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.