Complex dynamic modes of a population with age and sex structures

Mathematical Modeling of Population Dynamics Based on Recurrent Equations: Results and Prospects. Part I

Abstract

Approaches to modeling population dynamics using discrete-time models are described in this two-part review. The development of the scientific ideas of discrete time models, from the Malthus model to modern population models that take into account many factors affecting the structure and dynamics, is presented. The most important and interesting results of recurrent equation application to biological system analysis obtained by the authors are given. In the first part of this review, the population dynamic effects that result from density-dependent regulation of population, the age and sex structures, and the influence of external factors are considered.

Changing of the dynamics modes in populations with age and sex structure

Using a mathematical model of the number dynamics in a population with age and sex structure, the relationships were studied between the number of newborns and that of mature females and males. In addition, we studied transitions between different modes of population dynamics which were dependent on changes in not only the population parameters, such as reproduction intensity, survival, and self-regulation, but also in mating pair formation. Population dynamics was found to manifest multistability; i.e., the alternative dynamics modes were observed at the same values of the population parameters but under different initial conditions.

Chaos and Hyperchaos in a Model of Ribosome Autocatalytic Synthesis

Any vital activities of the cell are based on the ribosomes, which not only provide the basic machinery for the synthesis of all proteins necessary for cell functioning during growth and division, but for biogenesis itself. From this point of view, ribosomes are self-replicating and autocatalytic structures. In current work we present an elementary model in which the autocatalytic synthesis of ribosomal RNA and proteins, as well as enzymes ensuring their degradation are described with two monotonically increasing functions. For certain parameter values, the model, consisting of one differential equation with delayed argument, demonstrates both stationary and oscillatory dynamics of the ribosomal protein synthesis, which can be chaotic and hyperchaotic dependent on the value of the delayed argument. The biological interpretation of the modeling results and parameter estimation suggest the feasibility of chaotic dynamics in molecular genetic systems of eukaryotes, which depends only on the internal characteristics of functioning of the translation system.