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Multiscaling analysis of a slowly varying single species population model displaying an Allee effect. (English) Zbl 1306.34077

The authors consider the dynamics of a single species population which grows in a logistic fashion, while being also subject to an Allee effect. Assuming that the model parameters are slowly varying functions of time, a multiscaled perturbation analysis based on two time scales is performed to give analytic approximations of the solution. Although these approximations are given in an implicit form this form is relatively simple and easily representable, a good agreement between the leading order approximation and the numerical approximation being observed for a wide range of initial populations.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34E13 Multiple scale methods for ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
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References:

[1] Stephens, What is the Allee effect?, Oikos 87 pp 185– (1999) · doi:10.2307/3547011
[2] Courchamp, Inverse density dependence and the Allee effect, Trends in Ecology and Evolution 14 pp 405– (1999) · doi:10.1016/S0169-5347(99)01683-3
[3] Bazykin, Nonlinear Dynamics of Interacting Populations (1998) · doi:10.1142/2284
[4] Gonzalez-Olivares, Procceedings of the 2006 International Symposium on Mathematical and Computational Biology: BIOMAT 2006 pp 53– (2007)
[5] Idlango, Harvesting a logistic population in a slowly varying environment, Applied Mathematics Letters 25 pp 81– (2012) · Zbl 1229.92075 · doi:10.1016/j.aml.2011.07.015
[6] Shepherd, Analysis of the power law logistic population model with slowly varying coefficients, Mathematical Methods in the Applied Sciences 35 pp 238– (2012) · Zbl 1243.34069 · doi:10.1002/mma.1561
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