Summary: As the end of the title indicates, the book will concentrate on the ideas that go into the proofs and highlight the most important techniques: duality, the block construction, and the convergence of rescaled particle systems to ODE, PDE, and IDE. Chapters 3-8 are the core: they introduce our main techniques for proving results and apply them to various examples from ecology, genetics, and evolutionary games. Applications are mostly at a conceptual level, i.e., identifying the features of a system that are responsible for its observed behavior. Chapter 1 introduces important background material. Chapter 2 describes results for the voter model and contact process, which are the building blocks for many of our models. The final chapter tries to break new ground on a problems I have struggled with for my whole career: finding finite range particle system with positive rates and two transaltion invariant stationary distributions, and understading the exotic behavior of growth modesl that are not additive.
1.1. A few words about Markov processes
1.2. Harris construction for finite range models
1.3. Graphical representations, additive processes
1.4. Attractive processes, correlation inequalities
1.5. Oriented percolation
2.1. Basic voter model results
2.1.1. Clustering vs. Coexistence
2.1.2. Renormalizing the voter model in d ≥ 3
2.1.3. Asynmptotics for CRW densities
2.1.4. CRW on the complete graph
2.1.5. CRW on the torus
2.1.6. Voter model on the torus
2.1.7. Clustering in d=2
2.2. Basic contact process results
2.2.1. Edge speeds characterize the critical value in d=1
2.2.2. Block construction, exponential estimates in d=1
2.2.3. The critical contact process dies out
2.2.4. Exponential estimates: supercritical case
2.2.5. Subcritical estimates on Zd
2.2.6. Critical exponents
3.1. A simple prototype
3.2. Long range contact process
3.3. Models with a dominant species
3.4. Threshold models
3.5. Interlude: Weinberger (1982)
3.6. Quadratic contact process
3.7. Staver-Levin forest model
4.1. Competing contact processes
4.2. Heuristics of Durrett and Levin (1994)
4.3. Mutual invadability implies coexistence, I
4.4. Lotka-Volterra systems
4.5. Rock-paper-scissors
5.1. Four simple examples
5.2. From heuristics to proofs
5.3. Predator-prey systems
5.4. Catalytic surfaces
5.5. Mutual invadability implies coexistence, II
5.6. Lyapunov functions for Lotka-Volterra systems
5.7. Two prey, one predator
6.1. Basic theory
6.2. Hydrodynamic limit
6.3 General coexistence and extinction results
6.4. Limiting behavior of PDE
6.5. Lotka-Volterra systems
6.6. Evolution of cooperation
6.7. General 2 by 2 games
6.8. Three strategy games
6.9. A public goods game in pancreatic cancer
6.10. Contact process with fast voting
7.1. Voter model perturbations on the torus
7.2. Latent voter model
7.3. q-voter model
7.4. Darling and Norris' convergence theorem
7.5. The voter model of Molofsky et al
7.6. Interlude: Ising models
7.7. convergence to motion by mean curvature
7.8. A nonergodic particle system with positive rates?