Modeling Evolution of Heterogeneous Populations: Theory and Applications

Description

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. 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