Growth-Fragmentation Dynamics in Structured Populations

Growth-fragmentation dynamics describe the processes by which individual entities within a population increase in size or mass and subsequently split into daughter units. In structured populations, individuals are differentiated by a characteristic such as size, age or physiological state, and the interplay between growth rates and fragmentation rules determines the evolution of the population’s distribution over these traits. Mathematical formulations often take the form of partial differential or integro-differential equations, in which a transport term captures continuous growth and a non-local term models fragmentation events. Analytical tools including eigenvalue analysis, entropy methods and semigroup theory have been developed to characterise existence, uniqueness and long-term behaviour of solutions. Such frameworks yield insight into phenomena as diverse as microbial cell division, tumour progression, polymer breakage and animal group fission, and inform applications in biotechnology, epidemiology and ecosystem management. Fundamental results include criteria for asynchronous exponential growth—whereby the population converges towards a stable size distribution independent of initial conditions—and the identification of self-similar or periodic asymptotic regimes.

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Recent advances have extended discrete and continuous formulations to capture more realistic fragmentation kernels and growth laws. In one study, a discrete growth–decay–fragmentation system was shown to generate an analytic and compact semigroup, leading to an asynchronous exponential growth property and a unique stable population distribution determined by the dominant eigenmode. Parallel work on homogeneous fragmentation kernels established necessary and sufficient conditions for convergence towards a unique self-similar solution, clarifying how kernel homogeneity and mass conservation constrain large-time behaviour. A complementary line of inquiry provided an explicit formula for a critical growth-fragmentation equation, revealing the emergence of periodic asymptotic patterns in the absence of a stationary or self-similar profile; this analysis elucidated how transport and splitting parameters govern transitions between steady-state and oscillatory regimes. Together, these contributions have deepened understanding of how structural heterogeneity and fragmentation rules shape both transient dynamics and long-term patterns in size-structured populations.

Growth-Fragmentation Dynamics in Structured Populations publication trend

The graph below shows the total number of articles in growth-fragmentation dynamics in structured populations across all publications each year (not limited to Nature Index journals).

Technical terms

Growth-Fragmentation Equation: A mathematical model, often a partial differential or integro-differential equation, that represents population dynamics wherein individuals grow continuously and periodically split into smaller units.

Structured Population: A population model that classifies individuals by an intrinsic characteristic (size, age or physiological state), enabling the study of heterogeneity in growth and fragmentation processes.

Self-similar Solution: A solution profile that evolves purely by scaling in time and space, preserving its shape up to a rescaling of variables.

Asynchronous Exponential Growth: The phenomenon by which solutions converge exponentially fast to a dominant eigenfunction, yielding a stable population distribution independent of initial conditions.

Eigenvalue Problem: A formulation in which characteristic growth or decay rates are identified as eigenvalues of an operator acting on population density functions, with corresponding eigenfunctions describing modal distributions.

References

1. Discrete growth–decay–fragmentation equation: well-posedness and long-term dynamics. Journal of Evolution Equations (2019).
2. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B (2018).
3. Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation. ESAIM Proceedings and Surveys (2018).