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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling
Published on: July 4, 2007
Itô versus Stratonovich calculus in random population growth
1Department of Mathematics, Universidade de Evora, Rua Romão Ramalho 59, PT-7000-671 Evora, Portugal. braumann@uevora.pt
Mathematical Biosciences
|October 11, 2005
Summary
This study resolves the stochastic differential equation (SDE) calculus controversy by clarifying that Itô and Stratonovich calculi yield identical population growth models when using arithmetic and geometric averages, respectively.
Area of Science:
- Ecology
- Mathematical Biology
- Stochastic Processes
Background:
- Population growth models often use stochastic differential equations (SDEs) to represent environmental fluctuations.
- Interpreting SDEs involves Itô calculus and Stratonovich calculus, which can lead to different predictions.
- A controversy exists regarding the appropriate calculus for population dynamics models.
Purpose of the Study:
- To resolve the controversy surrounding the use of Itô versus Stratonovich calculus in population growth SDEs.
- To demonstrate that the apparent differences between the calculi are semantic, stemming from the interpretation of 'average' growth rate.
Main Methods:
- Analysis of the general stochastic differential equation (SDE) model for population growth: dN/dt=N(g(N)+sigmaepsilon(t)).
- Formal definition and comparison of the 'average' growth rate within Itô and Stratonovich calculus frameworks.
- Mathematical proof showing the equivalence of solutions when averages are properly defined.
Main Results:
- Itô calculus corresponds to an arithmetic average growth rate (R(a)(x)).
- Stratonovich calculus corresponds to a geometric average growth rate (R(g)(x)).
- When defined in terms of their respective averages, both calculi yield identical solutions to the SDE.
Conclusions:
- The controversy between Itô and Stratonovich calculus in population dynamics is semantic.
- Properly defining the 'average' growth rate resolves the apparent discrepancies between the two calculi.
- Both calculi provide equivalent predictions for population growth when the average is correctly interpreted.
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