Steady states and outbreaks of two-phase nonlinear age-structured model of population dynamics with discrete time delay

Abstract

In this paper we study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality. Both phases are modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE. The obtained system is studied both theoretically and numerically. Three different regimes of population dynamics for asymptotically stable states of autonomous systems are obtained in numerical experiments for the different initial values of population density. The quasi-periodical travelling wave solutions are studied numerically for the autonomous system with the different values of time delays and for the system with oscillating death rate and birth modulus. In both cases it is observed three types of travelling wave solutions: harmonic oscillations, pulse sequence and single pulse.

Stability Analysis of Delayed Age-structured Resource-Consumer Model of Population Dynamics with Saturated Intake Rate

“…2, 3). Similar dynamical regimes were obtained and described in works [6], [7] for the age-structured model with density-dependent delayed death rate and discussed in work [8] for the two-compartment age-structured model of locust population dynamics.…”

“…(86)) we observe the oscillatory regime of the system with asymptotic convergence of solution to the steady state (curve 3 in Figs.1a -1c). The existence of such periodic solutions of some Lotka-Volterra prey-predator models was proved in theoretical work [45] and was observed in numerical experiments in [6], [7]. Further increasing of basic reproduction number by parameter 0 θ causes the consumer population outbreaks (special dynamical regimes of population, see [1], [7], [9]).…”

“…We assume that the consumer fertility rate is an increasing with saturation monotone function and death rate is a decreasing with extinction monotone function of calorie intake rate satisfied Eqs. (7), (8). They are defined on the parametrized classes of algebraic functions:…”

“…Depending on the reproduction number of consumers, trajectories of system are unstable, oscillate in the vicinities of the trivial and semi-trivial equilibria or converge asymptotically to the semi-trivial equilibrium. The further increasing of consumer's basic reproduction number or time delay parameter leads to the consumer population outbreaks in form of pulse sequence, which are classified in the quantitative population ecology as the populations with cyclical eruption dynamics [1], [7]. The results of simulations illustrated the properties of the outbreak solutions are presented in Section 5.2.…”

“…If all non-depleted patches have only lower activity resources with . (23) it follows that the infimum of equilibrium intake rate (7)…”