D-unstable Cores Counterexample

No D-unstable cores. The full network can still have growing oscillations.

Growing Oscillations Four linearized species deviations over 700 model time units. The oscillations double their amplitude about every 346.6 units. Growing oscillations · growth rate 0.0020 Deviation from Equilibrium (scaled) -12-8-404812 X0X1X2X3 0100200300400500600700 Model Time
111.182

The Reaction Network

The four species and five reactions form a stoichiometric matrix SS. Rows represent species; columns represent reactions. Each entry gives the net change per reaction: negative means consumed, positive means produced.

r0:X0+2X3⟶4X2r1:4X1+2X2⟶∅r2:4X2⟶2X3r3:2X3⟶X0+6X1+X2r4:∅⟶2X2+2X3\begin{aligned} r_0:&\quad X_0+2X_3\longrightarrow4X_2\\ r_1:&\quad4X_1+2X_2\longrightarrow\varnothing\\ r_2:&\quad4X_2\longrightarrow2X_3\\ r_3:&\quad2X_3\longrightarrow X_0+6X_1+X_2\\ r_4:&\quad\varnothing\longrightarrow2X_2+2X_3 \end{aligned}
S=r0r1r2r3r4X0−10010X10−4060X24−2−412X3−202−22S=\begin{array}{c|rrrrr} &r_0&r_1&r_2&r_3&r_4\\\hline X_0&-1&0&0&1&0\\ X_1&0&-4&0&6&0\\ X_2&4&-2&-4&1&2\\ X_3&-2&0&2&-2&2 \end{array}

Here ∅\varnothing means material enters or leaves the model. At equilibrium, reaction rates (2,3,2,2,2)(2,3,2,2,2) balance every species: S(2,3,2,2,2)T=0S(2,3,2,2,2)^{\mathsf T}=0.

Child Selections and D-unstable Cores

We extract square matrices from the stoichiometric matrix SS by choosing species and assigning each a different reaction that consumes it. Keep those species’ rows and the assigned columns, in the same order. These are child-selection matrices; they may overlap. For example:

X0↦r0X_0\mapsto r_0X1↦r1X_1\mapsto r_1
A=[−100−4],AD=[−d000−4d1],d0,d1>0A=\begin{bmatrix}-1&0\\0&-4\end{bmatrix},\qquad AD=\begin{bmatrix}-d_0&0\\0&-4d_1\end{bmatrix},\quad d_0,d_1>0

The DD means we may multiply each column by any positive number. A selection is D-stable if every such scaling leaves all eigenvalues with negative real parts: disturbances in that linear system eventually decay. The example above is D-stable because its eigenvalues are always −d0-d_0 and −4d1-4d_1.

A selection is D-unstable if some positive scaling gives an eigenvalue with positive real part. A D-unstable core is a selection with that property that loses it whenever we remove matching rows and columns to make a smaller selection.

For this counterexample, 22 of the 24 selections are D-stable. The other two always retain a zero eigenvalue. All 24 are therefore D-nonunstable: none can acquire a positive real part under positive column scaling. There is no D-unstable core.

Kinetics: From Reaction Amounts to Motion

The stoichiometric matrix SS tells us how much each reaction consumes and produces, but not how fast it runs. To describe motion, we add kinetics: reaction rates v(x)v(x) that depend on the species concentrations xx.

dxdt=S v(x),Rji=∂vj∂xi∣x=x∗,G=SR\frac{dx}{dt}=S\,v(x),\qquad R_{ji}=\left.\frac{\partial v_j}{\partial x_i}\right|_{x=x^*},\qquad G=SR

The reaction-response matrix RR contains sensitivities, not the rates themselves. Entry RjiR_{ji} says how much reaction rjr_j speeds up when concentration XiX_i increases slightly at equilibrium. Multiplying SS by RR gives GG, the matrix governing small deviations from equilibrium.

R=[100E01/501/10000005/110000F0000]R=\begin{bmatrix}1&0&0&E\\0&1/50&1/1000&0\\0&0&5/11&0\\0&0&0&F\\0&0&0&0\end{bmatrix}

This example uses adjustable power-law rates. For instance, v0(x)=2x01/2x3E/2v_0(x)=2x_0^{1/2}x_3^{E/2} equals 2 at the equilibrium x∗=(1,1,1,1)x^*=(1,1,1,1), while its sensitivity to X3X_3 is EE. The slider changes this sensitivity while preserving the reactions and their equilibrium rates. These kinetic exponents are adjustable; they are not fixed by the stoichiometric coefficients.

Why the Full Network Can Oscillate and Grow

Reaction r0r_0 responds to both X0X_0 and X3X_3; reaction r1r_1 responds to both X1X_1 and X2X_2. A child selection cannot assign the same reaction twice. The full matrix G=SRG=SR includes these responses together. Their combined effect can generate growing oscillations even though none of the selected matrices is D-unstable.

That is the chart’s transition. With F≈0.6044F\approx0.6044 fixed, increasing EE moves a complex eigenvalue pair across zero real part, near E=81.867E=81.867. Negative real part gives decaying oscillations; zero gives constant-amplitude oscillations in the linear approximation; positive gives oscillatory growth. The transition does not establish a stable, self-sustaining cycle in the nonlinear system.

The counterexample shows that the absence of a D-unstable core does not guarantee a stable full network. The selections test particular response combinations, not independent physical subsystems: passing every selection test can miss instability arising from the combined kinetics.

All 24 Child Selections

Every conclusion holds for every positive column scaling. Two selections permit zero eigenvalues; none permits a positive real part.

