System Of Diffusion Equations Research Articles

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.

Large diffusivity stability of attractors in the C(Ω) — topology for a semilinear reaction and diffusion system of equations

Large diffusivity stability of attractors in the C(Ω) — topology for a semilinear reaction and diffusion system of equations

Upper semicontinuity of attractors for a semilinear reaction and diffusion system of equations, in sup norm

Upper semicontinuity of attractors for a semilinear reaction and diffusion system of equations, in sup norm

Ion and transmitter movements during spreading cortical depression.

A reaction--diffusion system of equations whose components are the extracellular concentrations of K+, Ca++, Na+, Cl-, an excitatory neurotransmitter and an inhibitory neurotransmitter, is developed in order to model the movements of these substances through various kinds of membrane in brain structures. Expressions are derived from probabilistic arguments for the conductances induced in subsynaptic membrane by transmitter substances at various concentrations, for one-way active transport rates and for an exchange pump rate. These expressions are employed in the reaction terms of the system. The meaning of the many constants is explained and, with appropriate choices for their values, the model predicts subthreshold responses to small enough local elevations of KCl or glutamate and stable propagating SD waves if the local elevations of these chemicals is sufficient. The SD waves consist of elevated K+ and transmitter substances and diminished Ca++, Na+ and Cl-, the velocity of propagation in cortex being about 0.6 mm/min. This is in the experimental range, the K+-amplitude being 17 mM relative to a baseline of 3 mM, as the model developed ignores the effects of action potentials. There is no SD response to either NaCl or GABA. The effects of no K+ and no glutamate diffusion are investigated, both being manifest as a failure in propagation of the stable SD waves. The wave solutions are analysed in terms of phase portraits. The roles of various amino acid uptake and release processes by neurons and glia are discussed, as are the complications with regard to their incorporation in a model for SD. The roles of neurons and glia are analysed and the six basic fluxes of K+ are outlined. It is postulated that under some circumstances, in cortex treated with TTX, there may be practically no transmitter release, but SD may propagate if TTX does not completely abolish gNa for nonsynaptic membrane, a corresponding system of model equations being developed. The data and ideas of K+-based and glutamate-based SD of Van Harreveld are discussed and interpreted in terms of which reaction terms are operative in the K+ equation. An appendix contains the values of the parameters used in the numerical calculations.