Résumé :
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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, s2 = 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 inconvenience, 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 greater than or equal to 1, h greater than or equal to 1). The parameter h indicates the dispersal rate of organisms. We then obtain Taylor's power law with b = {In[E(h)] + ln[E(nu(2))]}/{ln[E(h)] + In[E(nu)]}, 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 = In[E(h)l]/{ln[E(h)] + In[E(nu)]}. 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.
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