Continuous Monocyclic and Polycyclic Age Structured Models of Population Dynamics

Abstract

This paper focuses on the study of continuous age-structured models, or more general, physiologically structured models, which used for detailed and accurate study of population dynamics in many ecological, biological applications and medicine. In contrast to simpler unstructured models, these models allow us to relate the individual life-histories described as fertility and mortality rates of an individual at a given age with population dynamics. Depending from the particularity of reproduction mechanism continuous age-structured models are divided into monocyclic (reproduction occurs only at the one fixed age of individuals) and polycyclic (reproduction occurs with age-dependent probability at some age reproductive window) models. The linear monocyclic age-structured models are used often in cell cycles modelling, in population dynamics of plants, etc. In this case continuous age-structured models allow for obtaining the exact analytical solution. Since the linear and non-linear polycyclic age-structured models are more general then monocyclic models, they cover wider range of applications in life science. But in this case solution of model can be obtained only in the form of recurrent formulae and can be used only in numerical algorithms. Both solutions obtained in this work allow us to study numerically the important dynamical regimes-population outbreaks of three types: oscillations with large magnitude, pulse sequence and single pulse. Thus, analysis of continuous age-structured models of population dynamics provides insight into features and particularities of complex dynamical regimes of populations in many applications in biology, ecology and medicine.

Age-structured delayed SIPCV epidemic model of HPV and cervical cancer cells dynamics I. Numerical method

“…Using the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the SIPCV epidemic model. This solution is given in form of the recurrent formulae (like in works [3], [4], [8], [55]) in which the densities of all subpopulations are defined through the integrals from solution taken at previous instance of time.…”