Differential and Integral Equations

A sharp condition for the chaotic behaviour of a size structured cell population

S. EL Mourchid, A. Rhandi, H. Vogt, and J. Voigt

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We show that the condition $0\le \beta \le \frac{1}{2\ln 2}$ is necessary for the chaoticity of the solution of the cell population model \begin{equation}\label{eq1} \left\{\begin{array}{ll} \frac{\partial u(t,x)}{\partial t}=-\frac{\partial(x u(t,x))}{\partial x}+\gamma u(t,x)-\beta u(t,x)+4\beta u(t,2x)\chi_{(0,\frac{1}{2})}(x),\\ u(0,\cdot)= f \in L^1(0,1). \end{array} \right. \end{equation} (If $\gamma -3\beta >0$, then this condition is known to be sufficient.) The analysis depends on solving a forward delay equation.

Article information

Differential Integral Equations Volume 22, Number 7/8 (2009), 797-800.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37N25: Dynamical systems in biology [See mainly 92-XX, but also 91-XX]
Secondary: 35F10: Initial value problems for linear first-order equations 37D45: Strange attractors, chaotic dynamics 92D25: Population dynamics (general)


EL Mourchid, S.; Rhandi, A.; Vogt, H.; Voigt, J. A sharp condition for the chaotic behaviour of a size structured cell population. Differential Integral Equations 22 (2009), no. 7/8, 797--800. https://projecteuclid.org/euclid.die/1356019549.

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