X0↦r0X_{0} \mapsto r_{0}
[−1]\begin{bmatrix}-1\end{bmatrix}

All real parts strictly negative.

X1↦r1X_{1} \mapsto r_{1}
[−4]\begin{bmatrix}-4\end{bmatrix}

All real parts strictly negative.

X2↦r1X_{2} \mapsto r_{1}
[−2]\begin{bmatrix}-2\end{bmatrix}

All real parts strictly negative.

X2↦r2X_{2} \mapsto r_{2}
[−4]\begin{bmatrix}-4\end{bmatrix}

All real parts strictly negative.

X3↦r0X_{3} \mapsto r_{0}
[−2]\begin{bmatrix}-2\end{bmatrix}

All real parts strictly negative.

X3↦r3X_{3} \mapsto r_{3}
[−2]\begin{bmatrix}-2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X1↦r1X_{1} \mapsto r_{1}
[−100−4]\begin{bmatrix}-1 & 0 \\ 0 & -4\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X2↦r1X_{2} \mapsto r_{1}
[−104−2]\begin{bmatrix}-1 & 0 \\ 4 & -2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X2↦r2X_{2} \mapsto r_{2}
[−104−4]\begin{bmatrix}-1 & 0 \\ 4 & -4\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X3↦r3X_{3} \mapsto r_{3}
[−11−2−2]\begin{bmatrix}-1 & 1 \\ -2 & -2\end{bmatrix}

All real parts strictly negative.

X1↦r1X_{1} \mapsto r_{1}X2↦r2X_{2} \mapsto r_{2}
[−40−2−4]\begin{bmatrix}-4 & 0 \\ -2 & -4\end{bmatrix}

All real parts strictly negative.

X1↦r1X_{1} \mapsto r_{1}X3↦r0X_{3} \mapsto r_{0}
[−400−2]\begin{bmatrix}-4 & 0 \\ 0 & -2\end{bmatrix}

All real parts strictly negative.

X1↦r1X_{1} \mapsto r_{1}X3↦r3X_{3} \mapsto r_{3}
[−460−2]\begin{bmatrix}-4 & 6 \\ 0 & -2\end{bmatrix}

All real parts strictly negative.

X2↦r1X_{2} \mapsto r_{1}X3↦r0X_{3} \mapsto r_{0}
[−240−2]\begin{bmatrix}-2 & 4 \\ 0 & -2\end{bmatrix}

All real parts strictly negative.

X2↦r1X_{2} \mapsto r_{1}X3↦r3X_{3} \mapsto r_{3}
[−210−2]\begin{bmatrix}-2 & 1 \\ 0 & -2\end{bmatrix}

All real parts strictly negative.

X2↦r2X_{2} \mapsto r_{2}X3↦r0X_{3} \mapsto r_{0}
[−442−2]\begin{bmatrix}-4 & 4 \\ 2 & -2\end{bmatrix}

One zero eigenvalue; others negative.

X2↦r2X_{2} \mapsto r_{2}X3↦r3X_{3} \mapsto r_{3}
[−412−2]\begin{bmatrix}-4 & 1 \\ 2 & -2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X1↦r1X_{1} \mapsto r_{1}X2↦r2X_{2} \mapsto r_{2}
[−1000−404−2−4]\begin{bmatrix}-1 & 0 & 0 \\ 0 & -4 & 0 \\ 4 & -2 & -4\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X1↦r1X_{1} \mapsto r_{1}X3↦r3X_{3} \mapsto r_{3}
[−1010−46−20−2]\begin{bmatrix}-1 & 0 & 1 \\ 0 & -4 & 6 \\ -2 & 0 & -2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X2↦r1X_{2} \mapsto r_{1}X3↦r3X_{3} \mapsto r_{3}
[−1014−21−20−2]\begin{bmatrix}-1 & 0 & 1 \\ 4 & -2 & 1 \\ -2 & 0 & -2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X2↦r2X_{2} \mapsto r_{2}X3↦r3X_{3} \mapsto r_{3}
[−1014−41−22−2]\begin{bmatrix}-1 & 0 & 1 \\ 4 & -4 & 1 \\ -2 & 2 & -2\end{bmatrix}

All real parts strictly negative.

X1↦r1X_{1} \mapsto r_{1}X2↦r2X_{2} \mapsto r_{2}X3↦r0X_{3} \mapsto r_{0}
[−400−2−4402−2]\begin{bmatrix}-4 & 0 & 0 \\ -2 & -4 & 4 \\ 0 & 2 & -2\end{bmatrix}

One zero eigenvalue; others negative.

X1↦r1X_{1} \mapsto r_{1}X2↦r2X_{2} \mapsto r_{2}X3↦r3X_{3} \mapsto r_{3}
[−406−2−4102−2]\begin{bmatrix}-4 & 0 & 6 \\ -2 & -4 & 1 \\ 0 & 2 & -2\end{bmatrix}

All real parts strictly negative.

X0↦r0X_{0} \mapsto r_{0}X1↦r1X_{1} \mapsto r_{1}X2↦r2X_{2} \mapsto r_{2}X3↦r3X_{3} \mapsto r_{3}
[−10010−4064−2−41−202−2]\begin{bmatrix}-1 & 0 & 0 & 1 \\ 0 & -4 & 0 & 6 \\ 4 & -2 & -4 & 1 \\ -2 & 0 & 2 & -2\end{bmatrix}

All real parts strictly negative.

Example Code