We will start by deriving a simple model of intracellular dynamics of the signaling molecule IP₃, introducing conservation laws that represent fundamental scientific principles and constitutive relations that are empirically determined functions that form terms within differential equations of the system.

We will review geometric- and linear stability analysis, and then extend our discussion of the stability of fixed points to include potential, which can also be used to assess stability .

The concentration of IP₃ is denoted by variable I. We start by noting the conservation law I′ = j_prod - j_deg, where j are production and degradation rates. In particular, j_prod is a constant production rate we simply refer to as j, whereas j_deg is a function of current concentration of IP₃, that is f(I) =kI. Thus the system is:

I′ = j - kI

We draw the vector field I′ vs. I, and find a fixed point for the system at I* = j/k, as well as the fact that the vector field is a line with slope -k and intersects the I′ axis at j. By graphical methods the fixed point at j/k is stable.

We can go through a very similar analysis of the logistic equation for population dynamics: N′ = rN(1 - N/K), except that in this example there are two fixed points N* = 0 or K. By graphical methods we find that N*=0 is unstable, and N*=K is stable.

Linear stability analysis is a more quantitative way to determine the stability of fixed points, which is generally more useful in higher dimensional systems, i.e., more state variables. Strogatz gives all the explanation you need on pp.24 - 26. Another method to visualize the dynamics of systems and analyze their critical points is based on the idea of potential energy, which is explained on pp. 30 - 31.

We go through a few examples, applying our general approach to understanding dynamics by the following systematic approach:

The example systems include:

If you have trouble analyzing any of these systems, then please write me email, call me (x17808), or come by my office MS 303 or lab MS 318.

(1) Draw the vector field for the following systems showing all fixed points. Determine the stability of the fixed points either geometrically or using linear stability analysis, and sketch in arrows on the x-axis to indicate the phase portrait:

(2) Create an XPP input file for the differential equation x′ = x² - 6. Run a sequence of simulations showing x vs. t and print out enough trajectories with appropriately scaled x and t axes to demonstrate the presence of the fixed points and their stability. Be sure to label your graph such that stable and unstable fixed points are clearly indicated.

(3) Determine the characteristic time scale for the following systems (NOTE: a and b are constants):

(4) Find the analytical solution for IP₃ dynamics for the system: I′ = j - kI. Assume the initial IP₃ concentration is zero, i.e., I(0)=0. Draw the phase portrait for this sytem and determine the stability of the fixed point(s).

(5) What does the above example have in common with Strogatz' example 2.2.2 (p. 20)? What does this analytical solution have in common with Exercise 2.2.11?

(6-11) Do exercises 3.1.1, 3.2.1, 3.4.1, 3.4.3, 3.4.14 in Strogatz.