Abstract
The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.
Highlights
Imagine a population of individuals that grow and reproduce as time proceeds, in such a way that the evolution of each individual is independent from the others.The growth-fragmentation equation is the key equation that has been used in the field of structured population dynamics to model such systems
It was first introduced to describe cells dividing by fission [6] and, sequently, it has been used to model neuron networks [28], polymerization [14, 39], the TCP/IP window size protocol for the internet [1] and many other systems sharing the dynamics described above
The common point is that the “particles” under concern are well-characterized by their mass, i.e., a one-dimensional quantity that grows over time at a certain rate and that is distributed among the offspring when a dislocation event occurs
Summary
Imagine a population of individuals that grow and reproduce as time proceeds, in such a way that the evolution of each individual is independent from the others. The common point is that the “particles” under concern (cells, polymers, dusts, etc.) are well-characterized by their mass (or “size”), i.e., a one-dimensional quantity that grows over time at a certain rate (depending on the mass) and that is distributed among the offspring when a dislocation event occurs. We do not assume conservation of mass at dislocation events. This means that some of the mass may be lost or gained during a dislocation. The growth rate x>0 τ : [0, ∞) → (0, ∞) is a continuously differentiable function, the fragmentation rate. We rather deal with the weak form of the growth-fragmentation equation (1.1).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
