Determining Dynamics

What have I been up to since my last post?

Well…a lot more math.  We finished calculating the sensitivities of the model with respect to each parameter: development, mortality, and birth.  These terms are dependent on temperature so then we calculated the sensitivity with respect to temperature.  Sensitivity tells you how the eigenvalues of the model change as the parameters change.  The eigenvalues are indicators of the behavior of the model.  Basically we were using the techniques we learned with the smaller models from last time to understand the dynamics of the zooplankton model. We also looked at how the parameters change over time, so that we can make connections between the sensitivities and the model’s dynamics.

We worked some more on condensing our 13-stage class model into something more manageable.  During the semester, we condensed it to 4-stage classes, but it wasn’t perfect.  Last week, we condensed it to 6-stage classes and with some error correction it seems to work better, but 6-stages may be too many to do the analysis we want to do on it.

We began typing up all our notes from everything my partner and I did since we began working on this project last semester.  We’re using a program called LaTeX, which makes inserting figures and equations into a document super easy (at least when compared to Word).  You also have a lot more control over the formatting of your document and it can be outputted as a PDF.  With all our notes up until the beginning of this week, the document was already 27 pages (wow!).  It may include a lot of figures and equations, but it made me realize how much work we already done on this project, and it looks impressive when the parents ask what you’ve done this summer.

This week our professors/mentors are out of town, so we’ve been given some more smaller models to play with.  The models have more complicated dynamics than the first ones, because they include predator-prey interactions, since the mortality of the zooplankton in our model is a combination of predatory and non-predatory mortality.  We’ve spent the week finding fixed points and their stability and eigenvalues on four models, becoming more comfortable with predicting the dynamics of simple models so we can apply it to the zooplankton model.  With these models and the 2-stage class one from a few weeks ago, we also have been doing interval analysis.  We’ve been trying to figure out how much we can vary one parameter when the other remain fixed and still make the same assumptions about stability.  We look at when the eigenvalues change from positive to negative or real to complex.  This can usually be done algebraically with the linear models but gets a lot more complicated with the non-linear models.  And that’s when we turn to MATLAB.

No new cupcake recipes to share, but hopefully I’ll have one soon…if I don’t melt this weekend with the 100°F temperatures.