Velocity–Amplitude Relationship in the Gray–Scott Autowave Model in Isolated Conditions

Abstract

Velocity and amplitude are two basic characteristics of any autowave, and their relationship reflects the internal regulation of the autowave system. This study proposes an approach to approximately estimate steady velocity–amplitude (VA) relation without deriving separate formulas for V and A. The approach presumes constructing an ansatz which represents the “petal” form of phase trajectory and contains V, A, and a free parameter (parameters). After substituting this ansatz, integration of model equations leads to a VA relation analytically. A free parameter (parameters) can be determined by comparing the analytical VA relation to the numerical data. As an example, we used the simplest autowave model possessing threshold, that is, the Gray–Scott model (cubic autocatalysis with linear inhibition) in isolated conditions with an immobilized precursor and a diffusible reactant. For all values of the inhibition rate constant allowing autowave solution (i.e., except approaching zero), the free parameter of ansatz belongs to a narrow interval has little effect on VA relation and can be regarded as fixed. Assumption of precursor immobilization does not influence the results qualitatively. This approach will be useful in investigations of more complex active media systems in biochemistry, combustion, and disease control.

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“…Solution to the reaction–diffusion problems. To integrate Equations ( 1 ) and ( 3 ), we used the previously developed package [ 10 , 19 ] based on the numerical methods described in [ 20 ]: Störmer–Encke’s method for space discretization (uniform mesh with 401 nodes, zero Neumann boundary conditions at and ) and the embedded Runge–Kutta–Fehlberg method of order 2(3) with automatic step size control for integration in time (mixed local error estimation with max norm, , , ). The activation zone was , and in this zone was multiplied by to reduce gradients at , (thus, the average value of in equaled ).…”