×

Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge. (English) Zbl 1258.34161

Summary: This paper describes a prey-predator model with Holling type II functional response incorporating prey refuge. The equilibria of the proposed system are determined and the behavior of the system is investigated around equilibria. Density-dependent mortality rate for the predator is considered as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibrium point. Discrete-type gestational delay of predators is also incorporated on the system. The dynamics of the delay induced prey-predator system is analyzed. Delay preserving stability and direction of the system is studied. Global stability of the delay preserving system is shown. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
93C15 Control/observation systems governed by ordinary differential equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139, 311-326 (1989)
[2] Barbalat, I., System d’equations differentielle d’oscillations non linearies, Revue Roumaine De Mathématiques Pures ET Appliquées, 4, 267-270 (1959) · Zbl 0090.06601
[3] Birkoff, G.; Rota, G. C., Ordinary differential equations, Ginn (1982)
[4] Chakraborty, K.; Chakraborty, M.; Kar, T. K., Optimal control of harvest and bifurcation of a prey predator model with stage structure, Applied Mathematics and Computation, 217, 21, 8778-8792 (2011) · Zbl 1215.92059
[5] Chakraborty, K.; Chakraborty, M.; Kar, T. K., Regulation of a prey-predator fishery incorporating prey refuge by taxation: A dynamic reaction model, Journal of Biological Systems, 19, 3, 417-445 (2011) · Zbl 1258.91173
[6] Chen, L.; Chen, F.; Chen, L., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11, 246-252 (2010) · Zbl 1186.34062
[7] Costa, M. I.S.; Kaszkurewicz, E.; Bhaya, A.; Hsu, L., Achieving global convergence to an equilibrium population in predator-prey systems by the use of a discontinuous harvesting policy, Ecological Modelling, 128, 89-99 (2000)
[8] Cressmana, R.; Garay, J., A predator-prey refuge system: evolutionary stability in ecological systems, Theoretical Population Biology, 76, 248-257 (2009) · Zbl 1403.92238
[9] Gao, S.; Chen, L.; Teng, Z., Hopf bifurcation and global stability for a delayed predator prey system with stage structure for predator, Applied Mathematics and Computation, 202, 721-729 (2008) · Zbl 1151.34067
[10] Gause, G. F., The Struggle for Existence (1934), Williams and Wilkins: Williams and Wilkins Baltimore
[11] Gause, G. F.; Smaragdova, N. P.; Witt, A. A., Further studies of interaction between predators and prey, The Journal of Animal Ecology, 5, 1-18 (1936)
[12] González-Olivares, E.; Ramos-Jiliberto, R., Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166, 135-146 (2003)
[13] Gopalsamy, K., Stability and Oscillation in Delay Different Equations in Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0752.34039
[14] Hassard, B.; Kazarinoff, D.; Wan, Y., Theorey and Application of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[15] Holling, C. S., The functional response of predators to prey density and its role in mimicry and population regulation, Memories of Entromological Society of Canada, 45, 1-60 (1965)
[16] Huang, Y.; Chen, F.; Zhong, L., Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182, 672-683 (2006) · Zbl 1102.92056
[17] Kar, T. K., Stability analysis of a prey-predator model incorporation a prey refuge, Communications in Nonlinear Science and Numerical Simulation, 10, 681-691 (2005) · Zbl 1064.92045
[18] Kar, T. K., Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, Journal of Computational and Applied Mathematics, 185, 19-33 (2006) · Zbl 1071.92041
[19] Kar, T. K.; Ghorai, A.; Batabyal, A., Global dynamics and bifurcation of a tri-trophic food chain model, World Journal of Modelling and Simulation, 8, 1, 66-80 (2012)
[20] Ko, W.; Ryu, K., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, Journal of Differential Equations, 231, 534-550 (2006) · Zbl 1387.35588
[21] Křivan, V., On the Gause predator-prey model with a refuge: a fresh look at the history, Journal of Theoretical Biology, 274, 67-73 (2011) · Zbl 1331.92128
[22] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1997), Springer
[23] Lotka, A. J., Elements of Physical Biology (1926), Williams and Wilkins: Williams and Wilkins Baltimore · JFM 51.0416.06
[24] Magalhães, S.; Rijn, P. C.J. V.; Montserrat, M.; Pallini, A.; Sabelis, M. W., Population dynamics of thrips prey and their mite predators in a refuge, Oecologia, 150, 557-568 (2007)
[25] Pal, A. K.; Samanta, G. P., Stability analysis of an eco-epidemiological model incorporating a prey refuge, Nonlinear Analysis: Modelling and Control, 15, 4, 473-491 (2010) · Zbl 1339.49033
[26] Poggiale, J. C.; Michalski, J.; Arditi, R., Emergence of donor control in patchy predator-prey systems, Bulletin of Mathematical Biology, 60, 1149-1166 (1998) · Zbl 0927.92033
[27] Tian, Y.; Sun, K.; Chen, L., Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects, Mathematics and Computers in Simulation, 82, 318-331 (2011) · Zbl 1236.92072
[28] Ton, T. V.; Hieu, N. T., Dynamics of species in a model with two predators and one prey, Nonlinear Analysis: Theory Methods and Application, 74, 14, 4868-4881 (2011) · Zbl 1222.34055
[29] Venkatsubramanian, V.; Schattler, H.; Zaborszky, J., Local bifurcation and feasibility regions in differential-algebraic systems, IEEE Transactions on Automatic Control, 40, 12, 1992-2013 (1995) · Zbl 0843.34045
[30] Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118, 558-560 (1926) · JFM 52.0453.03
[31] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator prey model, Journal of Mathematical Analysis and Applications, 262, 499-528 (2001) · Zbl 0997.34069
[32] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Global stability of a Lotka Volterra type predator prey model with stage structure and time delay, Applied Mathematics and Computation, 159, 863-880 (2004) · Zbl 1056.92063
[33] Xu, R.; Ma, Z., Stability and Hopf bifurcation in a ratio-dependent predator prey system with stage structure, Chaos, Solitons & Fractals, 38, 669-684 (2008) · Zbl 1146.34323
[34] Yongzhen, P.; Changguo, L.; Lansun, C., Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay, Mathematics and Computers in Simulation, 79, 2994-3008 (2009) · Zbl 1172.92038
[35] Yu, H.; Zhong, S.; Ye, M., Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay, Mathematics and Computers in Simulation, 80, 619-632 (2009) · Zbl 1178.92058
[36] Zhang, F.-F.; Jin, Z.; Sun, G.-Q., Bifurcation analysis of a delayed epidemic model, Applied Mathematics and Computation, 216, 753-767 (2010) · Zbl 1186.92042
[37] Zhao, M.; Wang, X.; Yu, H.; Zhu, J., Dynamics of an ecological model with impulsive control strategy and distributed time delay, Mathematics and Computers in Simulation, 82, 1432-1444 (2012) · Zbl 1251.92049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.