Example of well understood/modeled biological mechanism
I would point out that, assuming (as the other commenter noted) you are interested in analytic rather than statistical modelling (which seems a little unlikely to me given your area), most mathematical models are derived from first principles (often from physics) or other assumptions. 
This suggests (to me) that looking at models of totally unrelated systems is not so useful; the whole idea of such models is to formalize your domain knowledge!

But I'll give some examples anyway.

For example, consider 
 biophysical models of neurons or continuum mechanical models of cardiac tissue, both of which have rich histories of mathematical models developed over decades of work, based on both theoretical and empirical considerations (albeit largely from physics).
Perhaps slightly more related (or at least empirically driven) are the advances in systems biology, which tend to derive from some underlying laws of chemical kinetics, but utilize enormous amounts of (multivariate, temporal) data in fitting (often non-linear stochastic) systems of differential equations. Methods for computational protein structure prediction tend to use a mix of ad hoc rules (for computational time-cost reasons) and physics-based considerations in their models.
Evolutionary biology and population dynamics also have wonderful models (see the book from the commenter above).

My guess, given that your question does not have much detail, is that you would prefer statistical, i.e. probabilistic graphical or stochastic process, models instead. 

Let's suppose you have a bunch of variables over time (e.g. the actions or status of a person) but you want to estimate an unobservable quantity (e.g. say, the degree to which they are insane [not my field, sorry]).
One simple probabilistic model is the hidden Markov model.
It has many examples of applications in sequence biology.
It also lets you figure out, in a way, which variables are important, and let's you tune certain parameters (e.g. it's order, which here describes how much the person's state at time $t$ affects their state at time $t+\Delta t$). 
In terms of predictive modelling power, though, the state of the art is in deep learning models for time series.

Anyway, I'd consider what your goal is more carefully. If you indeed have a strong idea of an analytic model (motivated by some theory you know), then by all means formalize it. However, if you want the data itself to suggest a model for you, then use a statistical approach.

Note: apologies if this answer was too trivial, I was not sure what you did and did not know.