文章基本信息

标题：On a linear finite-difference model of a mixed-culture biological system arising in food safety studies
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作者：Jorge E Macías-Díaz
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2013
卷号：2013
期号：1
页码：357
DOI：10.1186/1687-1847-2013-357
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this work, we introduce a linear finite-difference methodology to approximate non-negative and bounded solutions of a coupled system of nonlinear parabolic partial differential equations that describes the growth of two different microbial colonies on a substrate of nutrients. Some simpler versions of the model of interest possess qualitative results that guarantee the existence and uniqueness of non-negative and bounded solutions. However, the exact determination of analytical solutions of our system (and even of those simpler versions of our model) corresponding to physically meaningful initial conditions, may be a difficult task, whence the need to design computational methods to approximate the solutions of our mathematical model is pragmatically justified. Our numerical technique has the advantage of conditionally preserving the non-negative and bounded characters of initial-boundary data. The most important analytical results of this work are summarized as a theorem of existence and uniqueness of non-negative and bounded numerical solutions, whose proof relies on the non-singularity property of M-matrices, and the fact that the entries of the inverses of these matrices are positive real numbers. We provide some illustrative simulations to evince the fact that our method preserves the mathematical characteristics of the solutions that we mentioned above. MSC:65Q10, 65M06, 39A14, 92D25.
关键词：computational modeling ; biological films ; positivity and boundedness ; linear finite-difference scheme ; food safety investigation