# American Institute of Mathematical Sciences

2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133

## Comparison between stochastic and deterministic selection-mutation models

 1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010, United States 2 Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8205, United States

Received  April 2006 Revised  November 2006 Published  February 2007

We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection-mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.
Citation: Azmy S. Ackleh, Shuhua Hu. Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 133-157. doi: 10.3934/mbe.2007.4.133
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