# American Institute of Mathematical Sciences

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

## Equilibrium locus of the flow on circular networks of cells

 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.

Citation: Yirmeyahu J. Kaminski. Equilibrium locus of the flow on circular networks of cells. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1169-1177. doi: 10.3934/dcdss.2018066
##### References:
 [1] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55. [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. [3] J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$, Springer, 2013. [4] T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numerica, 6 (1997), 399-436.  doi: 10.1017/S0962492900002749. [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. [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.

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##### References:
 [1] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55. [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. [3] J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$, Springer, 2013. [4] T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numerica, 6 (1997), 399-436.  doi: 10.1017/S0962492900002749. [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. [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.
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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