Asymptotically stable states of nonlinear age-structured monocyclic population model Ⅰ. Travelling wave solution

摘要

This paper is devoted to the study of evolutionary dynamics of monocyclic age-structured population including effect of nonlinear mortality (population growth feedback) and proliferation. The total population is considered as partitioned by fixing age into two subpopulations. Individuals of first population are born, mature, die and can at the final fixed age give birth for some new individuals (with null age). Individuals of the second subpopulation are older than those of the first one. They can mature, die and do not have possibility to proliferate. This model was considered as a system of two initial-boundary value problems for nonlinear transport equations with non-local boundary conditions. We obtained explicit travelling wave solution provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee continuity and smoothness of solution. Explicit form of solution allowed us to perform numerical experiments with high accuracy using the set of parameterized algebraic functions which do not depend from the time. In all performed experiments solutions are attracted to some stationary functions for a long time period (asymptotically stable states of system). We indicated and studied three different regimes of population dynamics. The first is quasi-equilibrium regime, when the maximum value of population density by age, as a function of time, is attracted to the point from the neighbourhood of initial value. This is a result of balance between effects of proliferation and nonlinear mortality (like behaviour of microorganism population in the cases of asymptomatic or healthy carriers). The second and third regimes are characterized by increasing (decreasing) maximum by age values of population density with following attracting to values higher (lower) than initial one. We studied also the impact of parameters of nonlinear death rate on the tremendous growth of population density followed by transition to asymptotically stable states (like infection generalization process in living organisms).