Oscillatory and isochronous rate equations possibly describing chemical reactions

We study a simple mathematical model that can be interpreted as a description of the kinetics of the following four reactions involving the two chemicals U and W: (i) U + U → U with rate α, (ii) U + W → U with rate β, (iii) W + W → U with rate γ and (iv) W + W → W + W + W with rate δ + 2γ. The model can be generally solved by quadratures, and in the special case β = 2α, explicitly in terms of elementary functions.We focus on the case characterized by the two inequalities γβ2 > αδ2 and 2βγ > δ2, and we show that in this case the solutions vanish asymptotically at large times. But if a constant decay with rate θ of chemical U is added, then a nonvanishing equilibrium configuration arises. Moreover, for arbitrary strictly positive initial conditions, the solutions remain bounded. They either tend asymptotically (in the remote future) to this nonvanishing equilibrium configuration, or are periodic, or tend to a limit cycle. Indeed, we find that this system goes through a standard supercritical Hopf bifurcation at an appropriate value of the parameters. Another interesting case arises when, in addition to the original reaction, a negative constant term is added to the equation corresponding to chemical U, corresponding to siphoning out a constant amount of chemical U per unit time, independent of its concentration. A very remarkable feature of this (possibly not very realistic) model is the following: in the special case β = 2α, we again find an explicit solution in terms of elementary functions, which oscillates at a fixed frequency, independent of the initial condition. In other words, it is an isochronous system. If β ≠ 2α, however, no periodic orbits exist, implying that the nature of the bifurcation at β = 2α is rather peculiar