Modeling Evolution of Heterogeneous Populations: Theory and Applications

Front matter
Copyright
Dedication
Using mathematical modeling to ask meaningful biological questions
	Introduction
	General strategy
	Advantages and drawbacks of the Reduction theorem
Inhomogeneous models of Malthusian type and the HKV method
	Models of Malthusian type for inhomogeneous populations and a simplified version of the HKV method
	Dynamics of different initial distributions
	Inhomogeneous Malthusian model
	Logistic equation with distributed Malthusian parameter
	Inhomogeneous Allee-type models
		Distribution of growth parameter a
		Distribution of carrying capacity b
		Distribution of parameter m
		Comparison of the three distributed Allee models
Some applications of inhomogeneous population models of Malthusian type
	Example 3.1. Conceptual models of global demography
	Example 3.2. Modeling the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations
	Example 3.3. Models of forest stand self-thinning
Selection systems and the Reduction theorem
	Selection systems
	Dynamics of specific distributions
	Selection systems with self-regulations
	Reduction theorem for inhomogeneous models of populations
	Reduction theorem for inhomogeneous models of communities
Some applications of the Reduction theorem and the HKV method
	How to solve selection systems
	Example 5.1 Birth-and-death equation with distributed birth rate and average death rate
	Example 5.2 A model of group selection and evolution of altruism
	Example 5.3 The principle of limiting factors in modeling of early biological evolution
	Example 5.4 Inhomogeneous Ricker equation with two distributed parameters
	Example 5.5 Inhomogeneous prey-predator Volterra model with three distributed parameters
	Example 5.6 Competition of two inhomogeneous populations
	Example 5.7 The Fisher-Haldane-Wright equation
	Example 5.8 Haldane principle for selection systems
	Example 5.9 The Price equation and Fisher's fundamental theorem
Nonlinear replicator dynamics
	Problem formulation: Power replicator equations
	Population heterogeneity as the reason for the power law growth dynamics
	Canonical form of the power model
	Inhomogeneous model for exponential equation
	Superexponential models: The second representation
	How to choose between F- and D-models?
	Summary
Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution
	Problem formulation and basic models
	Solution to the inhomogeneous logistic equation
	Generalized logistic inhomogeneous models with distributed Malthusian parameter
	Inhomogeneous Gompertz equation
	Logistic equation with distributed carrying capacity
	Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity
	Dynamics of distributions in inhomogeneous models and the speed of natural selection
	Some notes on internal time and the competitive exclusion principle
	Mathematical Appendix (by F. Berezovskaya): The Newton diagram method and asymptotic behavior of q(t) for inhomogeneous bir ...
		Basic equation in the new form
		The Newton diagram method
		Asymptotics of orbits of system (A.4)
		Asymptotics of probabilities
Replicator dynamics and the principle of minimal information gain
	Problem formulation
	MinxEnt algorithm and the Boltzmann distributions
	Selection systems and dynamical principle of minimal information gain
	MinxEnt principle and selection systems: Applications
		Information gain in inhomogeneous Malthusian model
		Information gain in the model of early biological evolution
		Information gain in models of tree stand self-thinning
		Quasi-species equation and linear systems
		Information gain in inhomogeneous birth-and-death models
		Information gain in inhomogeneous Ricker model
	``Conjugative´´ approach to selection system dynamics
	Information gain in models of inhomogeneous communities
	Discussion
Subexponential replicator dynamics and the principle of minimal Tsallis information gain
	Problem formulation: Subexponential power equations
	Population of freely growing parabolic replicators
	Dynamical principles of minimal information gain
	Population of parabolic replicators with constant total size and the principle of minimal information gain
	Discussion
Modeling extinction of inhomogeneous populations
	Mathematical and nonmathematical motivations
	Problem formulation
	Population of subexponentially decreasing clones
	Dynamical principles of minimal Tsallis information loss
	Parametrically inhomogeneous models of population extinction
	Power extinction models and inhomogeneous F-models
	The ``internal time´´ for F-models of extinction
	Dynamical principles of minimal Shannon information loss
	Application of the model to time perception
		Some background information on time perception
		Application of the proposed model to understanding time perception in a dying brain
	Discussion
From experiment to theory: What can we learn from growth curves?
	Problem formulation
	Approach and introductory example
	Generalized logistic equation
	Gompertz versus Verhulst
	Example 11.1 Logistic versus Gompertz curves
	Hyperbolic and hyperbolic-exponential growth
	Exponential-linear growth
	Virus-specific RNA replication and the three-stage model
	Fitting experimental data to different models
		Data set 1: Naumov et al. (2006) JNCI
		Data set 2: Rogers et al. (2014)
		Data set 3: Rogers et al. (2014)
		Data set 4: Benzekry et al. (2014)
	Simeoni model and exponential-linear growth
	Discussion
	Applications and implications
	Appendix: Supplementary material
Traveling through phase-parameter portrait
	Introduction
	Example 12.1. Sustainability: Using a parametrically heterogeneous model to investigate resource depletion, transitional re ...
		Question 1. How will such a system behave when the number of overconsumers in it changes?
		Question 2. Can we identify transitional regimes that can serve as warning signals of approaching collapse?
		Question 3. What, if any, intervention measures can be implemented to prevent the tragedy of the commons?
	Example 12.2. Natural selection in resource allocation strategies
		Question 1. If one allows for the possibility of resource overconsumption, which strategy is preferable for avoiding popula ...
		Question 2. Which strategy (allocating shared resources toward rapid proliferation, or toward slower proliferation but incr ...
	Example 12.3 Cancer and oncolytic viruses
		Question 1. What are the transitional regimes that occur as the cancer cell population gains resistance to the virus? Can w ...
		Question 2. Why are cytotoxic therapies effective in some patients and not others?
	Distributed susceptibility
	Distributed susceptibility and distributed cytotoxicity
	Conclusions
Evolutionary games: Natural selection of strategies
	Problem formulation
	The model and the main equations
	Solution to the replicator equation
	Equilibria of frequencies
	Dynamics of the distribution of strategies
		Replicator equation
		Prisoner's dilemma
		Coordination game or SH game
	Natural selection of strategies in a ``hawk-dove´´ game
	Natural selection of strategies and the principle of minimum information gain
	Discussion
Natural selection between two games with applications to game theoretical models of cancer
	Model description
		The model
		Solution to the model
	Natural selection between games: Hawk-Dove versus Prisoner's dilemma
	Mutual invasibility of both strategies and games
	Games tumors play
		Game 1: Metabolism and resource allocation
		Game 2: Motility versus stability
	Some conclusions
Discrete-time selection systems
	Main definitions
	Malthusian inhomogeneous maps
	Evolution of the main statistical characteristics of inhomogeneous maps
	Self-regulated inhomogeneous maps
	The Price equation and the Fisher fundamental theorem for maps
	Applications and examples
		Ricker model with discrete time
		Inhomogeneous logistic map
		Inhomogeneous Ricker map with two distributed parameters
		Selection in natural rotifer community
	Discussion
Conclusions
Moment-generating functions for various initial distributions
	Distributions on the entire line or half line
	Distributions on a finite interval
	Many-dimensional distributions
Bibliogrpahy
Index