There is a need for a new kind of maths course, to be taught, not to mathematics students, but to biologists with little or no maths background. There have been many recent calls for an upgrade to the mathematical background of biologists: undergraduate biology students need to understand the role of modeling and dynamics in understanding ecological systems, evolutionary dynamics, neuroscience, physiology, epidemiology, and the modeling that underlies the concept of climate change. They also need to understand the importance of feedback, both positive and negative, in creating dynamical systems in biology.

 Such a course is possible. The most important foundational development was the 20^th century replacement of the vague and unhelpful concept of a differential equation by the rigorous geometric concept of a vector field, a function from a multidimensional state space to its tangent space, assigning “change vectors” to every point in state space. This twentieth-century concept is not just more rigorous, but in fact makes for superior pedagogy. We also discuss the key nonlinear behaviors that biological systems display, such as switch-like behavior, robust oscillations and even chaotic behavior.

 This talk will outline such a course. It would have a significant effect on the conduct of biological research and teaching, and bring the usefulness of mathematical modeling to a wide audience.