Quantitative concepts for undergraduate biology students:
Lou Gross
The below are concepts that I would hope that all biology majors
would not only be exposed to during their undergraduate career, but
would have some conceptual understanding of as well. Every student
should for at least a few of these, also have the ability to
analyze issues arising in these contexts in some depth, using
either analytical methods (e.g. pencil and paper) or appropriate
computational tools.
1. Rate of change - specific (e.g. per capita) and total
Discrete - as in difference equations
Continuous - calculus-based
2. Scale - different questions arise on different scales
What is important to include depends on the scales of the
questions you are addressing.
Modeling is a process of "selective ignorance"
Trade-offs in modeling - generality. precision, realism
3. Equilibria - rate of change = 0
There can be more than one
These can be dynamic
Can arise in an average sense in periodic systems
4. Stability - notion of a perturbation and system response to this
Alternative definitions exist including not just whether a
a system returns to equilibrium but how it does so.
Multiple stable states can exist - initial conditions
and the nature of perturbations (history) can affect
long-term dynamics
5. Structure - effect of grouping components of a system
Choosing different aggregations (sex, age, size,
physiological state) can expand or limit the questions
you can address, and data availability can limit your
ability to investigate effects of structure.
The geometry of grouping can matter.
6. Interactions - a few key types exist, based upon local interactions
Some general properties can be derived based upon these
(2-species competitive interactions), but even relatively
few interacting system components can lead to compelx
dynamics.
Though ultimately everything is hitched to everything else,
significant effects are not automatically transfered through
a connected system of interacting components - locality
can matter. Sequences of interactions can determine
outcomes - program order matters.
7. Stochasticity - what counts as unpredictable
Alternative notions of probability and the relationship
to risk - what is significance in experiments?
When does stochasticity matter, under what circumstances
are averages not sufficient?
8. Visualizing - there are diverse methods to display data
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.
What math is needed to get the above across? In addition to K-12 level
training, students would need linear algebra, discrete models, some
calculus, exposure to the modeling process, basic probability and
statistics, basic notions of logic and programming.