Population Dynamics

The following sections are included:

• Dedication
• Contents
• Preface

In this chapter we give some minimal preliminaries from biology, motivations of studying of population dynamics, and define evolution operators and associated algebras related to free and bisexual populations. Moreover, we formulate the main problems in the consideration of the population dynamics. The next chapters of this book will be devoted to solutions of these problems.

In this chapter we give basic definitions related to abstract algebras. Moreover, we study cubic matrices and algebras of such matrices, because the coefficients of any evolution operator and matrix of structural constants of any evolution algebra studied in this book are defined by cubic matrices. The results of this chapter will be useful for next chapters.

In this chapter we systematically present genetic algebras. These algebras are used to model inheritance in genetics. The first section of the chapter contains some variations of these algebras (called baric, train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras etc.). In other sections we give dibaric and evolution algebras.

The genetic algebras discussed in the previous chapter concerned with inheritance which is symmetric with respect to sex, in that properties are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. In this chapter we consider asymmetric situations, and give algebras of bisexual populations. These algebras are complicated by their higher dimensions and by fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. We give some basic properties of the algebras of bisexual populations.

This chapter is devoted to flows (chains) of algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are algebras, the matrix (in general may be cubic matrix) of structural constants of which (depending on time) satisfy an analogue of the Kolmogorov–Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we fix a multiplication and then consider the KCE with respect to the fixed multiplication. The existence of a solution to the KCE provides the existence of a FA. The aim is to construct FAs with respect to several specially chosen multiplications of matrices (known as Maksimov’s multiplications). Moreover, some time-dependent behavior properties of such FAs are given.

In the first section of this chapter we consider Maksimov’s associative multiplications of cubic matrices. The notion of stochastic cubic matrix varies depending on the real models of application. The second section is devoted to a Markov interaction process which generalizes a Markov process. In the third section we define Markov processes of cubic stochastic (in a fixed sense) matrices which also called a quadratic stochastic process (QSP). Note that a QSP is a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov–Chapman equation (KCE) considered with respect to a fixed multiplication of cubic matrices. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. For some specially chosen notions of stochastic cubic matrices and their multiplications we construct corresponding QSPs. Some time-dependent behavior (dynamics) of such processes are studied. The fourth section gives applications to the biology, constructing a QSP which describes the dynamics of a population with the possibility of twin births.

In this chapter analogically as quadratic stochastic operators and processes, we define cubic stochastic operator (CSO) and cubic stochastic processes (CSP). We give a construction of a CSO and show that dynamical systems generated by such a CSO can be studied by studying of the behavior of trajectories of a Volterra CSO given on a finite dimensional simplex. We define a CSP and drive differential equations for such CSPs with continuous time.

In this chapter we present results related to trajectories of the discrete-time dynamical systems generated by quadratic stochastic operators (QSOs). The first section devoted to Volterra QSOs. There are many works related with such operators, we will only give some of results which have clear biological interpretations. Other sections of this chapter are devoted to a wide class of non-Volterra QSOs and their dynamical systems. Note that there are many (non-)Volterra QSOs which we do not present in this book, but we will give citations to them for interested readers.

This chapter is devoted to two-sex populations, the discrete-time dynamical systems of such populations have not been fully studied yet. We present some results related to Volterra QSO of two-sex populations and a class of non-Volterra QSOs with a preferences on a type of females and males. The last section presents a predator-prey model with the assumption that birth and death rates for male and female preys are different. Conditions for the global stability of the boundary equilibrium and for the permanence of the prey and predators are obtained. It is shown that the predation will not alter the sex ratio of the prey eventually when it is not sex-biased.

In this chapter we consider dynamical systems generated by gonosomal evolution operators of sex linked inheritance. The main biological system of this chapter is a hemophilia, which is a group of hereditary genetic disorders that impair the body’s ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. The evolution of such system is studied by a quadratic gonosomal operator. This operator is considered as a mapping from ℝⁿ, n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and prove that this point is globally attractive. Biologically the main result of this chapter means that the hemophilia illness asymptotically disappears.

This chapter is devoted to a discrete-time dynamical system, generated by an evolution operator of a mosquito population. Under some conditions on the parameters of the system it has two saddle fixed points. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the evolution operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied.

This chapter is devoted to a discrete-time dynamical system generated by a nonlinear operator of ocean ecosystem. Under some conditions on the parameters of the system its evolution operator reduced to a ℓ-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. It is show that (under some conditions on parameters) this ℓ-Volterra operator may have up to three or a countable set of fixed points. These fixed points may be attracting, repelling or saddle points. The limit behaviors of trajectories of the dynamical system are studied. It is shown that independently on values of parameters and on initial point all trajectories converge. Thus the quadratic operator is regular. Some biological interpretations of these results are given.

The following sections are included:

• Bibliography
• Index