Analysis of age-structured population growth for single species

Abstract

A wide variety of mathematical models have been proposed to investigate age-structured population growth for single species. Mathematical models allowing for more biotic and abiotic factors tend to better describe the complex behaviour of populations. In this thesis, we attempt to provide a comprehensive mathematical modelling framework and rigorous theoretical analysis for age-structured populations with the consideration of various factors regulating population growth such as seasonal variations, intra-specific competition, spatial movements and diapause. It is worth noting that developmental durations within each age group are assumed varying with time. Consequently, the model reduced from the classical Mckendrick-von Foerster equation takes the form of retarded delay differential or reaction-diffusion equations with time-dependent delays, which brings novel challenges to the theoretical analysis. By applying the well-developed theory of retarded functional differential or reaction diffusion equations and the theory of monotone dynamical system in periodic environment, we establish the well-posedness of the solutions and the global dynamics involving the global extinction, uniform persistence and global stability of the trivial and positive periodic solutions in terms of the basic reproduction number. We begin this thesis with a brief introduction for the development of early and advanced age-structured population models in Chapter 1. Then, the methodologies employed in the theoretical analysis of age-structured models are reviewed. Finally, the motivations of this thesis are illustrated in detail. In Chapter 2, we provide some requisite mathematical theories for this thesis, which refer to the theories related to monotone dynamical systems, uniform persistence and basic reproduction number in periodic environment. The work presented in Chapter 3 mainly involves the investigation of the age-structured population growth based on the assumptions that the birth and death rate functions are dependent on density and periodic in time. In this work, we propose a generalised hyperbolic age-structured model, and give a detailed proof for the existence and uniqueness of the solution by applying the contraction mapping theorem on the integral form solution obtained through integration along characteristics. By assuming time-varying developmental durations and age thresholds and using tick population as a motivative example, we deduce an age-structured model of four coupled periodic delay differential equations (DDE) with time-dependent delays. When the immature intra-specific competition is ignored, we obtain a new reduced periodic DDE model system, the adult system of which can be decoupled. Based on this decoupled periodic delay differential equations, we show the global existence and uniqueness of the solution, define the basic reproduction number R0 and prove the global stability of the positive periodic solution in terms of R0 by defining a periodic solution semiflow on a suitable phase space and employing the theory of monotone dynamical systems. Under the consideration of immature intra-specific competition, the threshold dynamics including population extinction and uniform persistence in terms of R0 is established. Chapter 4 is devoted to analyse an age-structured population model with the consideration of spatial movements, seasonal variations, intra-specific competition and time-varying maturation duration simultaneously. When the competition among immatures is negligible, the model takes the form of a system of reaction-diffusion equations with time-dependent delays, in which situation one equation for the adult population density is decoupled. The well-posedness of the decoupled system is established and the basic reproduction number R0 is defined and shown to determine the global attractivity of either the zero equilibrium (when R0 ≤ 1) or a positive periodic solution (R0 > 1) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is negligible, the model is neither cooperative (where the comparison principle holds) nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number R~0 as a threshold index. In Chapter 5, we propose a novel modelling framework to investigate the effects of diapause on seasonal population growth. Diapause, a period of arrested development caused by adverse environmental conditions, serves as a key survival mechanism for insects and other invertebrate organisms in temperate and subtropical areas. In this work, a novel modelling framework, motivated by mosquito species, is proposed to investigate the effects of diapause on seasonal population growth, where diapause period is taken as an independent growth process, during which the population dynamics are completely different from that in the normal developmental and post-diapause periods. More specifically, the annual growth period is divided into three intervals, and the population dynamics during each interval are described by different sets of equations. We formulate two models of delay differential equations (DDE) to explicitly describe mosquito population growth with a single diapausing stage, either immature or adult. These two models can be further unified into one DDE model, on which the well-posedness of the solution and the global stability of the trivial and positive periodic solution in terms of an index R are analysed. The seasonal population abundances of two temperate mosquito species with different diapausing stages are simulated to identify the essential role on population persistence that diapause plays and predict that killing mosquitoes during the diapause period can lower but fail to prevent the occurrence of peak abundance in the following season. Instead, controlling mosquitoes during the normal growth period is much more efficient to decrease the outbreak size. Our modelling framework may shed light on the diapause-induced variations in spatiotemporal distributions of different mosquito species. Chapter 6 gives the conclusions of the results presented in this thesis and the discussions of the future work.