Stability analysis of a population model with maturation delay and Ricker birth function. (English) Zbl 1470.34138

Zheng, Chongwu; Zhang, Fengqin; Li, Jianquan

Abstr. Appl. Anal. 2014, Article ID 136707, 8 p. (2014).

Summary: A single species population model is investigated, where the discrete maturation delay and the Ricker birth function are incorporated. The threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium is obtained. The necessary and sufficient conditions ensuring the local asymptotical stability of the positive equilibrium are given by applying the Pontryagin’s method. The effect of all the parameter values on the local stability of the positive equilibrium is analyzed. The obtained results show the existence of stability switch and provide a method of computing maturation times at which the stability switch occurs. Numerical simulations illustrate that chaos may occur for the model, and the associated parameter bifurcation diagrams are given for certain values of the parameters.

MSC:

34C60	Qualitative investigation and simulation of ordinary differential equation models
92D25	Population dynamics (general)
34D20	Stability of solutions to ordinary differential equations

References:

[1]	Cooke, K.; van den Driessche, P.; Zou, X., Interaction of maturation delay and nonlinear birth in population and epidemic models, Journal of Mathematical Biology, 39, 4, 332-352 (1999) · Zbl 0945.92016
[2]	Zhao, X.-Q.; Zou, X., Threshold dynamics in a delayed SIS epidemic model, Journal of Mathematical Analysis and Applications, 257, 2, 282-291 (2001) · Zbl 0988.92027
[3]	Fan, G.; Liu, J.; van den Driessche, P.; Wu, J.; Zhu, H., The impact of maturation delay of mosquitoes on the transmission of West Nile virus, Mathematical Biosciences, 228, 2, 119-126 (2010) · Zbl 1204.92057
[4]	Cooke, K. L.; Elderkin, R. H.; Huang, W., Predator-prey interactions with delays due to juvenile maturation, SIAM Journal on Applied Mathematics, 66, 3, 1050-1079 (2006) · Zbl 1090.92047
[5]	Ruan, S., Delay differential equations in single species dynamics, Delay Differential Equations and Applications. Delay Differential Equations and Applications, NATO Sci. Ser. II Math. Phys. Chem., 205, 477-517 (2006), Dordrecht, The Netherlands: Springer, Dordrecht, The Netherlands · Zbl 1130.34059
[6]	Jiang, Z.; Zhang, W., Bifurcation analysis in single-species population model with delay, Science China. Mathematics, 53, 6, 1475-1481 (2010) · Zbl 1195.34127
[7]	Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33, 5, 1144-1165 (2002) · Zbl 1013.92034
[8]	Wei, J.; Zou, X., Bifurcation analysis of a population model and the resulting SIS epidemic model with delay, Journal of Computational and Applied Mathematics, 197, 1, 169-187 (2006) · Zbl 1098.92055
[9]	LaSalle, J. P., The Stability of Dynamical Systems, v+76 (1976), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA
[10]	Hale, J., Theory of Functional Differential Equations. Theory of Functional Differential Equations, Applied Mathematical Sciences, 3 (1993), New York, NY, USA: Springer, New York, NY, USA