Individual behavior. A generation ago, the computational complexity associated with
     predicting individual behaviour made it an enormous task. The development of
     computational power has made behavioural prediction using stochastic dynamic
     programming, genetic algorithms and other optimization methods feasible. Furthermore, this
     is an area where collaboration between experiment and mathematical theory is particularly
     fruitful because the time scale of individual behavior is conducive to rapid collection of data.
     Even so, many mathematical challenges remain, ranging from problems of numerical analysis
     of interpolation at boundaries to overcoming the curse of dimensionality in problems with
     many state variables.

     Single population dynamics. The analysis of the population dynamics of single species has
     contributed to the development of nonlinear differential equations, the theory of chaos
     (through analysis of discrete maps), nonlinear diffusion theory (through analysis of equations
     such as the Fisher equation), and stochastic population theory. Many interesting problems
     remain. These include: i) determining the spectra of time series generated by nonlinear maps
     (a topic that received much coverage in high profile journals such as Nature, recently), ii)
     connecting nonlinear stochastic and deterministic models where closure problems similar to
     the ones in the theory of turbluence arise, and iii) the origins of diffusion models from
     discrete movement models, particularly when some fraction of the population may make
     large movements.

     Multi-species population dynamics and community ecology. The interactions of two or
     more species, as in predation, competition, mutualism and disease, present new kinds of
     mathematical challenges. These include the extension of phase plane analysis to more than
     two dimensions, the estimation of parameters for complicated nonlinear systems, the
     possibilities of large excursions (as occur in pest or disease outbreaks) and understanding
     the stability properties of large multidimensional systems of ordinary, partial, and stochastic
     differential equations.
 
 

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