AbstractsComputer Science

Reaction-diffusion-ODE systems: de-novo formation of irregular patterns and model reduction

by Steffen Johannes Härting

Institution: Universität Heidelberg
Year: 2016
Posted: 02/05/2017
Record ID: 2119925
Full text PDF: http://www.ub.uni-heidelberg.de/archiv/20550


Classical models of pattern formation in systems of reaction-diffusion equations are based on diffusion-driven instability (DDI) of constant stationary solutions. The destabilisation may lead to emergence of stable, regular Turing patterns formed around the destabilised equilibrium. In this thesis it is shown that coupling reaction-diffusion equations with ordinary differential equations may lead to de-novo formation of far from equilibrium steady states. In particular, conditions for so called (ε0 , A)-stability (resp. stability in epi-graph-topology) are given, yielding from bistability and hysteresis effects in the null sets of nonlinearities. A model exhibiting coexistence of Turing-type destabilisation and stable far from equilibrium steady states, is proposed. It is shown, under suitable conditions, that DDI and (in)stability can be derived from so called quasi-stationary model reduction. Moreover, similar to a result for ordinary differential equations, proved by Tikhonov, the dynamical behaviour of the reduced and the unreduced model are similar. It is shown that the spectral properties of the operators resulting from linearisation of the unreduced system, determining the long-term behaviour around a steady state, are reflected in the spectral properties of the operators resulting from linearisation of the reduced system. The given conditions are satisfied by a larger range of classical models, as illustrated by application to a degenerate version of the Lengyel-Epstein model. The dynamical behaviour of reaction-diffusion equations for large diffusion and on finite time intervals is essentially reflected by their so called shadow systems. In this hesis, existence and stability of steady states with jump-type discontinuity is investigated and compared for this reduction. The results show that, in case of static patterns, not only the short-term behaviour, but also the long-term behaviour of the reduced system is reflected in the unreduced system. Moreover, a result showing Turing-type destabilisation for such shadow systems, given in a joint-paper, is generalised. Finally, such shadow systems are reduced by application of a quasi-stationary model reduction leading to a scalar integro-differential equation. It is shown that the quasi-stationary model reduction is regular in the sense of Turing-type destabilisation and dynamical behaviour on finite time intervals. Hence, reaction-diffusion-ODE models may be reduced to scalar integro-differential equations in order to investigate the qualitative behaviour around homogeneous steady states and the qualitative behaviour on finite time intervals. A hypothesis is that the long-term behaviour is similar, but a proof is missing. The result shows that a link between reaction-diffusion-ODE systems and scalar integro-differential equations exists and that the mechanisms of pattern formation may be investigated based on the reduction. Klassische mathematische Modelle zur Beschreibung von Musterbildungsprozessen basieren auf Turing Instabilität: ein örtlich homogener… Advisors/Committee Members: Marciniak-Czochra, Anna (advisor).