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August  2020, 13(8): 2135-2143. doi: 10.3934/dcdss.2020182

## The stability analysis of brain lactate kinetics

 1 Université P.-M. Curie, Paris 6, Laboratoire Jacques–Louis Lions, UMR 7598 CNRS, 4 Pl. Jussieu, Tour 16-26, 75252 Paris, France 2 Shanghai Jiao Tong University, SJTU-ParisTech Elite Institute of Technology, 800 Dong Chuan Road, 200240 Shanghai, China

* Corresponding author: Jean-Pierre Francoise

Received  January 2019 Published  August 2020 Early access  November 2019

Our aim in this article is to study properties of a generalized dynamical system modeling brain lactate kinetics, with $N$ neuron compartments and $A$ astrocyte compartments. In particular, we prove the uniqueness of the stationary point and its asymptotic stability. Furthermore, we check that the system is positive and cooperative.

Citation: Jean-Pierre Françoise, Hongjun Ji. The stability analysis of brain lactate kinetics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2135-2143. doi: 10.3934/dcdss.2020182
##### References:
 [1] B. Abraham and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. [2] A. Aubert and R. Costalat, Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, J. Cereb. Blood Flow Metab., 25 (2005), 1476-1490.  doi: 10.1038/sj.jcbfm.9600144. [3] R. Costalat, J.-P. Françoise, C. Menuel, M. Lahutte, J.-N. Vallée, G. de Marco, J. Chiras and R. Guillevin, Mathematical modeling of metabolism and hemodynamics, Acta Biotheor., 60 (2012), 99-107.  doi: 10.1007/s10441-012-9157-1. [4] R. Guillevin, A. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electron. J. Differential Equations, 23 (2017), 1-16. [5] J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 1998. [6] M. Lahutte-Auboin, Modélisation biomathématique du métabolisme énergétique cérébral : réduction de modèle et approche multi-échelle, application à l'aide à la décision pour la pathologie des gliomes, Ph.D thesis, Université Pierre et Marie Curie, 2015. [7] M. Lahutte-Auboin, R. Costalat, J.-P. Françoise and R. Guillevin, Dip and buffering in a fast-slow system associated to brain lactate, kinetics, preprint, arXiv: math/1308.0486v1. [8] M. Lahutte-Auboin, R. Guillevin, J.-P. Françoise, J.-N. Vallée and R. Costalat, On a minimal model for hemodynamics and metabolism of lactate: Application to low grade glioma and therapeutic strategies, Acta Biotheor., 61 (2013), 79-89.  doi: 10.1007/s10441-013-9174-8. [9] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [10] A. Miranville, A singular reaction-diffusion equation associated with brain lactate kinetics, Math. Methods Appl. Sci., 40 (2017), 2454-2465.  doi: 10.1002/mma.4150. [11] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375.  doi: 10.1137/0146025. [12] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

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##### References:
 [1] B. Abraham and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. [2] A. Aubert and R. Costalat, Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, J. Cereb. Blood Flow Metab., 25 (2005), 1476-1490.  doi: 10.1038/sj.jcbfm.9600144. [3] R. Costalat, J.-P. Françoise, C. Menuel, M. Lahutte, J.-N. Vallée, G. de Marco, J. Chiras and R. Guillevin, Mathematical modeling of metabolism and hemodynamics, Acta Biotheor., 60 (2012), 99-107.  doi: 10.1007/s10441-012-9157-1. [4] R. Guillevin, A. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electron. J. Differential Equations, 23 (2017), 1-16. [5] J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 1998. [6] M. Lahutte-Auboin, Modélisation biomathématique du métabolisme énergétique cérébral : réduction de modèle et approche multi-échelle, application à l'aide à la décision pour la pathologie des gliomes, Ph.D thesis, Université Pierre et Marie Curie, 2015. [7] M. Lahutte-Auboin, R. Costalat, J.-P. Françoise and R. Guillevin, Dip and buffering in a fast-slow system associated to brain lactate, kinetics, preprint, arXiv: math/1308.0486v1. [8] M. Lahutte-Auboin, R. Guillevin, J.-P. Françoise, J.-N. Vallée and R. Costalat, On a minimal model for hemodynamics and metabolism of lactate: Application to low grade glioma and therapeutic strategies, Acta Biotheor., 61 (2013), 79-89.  doi: 10.1007/s10441-013-9174-8. [9] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [10] A. Miranville, A singular reaction-diffusion equation associated with brain lactate kinetics, Math. Methods Appl. Sci., 40 (2017), 2454-2465.  doi: 10.1002/mma.4150. [11] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375.  doi: 10.1137/0146025. [12] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.
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