Modeling Evolution of Heterogeneous Populations

Modeling Evolution of Heterogeneous Populations: Theory and Applications describes, develops and provides applications of a method that allows incorporating population heterogeneity into systems of ordinary and discrete differential equations without significantly increasing system dimensionality. The method additionally allows making use of results of bifurcation analysis performed on simplified homogeneous systems, thereby building on the existing body of tools and knowledge and expanding applicability and predictive power of many mathematical models.

• Introduces Hidden Keystone Variable (HKV) method, which allows modeling evolution of heterogenous populations, while reducing multi-dimensional selection systems to low-dimensional systems of differential equations
• Demonstrates that replicator dynamics is governed by the principle of maximal relative entropy that can be derived from the dynamics of selection systems instead of being postulated
• Discusses mechanisms behind models of both Darwinian and non-Darwinian selection
• Provides examples of applications to various fields, including cancer growth, global demography, population extinction, tragedy of the commons and resource sustainability, among others
• Helps inform differences in underlying mechanisms of population growth from experimental observations, taking one from experiment to theory and back

Biological scientists looking to expand their mathematical modelling toolbox. Advanced graduate and 1st year PhD students. The areas of applicability of the method involve microbiology, ecology, population biology, cancer, social sciences, and infectious diseases, among others

1. Using mathematical modeling to ask meaningful biological questions through combination of bifurcation analysis and population heterogeneity
   2. Inhomogeneous models of Malthusian type and the HKV method
   3. Some applications of inhomogeneous population models of Malthusian type
   4. Selection systems and the reduction theorem
   5. Some applications of the reduction theorem and the HKV methods
   6. Nonlinear replicator dynamics
   7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution
   8. Replicator dynamics and the principle of minimal information gain
   9. Subexponential replicator dynamics and the principle of minimal Tsallis information gain
   10. Modeling extinction of inhomogeneous populations
   11. From experiment to theory: What can we learn from growth curves?
   12. Traveling through phase-parameter portrait
   13. Evolutionary games: Natural selection of strategies
   14. Natural selection between two games with applications to game theoretical models of cancer
   15. Discrete-time selection systems
   16. Conclusions
   17. Moment-generating functions for various initial distributions