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
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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
17. Moment-generating functions for various initial distributions
Dr. Irina Kareva is a theoretical biologist, and the primary focus of her research involves using mathematical modeling to study cancer as an evolving ecosystem within the human body, where heterogeneous populations of cancer cells compete for limited resources (i.e., oxygen and glucose), cooperate with each other to fight off predators (the immune system), and disperse and migrate (metastases). In 2017 Dr. Kareva gave a TED talk on using mathematical modeling for biological research. Dr. Kareva's book Understanding cancer from a systems biology point of view: from observation to theory and back was published by Elsevier in 2018. Dr. Kareva is a Senior Scientist in Simulation and Modeling at EMD Serono, Merck KGaA, where she develops quantitative systems pharmacology (QSP) models to help understand and predict dynamics of new therapeutics.
Dr. Georgy Karev has significant research experience in various fields of applied mathematics, mathematical modeling, and mathematical biology. His research spans computational biology and bioinformatics, modeling of genome evolution, Markov models, mathematical genetics, ecological modeling and modeling of dynamics of biological populations and communities. Dr. Karev has developed three new directions in mathematical biology: 1) theory of inhomogeneous population dynamics with applications to models of early biological evolution, population extinction, global demography, and ecology; 2) stochastic modeling of population size dynamics; and 3) theory of multi-dimensional structural models with applications to hierarchical models of complex biological systems. His current research is devoted to problems of computational biology including genome evolution, free-scaling networks, evolution of horizontally transferred genes, conceptual cancer models, replicator dynamics, and general theory of selection. Dr. Karev is a member of the Evolutionary Genomics Research Group at NCBI.