Extinction and periodic oscillations in an age-structured population model in a patchy environment

We consider an age-structured single-species population model in a patch environment consisting of infinitely many patches. Previous work shows that if the nonlinear birth rate is sufficiently large and the maturation time is small, then the model exhibits the usual transition from the trivial equilibrium to the positive (spatially homogeneous) equilibrium represented by a traveling wavefront. Here we show that (i) if the birth rate is so small that a patch alone cannot sustain a positive equilibrium then the whole population in the patchy environment will become extinct, and (ii) if the birth rate is large enough that each patch can sustain a positive equilibrium and if the maturation time is moderate then the model exhibits nonlinear oscillations characterized by the occurrence of multiple periodic traveling waves.     

A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into N different spatial patches. We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is ε (0 < ε ⪡ 1). The main result decomposes the action of the semigroup associated to our problem into three parts:

• 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process;
• 2.(2) the semigroup associated to the transitory process which leads to the first part; and
• 3.(3) an operator, bounded in norm, of order ε.

     

An improved model of age-structured population dynamics

The classical model of age-dependent population dynamics is improved. Instead of the traditional renewal equation, a new approach is developed to describe the reproduction process of the population. The composition of a population is redefined to contain the pre-birth individuals, and the disadvantages of the classical model avoided. Moreover, the improved model turns out to be an initial value problem, which is mathematically more convenient to deal with. Existence and uniqueness results for the nonlinear nonautonomous system of model equations are obtained. It is shown that the classical model and its time delay generalization are two degenerate cases of the improved model.     

Nonlinear stability of traveling wavefronts in an age-structured population model with nonlocal dispersal and delay

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

On age-structured SIS epidemic model for time dependent population

1.IntroductionandPreliminaryAgestructureinepidemicmodelshasbeenconsideredbymanyauthors,becauseoftherecoghtionthattransmissiondynamicsofcertaindiseasescouldnotbecorrectlydescribedbythetraditionalepidemicmodelswithnoagedependence.Especially,Busenbergetal.II'2]giveacompleteanalysisofafairlygeneralSISmodelwithagestructureandasteady-statetotalpopulation,showingtheekistenceofathresholdforendemicstates.In[tis],theyassumethatthepopulationhasreacheditssteadystate.Althoughitisnotilladeqilatetoassumet…     

Hopf bifurcation of an age-structured compartmental pest-pathogen model

This paper investigates an age-structured compartmental pest-pathogen model by using the theory of integrated semigroup. We study the stability of the steady state of the model by analyzing the associated characteristic transcendental equation. It is shown that Hopf bifurcation occurs at a positive steady state as bifurcating parameter passes a sequence of critical values.     

Existence-uniqueness and monotone approximation for an erythropoiesis age-structured model

We develop a monotone approximation to the solution of an age-structured model which describes the regulation of erythropoiesis, the process in which red blood cells are developed. The convergence of this approximation to the unique solution of the model is also established.     

Bifurcation analysis in an age-structured model of a single species living in two identical patches

In this paper, we consider the age-structured model of a single species living in two identical patches derived in So et al. [J.W.-H. So, J. Wu, X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol. 43 (2001) 37–51]. We chose a birth function that is frequently used but different from the one used in So et al. which leads to a different structure of the homogeneous equilibria. We investigate the stability of these equilibria and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out for supporting the analytic results.     

Stability analysis of age-structured disabled population dynamics

In this note, we present a characteristic of an ordered Banach space with base and then give a necessary and sufficient condition for the disabled population system to decay exponentially.     

The survival analysis for a single-species population model in a polluted environment

This paper concentrates on studying the long-term behavior of a single-species population living in a polluted environment. A new mathematical model is derived assuming that a born organism takes with it a quantity of internal toxicant, and the amount of toxicant stored in each living organism which dies is drifted into the environment. Sufficient criteria for uniform persistence, weak persistence in the mean or extinction of the population are obtained. Also we find some sufficient conditions, depending on the parameters of the model and the clean up rate, under which the population will be persistent.     

Existence and uniqueness for a stochastic age-structured population system with diffusion

In this paper, a class of stochastic age-dependent population dynamic system with diffusion is introduced. Existence and uniqueness of strong solution for a stochastic age-dependent population dynamic system in Hilbert space are established. The analysis use Barkholder–Davis–Gundy’s inequality, Itô’s formula and some special inequalities for our purposes.     

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ^∞,2 (L ²) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions. Mathematics subject classifications (2000) 65M12, 65M25, 65M60, 92D25 M.-Y. Kim: This work was supported by Korea Research Foundation Grant (KRF-2001-041-D00037).