Fluctuations and correlations in chemical reaction kinetics and population dynamics

Abstract

This chapter provides a pedagogical introduction and overview of spatial and temporal correlation and fluctuation effects resulting from the fundamentally stochastic kinetics underlying chemical reactions and the dynamics of populations or epidemics. After reviewing the assumptions and mean-field type approximations involved in the construction of chemical rate equations for uniform reactant densities, we first discuss spatial clustering in birth-death systems, where non-linearities are introduced through either density-limiting pair reactions, or equivalently via local imposition of finite carrying capacities. The competition of offspring production, death, and non-linear inhibition induces a population extinction threshold, which represents a non-equilibrium phase transition that separates active from absorbing states. This continuous transition is characterized by the universal scaling exponents of critical directed percolation clusters. Next we focus on the emergence of depletion zones in single-species annihilation processes and spatial population segregation with the associated reaction fronts in two-species pair annihilation. These strong (anti-)correlation effects are dynamically generated by the underlying stochastic kinetics. Finally, we address noise-induced and fluctuation-stabilized spatio-temporal patterns in basic predator-prey systems, exemplified by spreading activity fronts in the two-species Lotka-Volterra model as well as spiral structures in the May-Leonard variant of cyclically competing three-species systems akin to rock-paper-scissors games.

28 pages, 3 figures; book chapter, to appear in "Chemical kinetics beyond the textbook", eds. K. Lindenberg, R. Metzler, and G. Oshanin (World Scientific Publ.)

References in corpus (33)

• The World of the Complex Ginzburg-Landau Equation
• Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States
• Universality classes in nonequilibrium lattice systems
• Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games
• Predator-prey cycles from resonant amplification of demographic stochasticity
• Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains
• Geometry-controlled kinetics
• Entropic transport: Kinetics, scaling and control mechanisms
• Robust ecological pattern formation induced by demographic noise
• Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models
• Optimal reaction time for surface-mediated diffusion
• Fluctuation-driven Turing patterns
• Experimental realization of directed percolation criticality in turbulent liquid crystals
• Spatial Rock-Paper-Scissors Models with Inhomogeneous Reaction Rates
• Mortality, Redundancy, and Diversity in Stochastic Search
• Reaction-Diffusion Processes from Equivalent Integrable Quantum Chains
• Fluctuations and Correlations in Lattice Models for Predator-Prey Interaction
• Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
• Exploration and Trapping of Mortal Random Walkers
• Towards a full quantitative description of single-molecule reaction kinetics in biological cells
• Mean first-passage time of surface-mediated diffusion in spherical domains
• Explicit Construction of the Brownian Self-Transport Operator
• Population oscillations in spatial stochastic Lotka-Volterra models: A field-theoretic perturbational analysis
• Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena
• Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries
• Measurement of a reaction-diffusion crossover in exciton-exciton recombination inside carbon nanotubes using femtosecond optical absorption
• Exact mean exit time for surface-mediated diffusion
• Non-equilibrium relaxation in a stochastic lattice Lotka-Volterra model
• On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model
• On the Robin problem in fractal domains
• Symmetry and species segregation in diffusion-limited pair annihilation
• Ligand-receptor binding kinetics in surface plasmon resonance cells: A Monte Carlo analysis
• A stochastic analysis of the spatially extended May-Leonard model

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• Reversible reactions controlled by surface diffusion on a sphere
• Long-term memory induced correction to Arrhenius law
• Coverage Fluctuations and Correlations in Nanoparticle-Catalyzed Diffusion-Influenced Bimolecular Reactions
• Diffusion towards a nanoforest of absorbing pillars
• Arrival time for the fastest among $N$ switching stochastic particles
• Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains
• The Steklov problem for exterior domains: asymptotic behavior and applications
• Adsorption and permeation events in molecular diffusion
• Concentration of Empirical First-Passage Times