On a Hypercycle System with Nonlinear Rate

We study a (N +1)−hypercyclical reaction-diffusion system with nonlinear reaction rate n. It is shown that there exists a critical threshold N0 such that for N ≤ N0 the system is stable while for N > N0 it becomes unstable. It is also shown that for large reaction rate n, N0 remains a constant: in fact for n ≥ n0 ∼ 3.35, N0 = 5 and for n < n0 ∼ 3.35, N0 = 4. Some more general reaction-diffusion systems of N + 1 equations are also considered.