Stochastic models of growth with quasi-discrete generations

Stochastic models of growth and evolution of cells such as bacteria or cancer cells usually assume exponential distribution of division times. Birth, death, mutations, and other processes can then be modelled similarly to chemical reactions, assuming that each process occurs with a certain (possibly state- and time-dependent) rate. This is convenient because it enables the model to be described by a Markov process, for which many powerful analytic and computational techniques exist. However, biological cells do not replicate in this way; the distribution of reproduction times is usually centred around a non-zero “doubling time”. In this talk I will discuss how three stochastic models: exponential growth, logistic growth, and a two-species predator-prey model are affected by non-exponential, narrow distribution of doubling times. Using computer simulations and analytic calculations I will show that the modified models exhibit much larger fluctuations of the number of cells. In particular, the predator-prey model shows large quasi-periodic oscillations caused by resonant amplification of coloured noise generated by replication events if the generation time is tuned to the natural oscillatory frequency of the system. I will also briefly discuss the relevance of these result for laboratory experiments on bacterial growth.