Mathematical Models and Experimental Nonlinear Population Dynamics

Abstract

Mathematical Models and Experimental Nonlinear Population Dynamics Jim M. Cushing Department of Mathematics, University of Arizona at Tucson, USA ?: I will give an over view of a two decade long, and ongoing collaboration among mathematicians, biologists and statisticians in a project designed to interface mathematical models, in a quantitatively predictive way, with the dynamics of a biological population. The broad goal is to study a variety of model predicted nonlinear phenomena and validate their occurrence in a real biological population, and to do this by means of controlled and replicated experiments. The organisms involved were species of the genus Tribolium (flour beetles). The nonlinear phenomena studied include equilibrium states, stability and destabilization, bifurcations, periodic cycles, invariant loops, transients & saddle structure in phase space, roles of demographic & environmental stochasticity, and routes-to-chaos. The route-to-chaos experiment (which took eight years to complete) has been called the first unequivocal demonstration of chaos in a biological population. Observations from the experiments led to many new ideas and spin-off projects, including: patterns in chaos, the effect of habitat size on complex dynamics, lattice effects due to a discrete state space, control of chaos by means of sensitivity to initial conditions, resonant & attenuated oscillations, dynamic and spatial consequences of interactions among lifecycle stages, and adaptation modeled by evolutionary game theoretic versions of the dynamic model.