Colony expansion model for describing the spatial distribution of populations

Abstract

Two equations have been used frequently to describe the relation between the sample variance (s ²) and sample mean (m) of the number of individuals per quadrat: Taylor's power law, s ² = am ^b, and Iwao's m ^*−m regression, s ² = cm + dm ², where a, b, c, and d are constants. We can obtain biological information such as colony size and the degree of aggregation of colonies from parameters c and d of Iwao's m ^*−m regression. However, we cannot obtain such biological information from parameters a and b of Taylor's power law because these parameters have not been described by simple functions. To mitigate such in-convenience, I propose a mechanistic model that produces Taylor's power law; this model is called the colony expansion model. This model has the following two assumptions: (1) a population consists of a fixed number of colonies that lie across several quadrats, and (2) the number of individuals per unit occupied area of colony becomes v times larger in an allometric manner when the occupied area of colony becomes h times larger (v≥ 1, h≥ 1). The parameter h indicates the dispersal rate of organisms. We then obtain Taylor's power law with b = {ln[E(h)] + ln[E(v ²)]}/{ln[E(h)] + ln[E(v)]}, where E indicates the expectation. We can use the inverse of the exponent, 1/b, as an index of dispersal of individuals because it increases with increasing E(h). This model also yields a relation, known as the Kono–Sugino relation, between the proportion of occupied quadrats and the mean density per quadrat: −ln(1 −p) = fm ^g, where p is the proportion of occupied quadrats, f is a constant, and g = ln[E(h)]/{ln[E(h)] + ln[E(v)]}. We can use g as an index of dispersal as it increases with increasing E(h). The problem at low densities where Taylor's power law is not applicable is also discussed.