Non-linearity and heterogeneity in modeling of population dynamics

Abstract

The study of population growth reveals that the behaviors that follow the power law appear in numerous biological, demographical, ecological, physical and other contexts. Parabolic models appear to be realistic approximations of real-life replicator systems, while hyperbolic models were successfully applied to problems of global demography and appear relevant in quasispecies and hypercycle modeling. Nevertheless, it is not always clear why non-exponential growth is observed empirically and what possible origins of the non-exponential models are. In this paper the power equation is considered within the frameworks of inhomogeneous population models; it is proven that any power equation describes the total population size of a frequency-dependent model with Gamma-distributed Malthusian parameter. Additionally, any super-exponential equation describes the dynamics of inhomogeneous Malthusian density-dependent population model. All statistical characteristics of the underlying inhomogeneous models are computed explicitly. The results of this analysis show that population heterogeneity can be a reasonable explanation for power law accurately describing total population growth.

Keywords: Distributed parameters; Heterogeneous population; Non-exponential growth; Prebiological evolution.