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

Abstract

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.