Solutions to systems of nonlinear reaction-diffusion equations.

Abstract

General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂ci/∂t = Di▿2ci + Qi(c) subject to boundary conditions of practical interest, where the enumerator index i runs 1 to n, ci = ci(x, t) denotes the concentration or density of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species, and Qi(c), an algebraic function of the n-tuple c = (c1,…, cn), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals of c which decrease monotonically with time can often be found, as examplified here for Volterra and Verhulst-Volterra n-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, lim ⁡ t → ∞ c i ( x , t ) = ĉ i ( x ) , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl of Q(c) in c-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.