Physics 380, 2011: Lecture 22

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We are continuing our short introduction to dynamical systems.

Main lecture

Once systems become more than 1d, other arrangements but fixed points are possible. One may have oscillations as well. Consider a system of two interacting genes: ${\begin{array}{l}{\frac {dX}{dt}}=C_{X}+{\frac {V_{X}}{1+(Y/K_{Y})^{2}}}-rX\\{\frac {dY}{dt}}=C_{Y}+{\frac {V_{Y}X^{2}}{1+(X/K_{X})^{2}}}-rY\end{array}}$ ${\begin{array}{l}{\frac {dX}{dt}}=C_{X}+{\frac {V_{X}}{1+(Y/K_{Y})^{2}}}-rX\\{\frac {dY}{dt}}=C_{Y}+{\frac {V_{Y}X^{2}}{1+(X/K_{X})^{2}}}-rY\end{array}}$ .
- The point of zero derivatives are: for X: $X=\left(C_{X}+{\frac {V_{X}}{1+(Y/K_{Y})^{2}}}\right)/r$ , and for Y, $Y=\left(C_{Y}+{\frac {V_{Y}X^{2}}{1+(X/K_{X})^{2}}}\right)/r$ .