Multiscale Dynamics of an Adaptive Catalytic Network
Abstract
We study the multiscale structure of the JainKrishna adaptive network model. This model describes the coevolution of a set of continuoustime autocatalytic ordinary differential equations and its underlying discretetime graph structure. The graph dynamics is governed by deletion of vertices with asymptotically weak concentrations of prevalence and then reinsertion of vertices with new random connections. In this work we prove several results about convergence of the continuoustime dynamics to equilibrium points. Furthermore, we motivate via formal asymptotic calculations several conjectures regarding the discretetime graph updates. In summary, our results clearly show that there are several time scales in the problem depending upon system parameters, and that analysis can be carried out in certain singular limits. This shows that for the JainKrishna model, and potentially many other adaptive network models, a mixture of deterministic and/or stochastic multiscale methods is a good approach to work towards a rigorous mathematical analysis.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 DOI:
 10.48550/arXiv.1903.00046
 arXiv:
 arXiv:1903.00046
 Bibcode:
 2019arXiv190300046K
 Keywords:

 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Adaptation and SelfOrganizing Systems;
 Quantitative Biology  Molecular Networks
 EPrint:
 21 pages