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References

1. Ingalls, B. P. (2013). Mathematical modeling in systems biology: an introduction. MIT press.
2. Strogatz, S. H. (2018). Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC press.

Stability of Fixed Points

For this part, I mainly talk about nonlinear gene networks. Check (En) Systems Biology for the context and examples.

Take a nonlinear network of species interacting; the system’s dynamics consists of species described as a system of ODEs. How do you mathematically solve (or describe) them? For this part, check Strogatz’s book (ref 2).

One-Dimensional Case

$\displaystyle\frac{dx(t)}{dt}=f(x(t))$ is an ODE. How to solve it is described in 微积分. As we can see, we can only solve some very specific ODEs analytically. If we can’t solve the problem analytically, sketching the evolution trend without a lot of detail is helpful.

If the solution of $\displaystyle\frac{dx(t)}{dt}=0$ is $x=x_i$ (if nonlinear, there can be multiple solutions, and we call them fixed points), then \(x=x_i\) might be a candidate for the system to finally rest at this point. Will this happen?

We focus on the local part near the fixed points. Place your state near a fixed point. Will it go to the fixed point or diverge further away? And then, can we sketch the how phase plane by this?

(Ingalls’ 4.2) The long-time (i.e., asymptotic) behavior of biochemical and genetic networks will be either

• convergence to a steady state; or
• convergence to a sustained periodic oscillation, referred to as limit-cycle oscillation. (Shi’s comment: sorry for the messy structure but this only works for multi-dimensional case)

Other dynamic behaviors (divergence and chaos, for example) do not often occur in systems biology models.

Keep only the first-order term, and we have $\displaystyle\frac{dx(t)}{dt}\vert_{x=x_i}=f’(x_i)(x(t)-x_i)$. As Example 2.2.1 shows in Strogatz’s book ($f(x)=x^2-1$), the stability of fixed points is determined by the symbols of $f(x_i)$. If negative, go to the fixed point (stable); if positive, diverge away (unstable). A trickier case that $f(x_i)=0$, see Strogatz’s book, not this post.

No limit cycles (Bendixson).

Multi-dimensional Case

For example, check Ingalls’ model (4.2). For a general expression, $\mathbf{x(t)}=(x_1(t),…,x_n(t))^{T}$ is an \(n\times 1\) vector, and $\mathbf{f}(\mathbf x(t))=(f_1(x_1(t)),…,f_n(x_n(t))^T$ is an $n\times 1$ vector, the system of ODEs is therefore $\displaystyle\frac{d}{dt}\mathbf{x}(t)=\mathbf{f}(\mathbf x(t))$. Fixed points are $\mathbf x_i$. Consider the local properties near the fixed points, only keep the first-order terms, and we have $\displaystyle\frac{d}{dt}\mathbf{x}(t)\vert_{\mathbf x=\mathbf x_i}=\mathbf{J}(\mathbf x_i)\cdot (\mathbf x(t)-\mathbf x_i)$, where $\displaystyle J_{mn}(\mathbf x_i)=[\frac{\partial f_m(\mathbf x)}{\partial x_n}]\vert_{\mathbf x=\mathbf x_i}$ is the Jacobian matrix.

As in 1-D case, the stability is determined by the symbols of the first derivatives of $f(x)$ at the fixed points, what criteria we use in the multi-dimensional case? The eigenvalues of the Jacobian matrix at the fixed points!

Read Ingalls’ 4.2 for the great mathematical explanation. To summarize,

Linearized Stability Criterion

• If both eigenvalues of the Jacobian have negative real part, then the state is stable.
• If either eigenvalue has positive real part, then the steady state is unstable.

… not addressed the case of eigenvalues with zero real part … rarely encountered…

Limit cylces are possible, e.g., Van der Pol.

Category of fixed points

• Node: real eigenvalues (no oscillation) with the same sign

  • Stable node: negative eigenvalues

  • Unstable node: posotive eigenvalues.

• Saddle: real eigenvalues (no oscillation) with different signs, not stable

• Spiral: contains conjungate complex eigenvalues (with oscillation, see Ingalls’ page 105)

  • spiral sink: all eigenvalues have negative real part
  • spiral source: with positive real part

Bifurcation

Check Strogatz’s related sections.

If we change the value of a parameter (parameters) in ODEs, the structure of the flow diagram can change. The number and the stability of fixed points can change. Qualitative changes in the dynamics are called bifurcations.

Example: Saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, hopf bifurcation…