A numerical framework for computing steady states of structured population models and their stability
Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, f...
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AIMS Press
2017-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2017049 |
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| author | Inom Mirzaev David M. Bortz |
| author_facet | Inom Mirzaev David M. Bortz |
| author_sort | Inom Mirzaev |
| collection | DOAJ |
| description | Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can \emph{also} be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU). |
| format | Article |
| id | doaj-art-f5c9b2b3d7d04e3aa4647b84a9a5fb23 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2017-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-f5c9b2b3d7d04e3aa4647b84a9a5fb232025-01-24T02:39:54ZengAIMS PressMathematical Biosciences and Engineering1551-00182017-07-0114493395210.3934/mbe.2017049A numerical framework for computing steady states of structured population models and their stabilityInom Mirzaev0David M. Bortz1Department of Applied Mathematics, University of Colorado, Boulder, CO, 80309-0526, United StatesDepartment of Applied Mathematics, University of Colorado, Boulder, CO, 80309-0526, United StatesStructured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can \emph{also} be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).https://www.aimspress.com/article/doi/10.3934/mbe.2017049stationary solutionsnumerical stability analysisnonlinear evolutionequationspopulation balance equationssize-structured population modeltrotter-kato theorem |
| spellingShingle | Inom Mirzaev David M. Bortz A numerical framework for computing steady states of structured population models and their stability Mathematical Biosciences and Engineering stationary solutions numerical stability analysis nonlinear evolution equations population balance equations size-structured population model trotter-kato theorem |
| title | A numerical framework for computing steady states of structured population models and their stability |
| title_full | A numerical framework for computing steady states of structured population models and their stability |
| title_fullStr | A numerical framework for computing steady states of structured population models and their stability |
| title_full_unstemmed | A numerical framework for computing steady states of structured population models and their stability |
| title_short | A numerical framework for computing steady states of structured population models and their stability |
| title_sort | numerical framework for computing steady states of structured population models and their stability |
| topic | stationary solutions numerical stability analysis nonlinear evolution equations population balance equations size-structured population model trotter-kato theorem |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2017049 |
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