Dynamics of a predator-prey ecological system with nonlinear harvesting rate

Liu, Weiyi; Li, Biwen; Fu, Chenglong; Chen, Boshan

doi:10.1007/s11859-015-1054-4

Cited by 3 publications

(6 citation statements)

References 13 publications

(20 reference statements)

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“…We confirm the complexity of the dynamic behavior by computing the largest Lyapunov exponents. This paper extends our previous works (see [8,9]) and provides a sufficiently general parameterization method for a wide range of discrete singular systems.…”

Section: Resultssupporting

confidence: 68%

“…We find the fixed point and its stability. Most interestingly, we have seen that our results reveal a far richer dynamics of the discrete model compared with the continuous one proposed in [9], including an invariant circle, cascades of period-doubling bifurcation and chaotic sets. We confirm the complexity of the dynamic behavior by computing the largest Lyapunov exponents.…”

Section: Resultsmentioning

confidence: 55%

“…Consider the scaled equation, describing a singular bioeconomic system (see [6,9,[17][18][19]) with nonlinear harvesting function, as…”

Section: Statement Of Model and Local Parameterizationmentioning

confidence: 99%

“…Thus G(x) now represents the population growth per day and h is the daily harvesting rate, both being measured in tonnes per day. Predator-prey models with harvesting is always a hot research topic (see [2][3][4][5][6][7][8][9]). The most general form of the two-dimensional predator-prey model is Kolmogorov's model [10]: ⎧ ⎨ ⎩ẋ = xf (x, y), y = yg(x, y), (1.3) here x and y denote the prey and predator populations at time t, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic behaviors of a discrete-time predator–prey bioeconomic system

2018

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Bifurcation and chaotic behavior of a discrete-time singular bioeconomic system are investigated. First, the traditional catch equation is modified after accounting for the handling time of the catch in a singular bioeconomic system. To discover the richer dynamics compared with the continuous form, the proposed system is considered difference scheme. Specially, the tangent space local parameterization condensed method for DAEs is generalized. The new local parameterization method is sufficiently general to be applicable to this type of discrete singular system. Also the dynamic behaviors of the system are investigated, by using normal form theory, center manifold theorem and bifurcation theory, it is shown that the system undergoes a Neimark-Sacker bifurcation and a flip bifurcation, on varying step-size in some range. In addition, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors. MSC: Primary 37G10; secondary 39A28; 39A30

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Section: Resultssupporting

confidence: 68%

Section: Resultsmentioning

confidence: 55%

“…Consider the scaled equation, describing a singular bioeconomic system (see [6,9,[17][18][19]) with nonlinear harvesting function, as…”

Section: Statement Of Model and Local Parameterizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic behaviors of a discrete-time predator–prey bioeconomic system

2018

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“…Many research efforts have been focused on the investigation of this sort of dynamics. In [13][14][15][16], the authors have studied the dynamical behaviour of a class of predator-prey ecosystems formulated from several differential equations and an algebraic equation. They have obtained interesting results, such as stability of interior equilibrium, Hopf bifurcation, limit cycle, singularity induced bifurcation, and its control, and so on.…”

Section: (Communicated By Moxun Tang)mentioning

confidence: 99%

Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting

Nadjah¹,

Abdelouahab²

2021

era

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<p style='text-indent:20px;'>The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue <inline-formula><tex-math id="M1">$ v $</tex-math></inline-formula>. We show that a positive equilibrium point is locally asymptotically stable when the profit <inline-formula><tex-math id="M2">$ v $</tex-math></inline-formula> is less than a certain critical value <inline-formula><tex-math id="M3">$ v^{*}_1 $</tex-math></inline-formula>, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.</p>

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Differential-algebraic equations of the multicriteria locally optimal trajectory of economic restructuring

Vaninsky

2018

Int. J. Dynam. Control

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Dynamics of a predator-prey ecological system with nonlinear harvesting rate

Cited by 3 publications

References 13 publications

Dynamic behaviors of a discrete-time predator–prey bioeconomic system

Dynamic behaviors of a discrete-time predator–prey bioeconomic system

Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting

Differential-algebraic equations of the multicriteria locally optimal trajectory of economic restructuring

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