Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays

Zhang, Guodong; Shen, Yi; Chen, Boshan

doi:10.1016/j.jmaa.2012.05.045

Cited by 22 publications

(21 citation statements)

References 17 publications

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“…The main purpose of this paper is to establish some new sufficient conditions on the existence of positive ω ‐periodic solutions of system by using the fixed point theory based on monotone operator. The result obtained by this paper improves the main result in (see Remark in Section 3). Moreover, by applying the comparison theorem and constructing a suitable Lyapunov functional, some conditions for the permanence and global attractivity of system are obtained.…”

Section: Introductionsupporting

confidence: 88%

“…By Corollary , system has at least one positive π‐periodic analytical solution. Remark In view of Example , it is easy to verify that

m {\bar{ρ}}_{1} - ā_{1} < 1 - 9 = - 8 < 0,

which implies that ( F ) ′ in Theorem is not satisfied. So it is impossible to obtain the existence of positive periodic solution of system by the result in . Therefore, the main results in this paper complement the previously known result. Example Consider the following non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects:

({, \begin{matrix} \frac{d x (t)}{d t} = x (t) [2 - (15 - | \sin (0.2 πt) |) x (t) - \frac{0.0001 y (t)}{y^{0.6} (t) + x (t)}] - x (t), \\ \frac{d y (t)}{d t} = y (t) [- 1 + \frac{(6 - 0.1 \overset{2}{\cos} (0.2 πt)) x (t)}{y^{0.6} (t) + x (t)}] - y (t), t \neq t_{k}, \\ x (t_{k}^{+}) = θ_{1 k} x (t_{k}), \\ y (t_{k}^{+}) = θ_{2 k} y (t_{k}), k \in Z^{+} \end{matrix})

…”

Section: Examplesmentioning

confidence: 57%

“…Considering the strong impact on dynamic evolution of a population (e.g., see ), Zhang et al . focused on varying harvesting rate in system and discussed the following non‐autonomous system:

({, \begin{matrix} \frac{d x (t)}{d t} = x (t) [r_{1} (t) - b (t) x (t) - \frac{a_{1} (t) y (t)}{m y^{γ} (t) + x (t)}] - c_{1} (t) x (t), \\ \frac{d y (t)}{d t} = y (t) [- r_{2} (t) + \frac{a_{2} (t) x (t)}{m y^{γ} (t) + x (t)}] - c_{2} (t) y (t), \end{matrix}) (0 < γ < 1)

where c 1 and c 2 are the harvesting coefficients of prey and predator, respectively. By applying the coincidence degree theorem, the authors obtained some new sufficient conditions for the existence of positive periodic solutions for system .…”

Section: Introductionmentioning

confidence: 99%

“…In 1969, Hassell and Varley [14] introduced a general predator-prey system, in which the functional response dependents on the predator density in different way. It is called a Hassell-Varley type functional response, which take the following form: Considering the strong impact on dynamic evolution of a population (e.g., see [17][18][19][20][21]), Zhang et al [21] focused on varying harvesting rate in system (1.1) and discussed the following non-autonomous system: where c 1 and c 2 are the harvesting coefficients of prey and predator, respectively. By applying the coincidence degree theorem, the authors [21] obtained some new sufficient conditions for the existence of positive periodic solutions for system (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

Huang

Zhang

2015

Math Methods in App Sciences

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This paper is concerned with a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects. By using the fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive periodic solution of the model. The existence result of this paper implies that the functional response on prey does not influence the existence of positive periodic solution of the model, which completes some results given in recent years. Further, by applying the comparison theorem in impulsive differential equations and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples and numerical simulations are given to illustrate the feasibility and effectiveness of the main results. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

Section: Introductionsupporting

confidence: 88%

“…By Corollary , system has at least one positive π‐periodic analytical solution. Remark In view of Example , it is easy to verify that

m {\bar{ρ}}_{1} - ā_{1} < 1 - 9 = - 8 < 0,

({, \begin{matrix} \frac{d x (t)}{d t} = x (t) [2 - (15 - | \sin (0.2 πt) |) x (t) - \frac{0.0001 y (t)}{y^{0.6} (t) + x (t)}] - x (t), \\ \frac{d y (t)}{d t} = y (t) [- 1 + \frac{(6 - 0.1 \overset{2}{\cos} (0.2 πt)) x (t)}{y^{0.6} (t) + x (t)}] - y (t), t \neq t_{k}, \\ x (t_{k}^{+}) = θ_{1 k} x (t_{k}), \\ y (t_{k}^{+}) = θ_{2 k} y (t_{k}), k \in Z^{+} \end{matrix})

…”

Section: Examplesmentioning

confidence: 57%

“…Considering the strong impact on dynamic evolution of a population (e.g., see ), Zhang et al . focused on varying harvesting rate in system and discussed the following non‐autonomous system:

({, \begin{matrix} \frac{d x (t)}{d t} = x (t) [r_{1} (t) - b (t) x (t) - \frac{a_{1} (t) y (t)}{m y^{γ} (t) + x (t)}] - c_{1} (t) x (t), \\ \frac{d y (t)}{d t} = y (t) [- r_{2} (t) + \frac{a_{2} (t) x (t)}{m y^{γ} (t) + x (t)}] - c_{2} (t) y (t), \end{matrix}) (0 < γ < 1)

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

Huang

Zhang

2015

Math Methods in App Sciences

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show abstract

“…It is well known that Mawhin's continuation theorem of coincidence degree theory is an important method to investigate the existence of positive periodic solutions of some kinds of non-linear ecosystems (see [3,4,5,6,7,8,15,16,22,25,33,34]). However, it is difficult to be used to investigate the existence of positive almost periodic solutions of non-linear ecosystems.…”

Section: Introductionmentioning

confidence: 99%

Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay

Zhang¹,

Liao²

2017

Kybernetika

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Stability, bifurcation analysis, and chaos control of a discrete bioeconomic model

Yousef

Rida

Arafat

et al. 2022

Math Methods in App Sciences

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In this paper, we propose a host‐parasitoid model with harvest effort. The existence and stability of a positive fixed point are analyzed. The period‐doubling and Neimark–Sacker bifurcations are studied. These analyses are achieved by applying the normal form of the difference‐algebraic system, bifurcation theory, and center manifold theorem. Furthermore, we apply a state‐delayed feedback control strategy to control the complex dynamics of the proposed model. Numerical examples and simulations are given to verify our findings. Owing to the framework of Nicholson–Bailey host‐parasitoid system, the proposed difference‐algebraic model shows rich dynamics compared with the continuous‐time models.

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Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays

Cited by 22 publications

References 17 publications

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay

Stability, bifurcation analysis, and chaos control of a discrete bioeconomic model

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