# zbMATH — the first resource for mathematics

Complex dynamics in a prey predator system with multiple delays. (English) Zbl 1243.92051
Summary: The complex dynamics is explored of a prey predator system with multiple delays. Holling type-II functional response is assumed for the prey dynamics. The predator dynamics is governed by a modified P.H. Leslie and J.C. Gower scheme [Biometrika 47, 219–234 (1960; Zbl 0103.12502)]. The existence of periodic solutions via Hopf-bifurcations with respect to both delays are established. An algorithm is developed for drawing two-parametric bifurcation diagrams with respect to two delays. The domain of stability with respect to $$\tau _{1}$$ and $$\tau _{2}$$ is thus obtained. The complex dynamical behavior of the system outside the domain of stability is evident from the exhaustive numerical simulations. Direction and stability of periodic solutions are also determined using normal form theory and center manifold arguments.

##### MSC:
 92D40 Ecology 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations 65C20 Probabilistic models, generic numerical methods in probability and statistics 37N25 Dynamical systems in biology
##### Keywords:
time delays; stability; Hopf bifurcation; chaos
Full Text:
##### References:
 [1] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Dekker New York · Zbl 0448.92023 [2] Kot, M., Elements of mathematical ecology, (2001), Cambridge University Press Cambridge [3] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press Princeton, NJ [4] Murray, J.D., Mathematical biology I. an introduction, (2002), Springer New York [5] Cosner, C.; Angelis, D.L.; Ault, J.S.; Olson, D.B., Effects of spatial grouping on the functional response of predators, Theor popul biol, 56, 6575, (1999) [6] Hsu, S.B.; Hwang, T.W.; Kuang, Y., Rich dynamics of ratio-dependent one prey two predators model, J math biol, 43, 377-396, (2000) · Zbl 1007.34054 [7] Kuang, Y., Rich dynamics of gause-type ratio-dependent predator-prey system, Fields inst commun, 21, 325-337, (1999) · Zbl 0920.92032 [8] Haque, M., Ratio-dependent predator – prey models of interacting populations, Bull math biol, 71, 430-452, (2009) · Zbl 1170.92027 [9] Leslie, P.H.; Gower, J.C., The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrica, 47, 219-234, (1960) · Zbl 0103.12502 [10] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley-Interscience New York · Zbl 0259.92001 [11] Aziz-Alaoui, M.A.; Daher Okiye, M., Boundedness and global stability for a predator – prey model with modified leslie – gower and Holling-type II schemes, Appl math lett, 16, 1069-1075, (2003) · Zbl 1063.34044 [12] Hale, J.K., Theory of functional differential equations, (1977), Springer New York [13] Hale, J.K.; Verduyn Lunel, S.M., Theory of functional differential equations, (1993), Springer New York · Zbl 0787.34002 [14] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic press Inc. Boston · Zbl 0777.34002 [15] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer New York [16] Macdonald, N., Time delay in prey – predator models, Math bio, 28, 321-330, (1976) · Zbl 0324.92016 [17] Macdonald, N., Biological delay systems: linear stability theory, (1989), Cambridge University press Cambridge · Zbl 0669.92001 [18] Gopalsamy, K., Harmless delays in model systems, Bull math biol, 45, 295-309, (1983) · Zbl 0514.34060 [19] Cook, K.; Grossman, Z., Discrete delay, distributed delay and stability switches, J math anal appl, 86, 592-627, (1982) · Zbl 0492.34064 [20] Cooke, K.; van den Driessche, P.; Zou, X., Interaction of maturation delay and nonlinear birth in population and epidemic models, J math biol, 39, 32-352, (1999) · Zbl 0945.92016 [21] Cushing, J.M., Integrodifferential equations and delay model in population dynamics, (1977), Springer Heidelberg · Zbl 0363.92014 [22] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependant parameters, SIAM J math anal, 33, 1144-1165, (2002) · Zbl 1013.92034 [23] Hastings, A., Delays in recruitment at different trophic levels: effect on stability, J math biol, 21, 35-44, (1984) · Zbl 0547.92014 [24] Ruan, S., The effect of delays on stability and persistence in plankton models, Nonlinear anal: theory methods appl, 24, 575-585, (1995) · Zbl 0830.34067 [25] Ruan, S.; Wei, J.; Martin, A.; Ruan, S., Predator – prey models with delay and prey harvesting, Dyn cont disc impulsive syst ser A: math anal, J math biol, 43, 247-267, (2001) · Zbl 1008.34066 [26] Aiello, W.G.; Freedman, H.I., A time-delay model of single species growth with stage structure, Math biosci, 101, 139-153, (1990) · Zbl 0719.92017 [27] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage-structured population growth with state dependent time delay, SIAM J appl math, 52, 855-869, (1992) · Zbl 0760.92018 [28] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0105.06402 [29] Nindjin, A.F.; Aziz-Alaoui, M.A.; Cadivel, M., Analysis of a predator – prey model with modified leslie – gower and Holling-type II schemes with time delay, Nonlinear anal., 7, 1104-1118, (2006) · Zbl 1104.92065 [30] Gazi, N.H.; Bandyopadhyay, M., Effect of time delay on a detritus-based ecosystem, Int J math math sci, 1-28, (2006) · Zbl 1127.92042 [31] Hu, G.; Li, W.; Yan, X., Hopf bifurcations in a predator – prey system with multiple delays, Chaos solitons fract, 42, 12731285, (2009) [32] He, X., Stability and delays in a predator – prey system, J math anal appl, 198, 355-370, (1996) · Zbl 0873.34062 [33] Yuan, S.; Song, Y., Bifurcation and stability analysis for a delayed Leslie gower predator – prey system, IMA J appl math, 74, 574-603, (2009) · Zbl 1201.34132 [34] Gakkhar, S.; Sahani, S.K.; Negi, K., Effects of seasonal growth on delayed prey – predator model, Chaos solitons fract, 39, 230-239, (2009) · Zbl 1197.34134 [35] Gakkhar, S.; Negi, K.; Sahani, S.K., Effects of seasonal growth on ratio dependent delayed prey – predator system, Commun nonlinear sci numer simulat, 14, 850-862, (2009) · Zbl 1221.34187 [36] Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002 [37] Song, Y.; Wei, J., Bifurcation analysis for chen’s system with delayed feedback and its application to control of chaos, Chaos solitons fract, 22, 75-91, (2004) · Zbl 1112.37303 [38] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Boston · Zbl 0752.34039 [39] Li, X.; Ruan, S.; Wei, J., Stability and bifurcation in delay-differential equations with two delays, J math anal appl, 236, 254-280, (1999) · Zbl 0946.34066 [40] Faria, T., Stability and bifurcation for a delayed predator – prey model and the effect of diffusion, J math anal appl, 254, 433-463, (2001) · Zbl 0973.35034 [41] Song, Y.; Han, M.; Peng, Y., Stability and Hopf bifurcations in a competitive lotkavolterra system with two delays, Chaos solitons fract, 22, 1139-1148, (2004) · Zbl 1067.34075 [42] Wu, S.; Ren, G., A note on delay-independent stability of a predator – prey model, J sound vibr, 275, 17-25, (2004) · Zbl 1236.34101 [43] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.