Oscillations in planar deficiency-one mass-action systems

Abstract

Whereas the positive equilibrium of a mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, with centers or multiple limit cycles.

Figures (4)

• Figure 1: For a positive equilibrium to exist, the point $P_4$ is located in the sector that is the intersection of the green and red open half-spaces (left panel). Equivalently, the sum of the two angles indicated is less than $180^\circ$ (right panel).
• Figure 2: The mass-action systems \ref{['eq:chain_ode']} with the substitution \ref{['eq:chain_reversible_abpq']} (left column), and the phase portraits of the corresponding scaled differential equation \ref{['eq:chain_ode_scaled_reversible']} (right column). The top row is for $p+q>0$, while the bottom row is for $p+q<0$. Notice that the union of closed orbits is bounded for $p+q>0$, and unbounded for $p+q<0$.
• Figure 3: Some reaction networks that all fall under \ref{['cor:three_reactions_reversible_center']}, along with the differential equation \ref{['eq:three_reactions_ode_scaled_reversible']}. Among the graphs, in the top row we have $p>0$ and $-p<q<p$, while in the bottom row we have $p<0$ and $p<q<-p$.
• Figure 4: Some reaction networks that all fall under \ref{['cor:three_reactions_lienard_center']}, along with the differential equation \ref{['eq:three_reactions_ode_scaled_lienard']}, and the corresponding phase portraits.

Theorems & Definitions (26)

• definition 1
• definition 2
• definition 3
• definition 4
• theorem 1
• theorem 2: Deficiency-Zero Theorem feinberg:1972, horn:1972, horn:jackson:1972
• theorem 3: Deficiency-One Theorem feinberg:1995
• proposition 1
• proof
• proposition 2