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On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity. (English) Zbl 1221.34192
Summary: We study the effect of the degree of habitat complexity and gestation delay on the stability of a predator--prey model. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. The qualitative dynamical behavior of the model system is verified with the published data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction. It is observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the gestation period exceeds some critical value. However, the fluctuations in the population levels can be controlled completely by increasing the degree of habitat complexity.

34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
92D25Population dynamics (general)
37N25Dynamical systems in biology
Full Text: DOI
[1] Sih, A.; Crowley, P.; Mcpeek, M.; Petranka, J.; Strohmeier, K.: Predation, competition, and prey communities, a review of field experiments, Ann. rev. Ecol. syst. 16, 269-311 (1985)
[2] Alstad, D.: Basic populas models of ecology, (2001)
[3] Anderson, T. W.: Predator responses, prey refuges and density-dependent mortality of a marine fish, Ecology 82, No. 1, 245-257 (2001)
[4] Savino, J. F.; Stein, R. A.: Predator -- prey interaction between largemouth bass and bluegills as influenced by simulated, submersed vegetation, Trans. am. Fish. soc. 111, 255-266 (1982)
[5] Savino, J. F.; Stein, R. A.: Behavioural interactions between fish predators and their prey: effects of plant density, Anim. behav. 37, 311-321 (1989)
[6] Savino, J. F.; Stein, R. A.: Behavior of fish predators and their prey: habitat choice between open water and dense vegetation, Environ. biol. Fishes 24, 287-293 (1989)
[7] Anderson, O.: Optimal foraging by largemouth bass in structured environments, Ecology 65, 851-861 (1984)
[8] Folsom, T. C.; Collins, N. C.: The diet and foraging behavior of the larval dragonfly anax junius (Aeshnidae), with an assessment of the role of refuges and prey activity, Oikos 42, 105-113 (1984)
[9] Persson, L.: Behavioral response to predators reverses the outcome of competition between prey species, Behav. ecol. Sociobiol. 28, 101-105 (1991)
[10] Christensen, B.; Persson, L.: Species-specific antipredator behaviours: effects on prey choice in different habitats, Behav. ecol. Sociobiol. 32, 1-9 (1993)
[11] Manatunge, J.; Asaeda, T.; Priyadarshana, T.: The influence of structural complexity on fish -- zooplankton interactions: A study using artificial submerged macrophytes, Environ. biol. Fishes 58, 425-438 (2000)
[12] Luckinbill, L.: Coexistence in laboratory populations of paramecium aurelia and its predator didinium nasutum, Ecology 54, 1320-1327 (1973)
[13] Gause, G. F.: The struggle for existence, (1934) · Zbl 60.1113.08
[14] Macdonald, M.: Biological delay systems: linear stability theory, (1989) · Zbl 0669.92001
[15] May, R. M.: Theoretical ecology: principles and applications, (1981)
[16] Gopalsamy, K.; He, X.: Delay-independent stability in bi-directional associative memory networks, IEEE trans. Neural netw. 5, 998-1002 (1994)
[17] Cao, J. D.: On stability analysis in delayed cellular neural network, Phys. rev. E 59, 5940-5944 (1999)
[18] Cao, J. D.: Periodic oscillation solution of biderictional associative memory networks with delay, Phys. rev. E 59, 1825-1828 (2000)
[19] Song, Y.; Han, M.; Wei, J.: Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D 200, 185-204 (2005) · Zbl 1062.34079 · doi:10.1016/j.physd.2004.10.010
[20] 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 · doi:10.1016/j.chaos.2003.12.075
[21] Yang, H.; Tian, Y.: Hopf bifurcation in REM algorithm with communication delay, Chaos, solitons fract. 25, 1093-1105 (2005) · Zbl 1198.93099 · doi:10.1016/j.chaos.2004.11.085
[22] Qu, Y.; Wei, J.: Bifurcation analysis in a time-delay model for prey -- predator growth with stage-structure, Nonlinear dyn. 49, 285-294 (2007) · Zbl 1176.92056 · doi:10.1007/s11071-006-9133-x
[23] Celik, C.: The stability and Hopf bifurcation of a predator -- prey system with time delay, Chaos, solitons fract. 37, 87-99 (2008) · Zbl 1152.34059 · doi:10.1016/j.chaos.2007.10.045
[24] Celik, C.: Hopf bifurcation of a ratio-dependent predator -- prey system with time delay, Chaos, solitons fract. 42, 1474-1484 (2009) · Zbl 1198.34149 · doi:10.1016/j.chaos.2009.03.071
[25] Sun, C.; Lin, Y.; Han, M.: Stability and Hopf bifurcation for an epidemic disease model with delay, Chaos, solitons fract. 30, 204-216 (2006) · Zbl 1165.34048 · doi:10.1016/j.chaos.2005.08.167
[26] Holling, C. S.: Some characteristics of simple types of predation and parasitism, Can. entomologist 91, 385-398 (1959)
[27] Winfield, I. J.: The influence of simulated aquatic macrophytes on the zooplankton consumption rate of juvenile roach, rutilus rutilus, rudd, scardinius erythrophthalmus, and perch, perca fluviatilis, J. fish biol. 29, 37-48 (1986)
[28] Kot, M.: Elements of mathematical ecology, (2001) · Zbl 1060.92058
[29] Culshaw, R. V.; Ruan, S.: A delay-differential equation model of HIV infection of CD4+ T-cells, Math. biosci. 165, 27-39 (2000) · Zbl 0981.92009 · doi:10.1016/S0025-5564(00)00006-7
[30] Dieudonne’, J.: Foundations of modern analysis, (1960)
[31] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and application of Hopf bifurcation, (1981) · Zbl 0474.34002
[32] Veilleux, B. G.: An analysis of the predator interaction between paramecium and didinium, J. anim. Ecol. 48, 487-803 (1979)
[33] Salt, G. W.: Predator and prey densities as controls of the rate of capture by the predator didinium nasutum, Ecology 55, 434-439 (1974)
[34] Reukauf, E.: Zur biologie von didinium nasutum, Z. vergl. Physiol. 11, 689-701 (1930)
[35] Jost, C.; Ellner, S. P.: Testing for predator dependence in predator -- prey dynamics: A non-parametric approach, Proc. R. Soc. lond. B. 267, 1611-1620 (2000)
[36] Butzel, H. M.; Bolten, A. B.: The relationship of the nutritive state of the prey organism paramecium aurelia to the growth and encystment of didinium nasutum, J. protozool. 15, 256-258 (1968)
[37] Harrison, G. W.: Comparing predator -- prey models to luckinbill’s experiment with didinium and paramecium, Ecology 76, No. 2, 357-374 (1995)