As regards interesting mathematics arising in biology:
A mathematically fascinating class of integro-PDEs arise in the study of age-structured population models. The independent variables are age $a$ and time $t$ ; the systems are first-order PDE; and the boundary conditions on the curve age=0 are given in terms of integrals of the dependent variables, $u$. That is, $u(0,t)= \int_{a=0}^T \phi(u(s,t)) ds$ where $\phi$ may be a nonlinear function. Such models arise frequently in physiology. It's my impression that this is a field with many interesting open mathematical questions to be asked.
PDEs arising in pattern-forming systems in biology exhibit interesting mathematical behavior; questions about long-time regularity of such PDE are mathematically interesting. One may wish, for example, to characterize finite-time blow-up, or development of geometric singularities on interfaces.
Dynamical systems with delays (functional differential equations) for the form $\frac{dy}{dt} = A(y(t-\tau),t)$ arise naturally in biology. This is a field which is not as mathematically developed as the theory of ODE.
