Delay differential equations are becoming increasing popular in mathematical modeling for biology. Generally, numerical approaches are chosen, since DDEs can be so complicated. Some analytical methods are available, however, and this has been the focus of my research in this area.
The characteristic equations of ordinary differential equations are polynomial, of the form
Therefore, each steady state has a finite number of eigenvalues. By contrast, delay differential equations have transcendental characteristic equations,
and infinitely many eigenvalues. I developed a method of determining changes in linear stability when the length of the delay is the bifurcation parameter. The method involves Sturm sequences. You can read all about it in the article here.
For the truly intrepid, you can see all of my work in this area in my thesis.
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