
Mathematical Biosciences and Engineering, 2017, 14(4): 933952. doi: 10.3934/mbe.2017049.
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A numerical framework for computing steady states of structured population models and their stability
. Department of Applied Mathematics, University of Colorado, Boulder, CO, 803090526, United States
Received: , Accepted: , Published:
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 TrotterKato 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 SinkoStreifer 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 wellknown Smoluchowski coagulationfragmentation 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).
Keywords: Stationary solutions; numerical stability analysis; nonlinear evolution; equations; population balance equations; sizestructured population model; TrotterKato theorem
Citation: Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences and Engineering, 2017, 14(4): 933952. doi: 10.3934/mbe.2017049
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Copyright Info: 2017, David M. Bortz, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
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