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December  2018, 11(6): 1169-1177. doi: 10.3934/dcdss.2018066

Equilibrium locus of the flow on circular networks of cells

Yirmeyahu J. Kaminski

Depart. of Applied Mathematics, Holon Institute of Technology, Holon, Israel

Received  June 2016 Revised  September 2016 Published  June 2018

We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is an embedded smooth and connected curve. For different values of the parameters, the curves are all isomorphic.

Moreover, we show how to build a homotopy between different curves obtained for different values of the parameter set. This procedure allows the efficient computation of the equilibrium point for each value of some first integral of the system. This point would have been otherwise difficult to be computed for higher dimensions. We illustrate this construction by some numerical experiments.

Eventually, we show that when considering the parameters as inputs, one can easily bring the system asymptotically to any equilibrium point in the reachable set, which we also easily characterize.

Keywords: Equilibrium, fiber spaces, homotopy continuation.
Mathematics Subject Classification: Primary: 57R99, 37C25; Secondary: 34H05.
Citation: Yirmeyahu J. Kaminski. Equilibrium locus of the flow on circular networks of cells. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1169-1177. doi: 10.3934/dcdss.2018066
References:
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C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55.   Google Scholar

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A. Raveh, Y. Zarai, M. Margaliot and T. Ruller, Ribosome flow model on a ring, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 12 (2015), 1429-1439.  doi: 10.1109/TCBB.2015.2418782.  Google Scholar

show all references

References:
[1]

C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55.   Google Scholar

[2]

D. Lazard and F. Rouillier, Solving parametric polynomial systems, Journal of Symbolic Computations, 42 (2007), 636-667.  doi: 10.1016/j.jsc.2007.01.007.  Google Scholar

[3]

J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$, Springer, 2013.  Google Scholar

[4]

T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numerica, 6 (1997), 399-436.  doi: 10.1017/S0962492900002749.  Google Scholar

[5]

J. Mather, Notes on topological stability, Bulletin of the American Mathematical Society, 49 (2012), 475-506.  doi: 10.1090/S0273-0979-2012-01383-6.  Google Scholar

[6]

A. Raveh, Y. Zarai, M. Margaliot and T. Ruller, Ribosome flow model on a ring, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 12 (2015), 1429-1439.  doi: 10.1109/TCBB.2015.2418782.  Google Scholar

Figure 1.  The homotopy path generated with the following parameters ${\mathit{\boldsymbol{\lambda}}}_1 = (1, 1, 1)$, ${\mathit{\boldsymbol{\lambda}}}_2 = ( 1.39328599, 8.30098374, 3.98355604)$ and $s = 1$. The first equilibrium point $E_0 = (1/3, 1/3, 1/3)$ is represented by a green ball, while the final equilibrium point $E_1 = ( 0.53112814, 0.1203633, 0.34850856)$ is rendered as a red triangle. Of course all over the path the constraint $e_1 + e_2 + e_3 = s$ holds.
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