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Closed form solution of a periodically forced logistic model

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Abstract

We give the exact closed form solution of the following ordinary differential equation:

$$\left\{ \begin{array}{ll} \dot{x}(t)&=a\,x(t)-b\,x^{2}(t)+c\, {\rm cos} (\omega t)-h,\\ x(0)&=x_{0} \end{array}\right.$$

which is a modified logistic one, wherein x(t) is the population of a homogeneous species x at time t. Other than integrating the above nonlinear differential equation by means of Mathieu functions of the first kind, we also provide a condition of a couple of inequalities involving a, b, c, h and x 0 whose fulfillment is sufficient to ensure that a bounded solution for x(t) there exists.

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Authors and Affiliations

  1. Dipartimento di Matematica per le scienze economiche e sociali, Università di Bologna, viale Filopanti, 5, 40126, Bologna, Italy

    Giovanni Mingari Scarpello & Daniele Ritelli

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  1. Giovanni Mingari Scarpello
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  2. Daniele Ritelli
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Correspondence to Daniele Ritelli.

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Mingari Scarpello, G., Ritelli, D. Closed form solution of a periodically forced logistic model. Ann. Univ. Ferrara 54, 85–94 (2008). https://doi.org/10.1007/s11565-008-0035-3

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  • DOI: https://doi.org/10.1007/s11565-008-0035-3

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