonomous impulsive semidynamical systems

Bonotto¹,

Demuner²

2020

Communications on Pure &Amp; Applied Analysis

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Complex Behavior in a Fish Algae Consumption Model with Impulsive Control Strategy

Cited by 10 publications

References 25 publications

Effects of awareness program for controlling mosaic disease in Jatropha curcas plantations

Effects of awareness program for controlling mosaic disease in Jatropha curcas plantations

Bifurcation Behavior Analysis in a Predator-Prey Model

Stability and forward attractors for non-autonomous impulsive semidynamical systems

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