Comparison between stochastic and deterministic selection-mutation models
We present a deterministic selection-mutation model with adiscrete trait variable. We show that for an irreducibleselection-mutation matrix in the birth term the deterministicmodel has a unique interior equilibrium which is globally stable.Thus all subpopulations coexist. In the pure selection case,...
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AIMS Press
2007-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.133 |
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| author | Azmy S. Ackleh Shuhua Hu |
| author_facet | Azmy S. Ackleh Shuhua Hu |
| author_sort | Azmy S. Ackleh |
| collection | DOAJ |
| description | We present a deterministic selection-mutation model with adiscrete trait variable. We show that for an irreducibleselection-mutation matrix in the birth term the deterministicmodel has a unique interior equilibrium which is globally stable.Thus all subpopulations coexist. In the pure selection case, theoutcome is known to be that of competitive exclusion, where thesubpopulation with the largest growth-to-mortality ratio willsurvive and the remaining subpopulations will go extinct. We showthat if the selection-mutation matrix is reducible, thencompetitive exclusion or coexistence are possible outcomes. Wethen develop a stochastic population model based on thedeterministic one. We show numerically that the mean behavior ofthe stochastic model in general agrees with the deterministic one.However, unlike the deterministic one, if the differences in thegrowth-to-mortality ratios are small in the pure selection case,it cannot be determined a priori which subpopulation will have thehighest probability of surviving and winning the competition. |
| format | Article |
| id | doaj-art-a9bf82dccfef4588870154d12df043ee |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-a9bf82dccfef4588870154d12df043ee2025-01-24T01:53:27ZengAIMS PressMathematical Biosciences and Engineering1551-00182007-01-014213315710.3934/mbe.2007.4.133Comparison between stochastic and deterministic selection-mutation modelsAzmy S. Ackleh0Shuhua Hu1Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010We present a deterministic selection-mutation model with adiscrete trait variable. We show that for an irreducibleselection-mutation matrix in the birth term the deterministicmodel has a unique interior equilibrium which is globally stable.Thus all subpopulations coexist. In the pure selection case, theoutcome is known to be that of competitive exclusion, where thesubpopulation with the largest growth-to-mortality ratio willsurvive and the remaining subpopulations will go extinct. We showthat if the selection-mutation matrix is reducible, thencompetitive exclusion or coexistence are possible outcomes. Wethen develop a stochastic population model based on thedeterministic one. We show numerically that the mean behavior ofthe stochastic model in general agrees with the deterministic one.However, unlike the deterministic one, if the differences in thegrowth-to-mortality ratios are small in the pure selection case,it cannot be determined a priori which subpopulation will have thehighest probability of surviving and winning the competition.https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.133coexistencestochastic differential equations.selection-mutation modelscompetitive exclusion |
| spellingShingle | Azmy S. Ackleh Shuhua Hu Comparison between stochastic and deterministic selection-mutation models Mathematical Biosciences and Engineering coexistence stochastic differential equations. selection-mutation models competitive exclusion |
| title | Comparison between stochastic and deterministic selection-mutation models |
| title_full | Comparison between stochastic and deterministic selection-mutation models |
| title_fullStr | Comparison between stochastic and deterministic selection-mutation models |
| title_full_unstemmed | Comparison between stochastic and deterministic selection-mutation models |
| title_short | Comparison between stochastic and deterministic selection-mutation models |
| title_sort | comparison between stochastic and deterministic selection mutation models |
| topic | coexistence stochastic differential equations. selection-mutation models competitive exclusion |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.133 |
| work_keys_str_mv | AT azmysackleh comparisonbetweenstochasticanddeterministicselectionmutationmodels AT shuhuahu comparisonbetweenstochasticanddeterministicselectionmutationmodels |