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Minimal Network Structure for Turing Instability
Millis, Brendan
Millis, Brendan
Abstract
Turing instability originates from diffusion-induced instability within biochemical systems and has been described as a critical component regarding the recognition of pattern formations. Turing instability is defined as the destabilization of an equilibrium solution within a spatially homogeneous reaction-diffusion system. Matrices can be used to model the interactions between nodes in these systems therefore knowing how the system becomes unstable may be desirable. Using matrix theory and graph theory we are able to classify matrix patterns with minimal number of nonzero entries so Turing instability and bifurcation are possible. For 2x2 and 3x3 matrices the minimum number of nonzero entries for the matrix to exhibit Turing instability is already known, we classify the type of bifurcation that occurs for all potentially stable matrices when a constant equilibrium in a reaction-diffusion system transitions to an unstable state. For 4x4 matrices we show that the minimum number of nonzero entries required to exhibit Turing instability is 6, and we identify the bifurcation that occurs to the reaction-diffusion system for the only potentially stable matrix.
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Date
2023-05-01
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Mathematics