387 research outputs found
Lie symmetries of the Shigesada-Kawasaki-Teramoto system
The Shigesada-Kawasaki-Teramoto system, which consists of two
reaction-diffusion equations with variable cross-diffusion and quadratic
nonlinearities, is considered. The system is the most important case of the
biologically motivated model proposed by Shigesada et al. A complete
description of Lie symmetries for this system is derived. It is proved that the
Shigesada-Kawasaki-Teramoto system admits a wide range of different Lie
symmetries depending on coefficient values. In particular, the Lie symmetry
operators with highly unusual structure are unveiled and applied for finding
exact solutions of the relevant nonlinear system with cross-diffusion
On a Hypercycle System with Nonlinear Rate
We study an (N+1)-hypercyclical reaction-diffusion system with nonlinear reaction rate n.
It is shown that there exists a critical threshold N_0 such that for N\leq N_0 the system is stable while
for N> N_0 it becomes unstable. It is also shown that for large reaction rate
n, N_0 remains a constant: in fact for n \geq n_0 \sim 3.35, N_0=5 and for n < n_0 \sim 3.35,
N_0=4. Some more general reaction-diffusion systems of N+1 equations are also considered
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