Fluctuations and correlations in chemical reaction kinetics and population dynamics

TL;DR

This chapter explores how stochastic effects influence spatial and temporal fluctuations in chemical reactions and population dynamics, highlighting phase transitions, correlation effects, and pattern formations.

Contribution

It provides a comprehensive pedagogical overview of fluctuation effects, phase transitions, and pattern formations in stochastic chemical and population systems, including new insights into non-equilibrium phenomena.

Findings

Identification of population extinction thresholds as non-equilibrium phase transitions.

Observation of depletion zones and spatial segregation in reaction-diffusion systems.

Analysis of noise-induced patterns such as activity fronts and spiral structures.

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.