Methods of small parameter in mathematical biology
In the introductory Chapter 1, the basic ideas of asymptotic analysis are presented with information on models of mathematical biology, chemical kinetics and Brownian motion, population dynamics, epidemiology, models of fluid dynamics with initial and boundary layer phenomena, to which asymptotic analysis methods and in particular small parameters methods shall be applied. The basic idea is the subdivision of the variables describing the model into two groups: variables that evolve slowly and variables that evolve quickly. A systematic way of asymptotic analysis application is suggested here: {\parindent=6mm\begin{itemize}\item[1.] Finding an appropriate set of aggregated macro-variables, which describes the dynamics of the model at this level. \item[2.] Proving that these aggregated macro-variables approximate the micro-variables. \item[3.] Determining the relations between the parameters at the macro-level and the original micro-variables. \end{itemize}} The subsequent chapters present several realizations of this program. Chapter 2 gives the presentation of the Chapman-Enskog-type asymptotic expansion, with the basic techniques for the proof of its convergence on linear ODEs with a preliminary survey of the necessary information from linear algebra to the theory of finite-dimensional dynamical systems. As a concrete application, the authors consider the simplified linear model of interacting populations that can reside in one of two patches with migration between them much faster than their demographic processes. The techniques developed here can be applied to nonlinear processes modeled by ODEs, PDEs and also integro-differential equations with a proof for each application. In Chapter 3 presents a theory which is based on some results due to \textit{N. A. Tikhonov} and \textit{A. B. Vasilyeva} (cf.\ [\textit{A. N. Tikhonov} et al., Differential equations. English edition. Berlin etc.: Springer-Verlag (1985; Zbl 0553.34001)]), which provides constrictive estimates of approximate error and turns out to be applicable in a general situation of asymptotic expansion of the solution. Its applications to various models are considered in Chapter 4. These are: {\parindent=6mm\begin{itemize}\item[{\(\bullet\)}] the Allee population model in which females are divided into two classes, searching for a mate and recently mated, with the assumption that the satiation period after mating is much shorter than the average lifespan of the females and their ratio can be taken as a small parameter \(\varepsilon\); \item[{\(\bullet\)}] the epidemiological SIS model with a basic age-structure, when the disease turnover is much faster then the demographic process; \item[{\(\bullet\)}] the Lotka-Volterra-type predator-prey model, where the prey population can move freely between two locations, while predators are confined to only one location, but the hunting takes place in only one patch, so that the other patch can serve as a refuge and demographic rates are slow in comparison with fast migration rates. \end{itemize}} By introducing the continuous age-structure into the model of interacting populations of Chapter 2, one obtains the McKendrick model, which is a system of PDEs with nonlocal boundary conditions. This model is investigated by the methods of Chapter 2, with the usage of functional analysis and operator semigroups. In Chapter 6, asymptotic limits of correlated and, in the case of constant coefficients of thin equations, uncorrelated random walks are considered with the aim to prove that the probabilistic densities describing the correlated random walk as solutions of the hyperbolic telegraph equation are approximated by the solutions of the specially constructed diffusion equation describing the miscorrelated random walk. In Chapter 7, the kinetic model for interacting agents, which can switch their direction in accordance with the majority of other agents in their neighborhood, is considered. The mean time between the changes of the motion directions is regarded here as a small parameter. As a result, it is shown that the development of this population is similar to a wave traveling in accordance with the movement of the majority of the initial agents. This model has some features parallel with the Boltzmann equation and can be considered as its simplified version. It provides a new approach to swarming phenomena. Chapter 8 can be considered as a general overview of multiscale descriptions of natural phenomena, illustrating a path from individual-based models, where the interactions between the agents are described by large systems of linear Markov semigroups, through the intermediate level governed by nonlinear nonlocal equations for the one-agent distributions function to the macro-models, in which the population is described by averaged densities satisfying appropriate reaction-diffusion systems.
Tikhonov-Vasilyeva theory with applications to Allee model
population problems with fast migrations
diffusion limit of the telegraph equation
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