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Harvesting in a random environment: Itô or Stratonovich calculus?

Authors
  • Braumann, Carlos A
Type
Published Article
Journal
Journal of theoretical biology
Publication Date
Feb 07, 2007
Volume
244
Issue
3
Pages
424–432
Identifiers
PMID: 17070851
Source
Medline
License
Unknown

Abstract

We extend to harvesting stochastic differential equation (SDE) models in a random environment our previous work on models without harvesting concerning the resolution of the Itô-Stratonovich controversy. The resolution is obtained for the very general class of models dN/dt=N (r(N)-h(N)+sigmaepsilon(t)), where N=N(t) is the population size at time t, r(N) is the (density-dependent) "mean" per capita growth rate, h(N) is the (density-dependent) harvesting effort, epsilon(t) is a standard white noise (representing environmental random fluctuations), and sigma is a noise intensity parameter. Itô and Stratonovich calculus in the resolution of SDEs apparently give different qualitative and quantitative results, leading to controversy on which calculus is more appropriate and creating an obstacle on the use of this modeling approach. We show that the apparent difference between the two calculi is due to a semantic confusion based on the fallacious assumption that we are working with the same type of mean rates. After clearing the confusion, the two calculi yield exactly the same results and we obtain important common conditions for extinction and for existence of a stationary density. The resolution of the controversy is intertwined with and sheds light on the estimation issues.

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