Methods of Linear Analysis

Abstract

In this chapter we first introduce the fundamental types of ecological associations such as predation, competition, cooperation (or mutualism) and then present several methods for studying the dynamical characteristics of linear systems. We begin with some remarks on models of single species dynamics. Most of the differential equation models of population dynamics have been derived starting from the following simple format

$$\frac{dN(t)}{dt}=\begin{Bmatrix} \textup{an \;individual's\:contribution}\\ \textup{to\:population\:change\:in\:unit\:time} \end{Bmatrix}N(t)$$

(3.1.1)

where N(t) denotes the density of a population (or biomass) of a single species at time t. Subsequently, one makes an assumption regarding the factor inside the brackets in (3.1.1). In particular, if one assumes that an individual’s contribution to the change in population in unit time is denoted by a function say f(t, N) defined suitably for all t > 0, N > 0, then one obtains from (3.1.1) the so called Kolmogorov formulation in the form

$$\frac{dN(t)}{dt}=f\left ( N(t) \right )N(t).$$

(3.1.2)