Math. Model. Nat. Phenom.
Volume 14, Number 4, 2019
Singular perturbations and multiscale systems
|Number of page(s)||18|
|Published online||02 April 2019|
Multiscale dynamics of an adaptive catalytic network
Technical University of Munich, Faculty of Mathematics,
Garching b. München, Germany.
2 External Faculty, Complexity Science Hub Vienna, Josefstädterstrasse 39, 1090 Vienna, Austria.
* Corresponding author: email@example.com
Accepted: 5 March 2019
We study the multiscale structure of the Jain–Krishna adaptive network model. This model describes the co-evolution of a set of continuous-time autocatalytic ordinary differential equations and its underlying discrete-time graph structure. The graph dynamics is governed by deletion of vertices with asymptotically weak concentrations of prevalence and then re-insertion of vertices with new random connections. In this work, we prove several results about convergence of the continuous-time dynamics to equilibrium points. Furthermore, we motivate via formal asymptotic calculations several conjectures regarding the discrete-time 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 Jain–Krishna 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.
Mathematics Subject Classification: 37C10 / 05C82 / 92B20
Key words: Adaptive network / co-evolutionary network / autocatalytic reaction / Jain–Krishna model / network dynamics / multiple time scale system / pre-biotic evolution / random graph
© EDP Sciences, 2019
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