Malthusian behaviour of the Leslie model for population growth

Abstract

The Leslie model considers age-specific birth and survival rates to describe how a population size and age distribution changes over time. This thesis investigates the long-term dynamics of the Leslie model for population growth, utilizing mathematical theorems such as Perron-Frobenius, Doeblin’s theorem, and Branching processes.

The Perron-Frobenius theorem guarantees the existence of a dominant eigenvalue. This dominant eigenvalue and its corresponding eigenvector represent the long-term behaviour of a population; The dominant eigenvalue indicates the long-term population growth, and the corresponding eigenvector indicates the long-term age distribution. Furthermore, the Perron-Frobenius theorem implies that a population asymptotically reaches a stable age distribution that is independent of its initial age structure. Once this stable age distribution is reached, the population continues to grow exponentially, exhibiting Malthusian behaviour. Doeblin’s theorem, although not directly applicable to the complete Leslie model, provides valuable insights into the long-term behaviour of Markov chains. As Doeblin‘s theorem can not be applied to the complete Leslie model, the reproduction process of the Leslie model is formulated as a Branching process. Introducing the Leslie model as a Branching process allows for the consideration of demographic stochasticity. Simulations reveal that for larger populations, the Branching process closely mirrors the Leslie model, while disparities become more pronounced in smaller populations. These results illustrate the impact of probabilistic factors in population dynamics, as well as the strength of the Leslie model for larger populations.