The Behavioural Basis of Redistribution. II. Simulations of the Delta -Model

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Abstract
The difference between total population number and density is discussed and its relevance to the .DELTA.-model (Taylor and Taylor 1977; R. A. J. Taylor 1981) defined. A computer simulation model was devised to investigate the fates of populations in which the movement of individuals is governed by the .DELTA.-model. The resulting spatial distributions were analyzed for conformity with L. R. Taylor''s (1961) Power Law. In a homogeneous environment occupied by an immortal population which reproduced by binary fission, the population spread out in nearly uniform annuli and maintained a near constant mean density just above the optimum defined in the .DELTA.-model (.rho.O), as the total number (Nt) increased exponentially. Non-selective mortality reduced the rate of increase of Nt, and may result in variations in the diameter of the population. Relaxing the assumption of homogeneity of environment broke up the regular annular pattern and reduced the rate of increase of Nt. Expansion of the population diameter was then no longer uniform; the center of population moved as new benign areas were discovered and clonized. Change in the environmental matrix introduced non-selective mortality. Mortality introduced the risk of population extinction when the environment was exceptionally hostile and the degree of mobility low. Highly mobile individuals could survive even in an environment changing rapidly and drastically. Dividing the simulation arena into regions analogous to demes resulted in a change of scale without qualitative changes in the populations'' behavior. Reproductive behavior was made more complex by simulating parthenogensis and hermaphroditism with the result that the total population number fluctuated widely with high reproductive rate. The density varied less and return time to the endemic population number after an epidemic was greater than predicted by the logistic equation. Build-ups could also be slower. Comparison of direct and delayed density-dependent spatial behavior showed that movement in response to parental density (or migration programmed by parents) results in more rapid buildup and higher peaks of population. All simulations, whether analyzed at an instant or continuously, eventually produced population distributions which conformed to Taylor''s Power Law. The behavior of the rate of aggregation parameter, b, and its standard error are consistent with the hypothesis that a species'' population density surface has a preferred modulation which it returns after disturbance.