Applications of Dynamical Systems to Statistical Physics, Geometry, and Population Biology/Demography

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ABSTRACTWeissThis proposal is to study several applications of dynamical systems tostatistical physics, geometry, and population biology/ecology. (i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, natural classes of Holder continuous potentials for certain one-dimensional lattice spin systems are determined by their free energy. We also plan to investigate striking similarities between the rigidity of free energy and fascinating rigidity problems in spectral geometry and number theory. (ii) Little is known about thedynamics of the geodesic flow on positively curved manifolds. The PIplans to continue studying the relations between positive curvature andcomplicated dynamics of the geodesic flow. (iii) The PI has started aprogram to systematically study the global dynamics and bifurcations fornonlinear Leslie models where the fertility rates and survivalprobabilities have various natural functional forms as functions of thepopulation size.(i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. Lattice spin systems provide an important and illuminating family of models in statistical physics, condensed matter physics, and chemistry. For instance, phase transitions correspond to non-differentiability for some derivative of free energy. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, potentials forone-dimensional lattice systems are determined by their free energy. Wehope this work will provide new insights into this important, yetmysterious, quantity. (ii) Essentially all demographic and animalpopulation models in current use are based on the linear Leslie model. Many population biologists, ecologists, and demographers are now lookingto nonlinear population models for more accurate population forecasting. The PI has started a program to systematically study the global dynamicsand bifurcations for nonlinear Leslie models where the fertility rates andsurvival probabilities have various natural functional forms as functionsof the population size. One of our ultimate goals is to create a``population modeling toolbox'' which could be used by a wide range ofpopulation modelers to more accurately predict animal populations.