Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T19:21:00.993Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Dynamical Behaviour in a Predator-Prey Model with Harvesting

Published online by Cambridge University Press:  02 May 2017

Wei Liu and
Yaolin Jiang
Wei Liu*
Affiliation:
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, ChinaSchool of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China
Yaolin Jiang*
Affiliation:
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
*
*Corresponding author. Email addresses:wliu2015@163.com (W. Liu), yljiang@xjtu.edu.cn (Y.L. Jiang)
*Corresponding author. Email addresses:wliu2015@163.com (W. Liu), yljiang@xjtu.edu.cn (Y.L. Jiang)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We investigate the stability and periodic orbits of a predator-prey model with harvesting. The model has a biologically-meaningful interior, an attractor undergoing damped oscillations, and can become destabilised to produce periodic orbits via a Hopf bifurcation. Some sufficient conditions for the existence of the Hopf bifurcation are established, and a stability analysis for the periodic solutions using a Lyapunov function is presented. Finally, some computer simulations illustrate our theoretical results.

Keywords

Predator-prey systemHolling type IV functional responseperiodic solutionbifurcationLyapunov function

MSC classification

Secondary:92D25: Population dynamics (general)

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017, pp. 376 - 395
Copyright
Copyright © Global-Science Press 2017 

References

[1] Chen, L.S., Mathematical Models and Methods in Ecology (in Chinese), Science Press, Beijing (1988).Google Scholar
[2] Yodzis, P., Predator-prey theory and management of multispecies fisheries, Ecol. Appl. 4, 51–58 (1994).Google Scholar
[3] Jost, C., Comparing predator-prey models qualitatively and quantitatively with ecological time-series data, PhD Thesis, Institut National Agronomique Paris-Grignon (1998).Google Scholar
[4] Lucas, W.F., Modules in Applied Mathematics: Differential Equation Models, Springer, New York (1983).Google Scholar
[5] Gordon, H.S., Economic theory of a common property resource: The fishery, J. Polit. Econ. 62, 124–142 (1954).Google Scholar
[6] Zhang, G.D., Shen, Y. and Chen, B.S., Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn. 73, 2119–2131 (2013).Google Scholar
[7] Zhang, G.D., Shen, Y. and Chen, B.S., Bifurcation analysis in a discrete differential-algebraic predator-prey system, Applied Math. Modelling 38, 4835–4848 (2014).Google Scholar
[8] Wu, X.Y. and Chen, B.S., Bifurcations and stability of a discrete singular bioeconomic system, Nonlinear Dyn. 73, 1813–1828 (2013).Google Scholar
[9] Chen, B.S. and Chen, J.J., Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comp. 219, 2371–2386 (2012).Google Scholar
[10] Teng, Z.D. and Rehim, M., Persistence in nonautonomous predator-prey systemswith infinite delays, J. Comput. Appl. Math. 197, 302–321 (2006).Google Scholar
[11] Muhammadhaji, A. and Teng, Z.D., Permanence and extinction analysis for a periodic competing predator-prey system with stage structure, International Journal of Dynamics and Control, In press, On-line vr. DOI:10.1007/s40435-015-0211-0 (2015).CrossRefGoogle Scholar
[12] Kar, T.K. and Pahari, U.K., Non-selective harvesting in prey-predator models with delay, Commun. Nonlinear Sci. Numer. Simul. 11, 499–509 (2006).Google Scholar
[13] Krise, S. and Choudhury, S.R., Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos Soliton Fract. 16, 59–77 (2003).Google Scholar
[14] Zhang, G.D. and Shen, Y., Periodic solutions for a neutral delay Hassell-Varley type predator-prey system, Appl. Math. Comput. 264, 443–452 (2015).Google Scholar
[15] Chen, B.S. and Chen, J.J., Complex dynamic behaviors of a discrete predator-prey model with stage-structure and harvesting, Int. J. Biomath. 10, 1750013 (2017).Google Scholar
[16] Chen, B.S., Liao, X.X. and Liu, Y.Q., Normal forms and bifurcations for the differential-algebraic systems (in Chinese), Acta Math. Appl. Sin. 23, 429–443 (2000).Google Scholar
[17] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983).Google Scholar
[18] Zhang, J.Y., Geometry Theory and Bifurcation Problems of Ordinary Differential Equations (in Chinese), Peking University Press, Beijing (1987).Google Scholar