Spread rate for a nonlinear stochastic invasion |
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Authors: | MA Lewis |
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Institution: | (1) Department of Mathematics, University of Utah, JWB 233, Salt Lake City, UT 84112, USA. e-mail: mlewis@math.utah.edu, US |
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Abstract: | Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated
using deterministic equations 29]. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature
or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising
from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic
manner). It has been long conjectured 27] that intrinsic stochastic factors associated with interacting individuals can slow
the spread of a population or disease, even in a uniform environment. While this conjecture has been borne out by numerical
simulations, we are not aware of a thorough analytical investigation.
In this paper we analyze the effect of intrinsic stochastic factors when individuals interact locally over small neighborhoods.
We formulate a set of equations describing the dynamics of spatial moments of the population. Although the full equations
cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower
bounds for the expected density of individuals. Analysis of the upper solution gives a bound on the rate of spread of the
stochastic invasion process which lies strictly below the rate of spread for the deterministic model. The slow spread is most
evident when invaders occur in widely spaced high density foci. In this case spatial correlations between individuals mean
that density dependent effects are significant even when expected population densities are low. Finally, we propose a heuristic
formula for estimating the true rate of spread for the full nonlinear stochastic process based on a scaling argument for moments.
Received: 19 October 1998 / Revised version: 1 September 1999 / Published online: 4 October 2000 |
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