Fixed Points

Fixed points are the points where and equation's rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).

Why are fixed points important?

fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as fzero in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.

Results

We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.