On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity.

*(English)*Zbl 1221.34192Summary: 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.

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

92D25 | Population dynamics (general) |

37N25 | Dynamical systems in biology |

##### Keywords:

predator; prey interaction; habitat complexity; limit cycle; stability and direction of Hopf-bifurcation
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\textit{N. Bairagi} and \textit{D. Jana}, Appl. Math. Modelling 35, No. 7, 3255--3267 (2011; Zbl 1221.34192)

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##### References:

[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), Prentice Hall, Inc. NJ |

[3] | Anderson, T.W., Predator responses, prey refuges and density-dependent mortality of a marine fish, Ecology, 82, 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), Williams and Wilkins Baltimore |

[14] | MacDonald, M., Biological delay systems: linear stability theory, (1989), Cambridge University Press Cambridge · Zbl 0669.92001 |

[15] | May, R.M., Theoretical ecology: principles and applications, (1981), Blackwell Scientific Publications Oxford · Zbl 1228.92076 |

[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 |

[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 |

[21] | Yang, H.; Tian, Y., Hopf bifurcation in REM algorithm with communication delay, Chaos, solitons fract., 25, 1093-1105, (2005) · Zbl 1198.93099 |

[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 |

[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 |

[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 |

[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 |

[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), Cambridge Univ. Press Cambridge |

[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 |

[30] | Dieudonne’, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201 |

[31] | Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.H., Theory and application of Hopf bifurcation, (1981), Cambridge University Cambridge · 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, 2, 357-374, (1995) |

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