Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation

Abstract

We prove, in the framework of measure solutions, that the equal mitosis equation present persistent asymptotic oscillations. To do so we adopt a duality approach, which is also well suited for proving the well-posedness when the division rate is unbounded. The main difficulty for characterizing the asymptotic behavior is to define the projection onto the subspace of periodic (rescaled) solutions. We achieve this by using the generalized relative entropy structure of the dual problem.

Long-time behavior and darwinian optimality for an asymmetric size-structured branching process

“…We can see in the proof that the change of variable is no longer possible because d Þ Ñ e dˆ0 is constant. As shown in [4,21], in such a model, the distribution of the process is concentrated in a comb that depends on the initial conditions and then does not verify the Doeblin assumption. This is why eigenelements exist (see [18]) but the convergence does not hold (see [22]).…”

“…The second assumption is the physiological asymmetry assumption which is necessary to avoid oscillation of the size distribution depending on the initial state as shown in [4,21].…”

“…The population grows exponentially fast but the size distribution does not stabilize. It is shown in [4,21] that the latter oscillates at frequencies that depend on the initial configuration. It is because of this atypical property that our demonstration of Malthusian behaviour is delicate.…”

Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle

“…The best intuition on what happens here comes from the underlying stochastic branching tree: all descendants of a given cell of size x 0 at time 0 live on the countable set of curves x 0 e t 2 −n , due to the very specific relation between growth and division, whereas for other growth or division the times of division account, leading to a kind of dissipativity. This case has been studied by semigroup theory for compact support in size by G. Greiner and R. Nagel [86], and extended and revisited in [87] where the following explicit asymptotic result has been proved -see also [88] for extension to measure solutions.…”