Discrete growth–decay–fragmentation equation: well-posedness and long-term dynamics

  • J. BanasiakEmail author
  • L. O. Joel
  • S. Shindin
Open Access


In this paper, we consider discrete growth–decay–fragmentation equations that describe the size distribution of clusters that can undergo splitting, growth and decay. The clusters can be for instance animal groups that can split but can also grow, or decrease in size due to birth or death of individuals in the group, or chemical particles where the growth and decay can be due to surface deposition or erosion. We prove that for a large class of such problems, the solution semigroup is analytic and compact and thus has the asynchronous exponential growth property; that is, the long-term behaviour of any solution is given by a scalar exponential function multiplied by a vector, called the stable population distribution, that is independent on the initial conditions.


Discrete fragmentation Birth-and-death process \(C_0\)-Semigroups Long-term behaviour Asynchronous exponential growth Spectral gap Numerical simulations 

Mathematics Subject Classification

34G10 35B40 35P05 47D06 45K05 80A30 



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Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa
  2. 2.Institute of MathematicsTechnical University of ŁódźŁódźPoland
  3. 3.School of Mathematics, Statistics and Computer ScienceUniversity of Kwazulu-NatalDurbanSouth Africa

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