Branching Processes in Random Environments

Branching processes in random environments (BPREs) extend classical Galton–Watson models by allowing the reproduction law to vary according to a sequence of random factors. In each generation, the offspring distribution is drawn from an environmental distribution, which may be independent and identically distributed or exhibit correlation across time. This framework captures the influence of fluctuating resources, climate variability, or external stressors on population dynamics. Key theoretical developments address survival and extinction probabilities under subcritical, critical and supercritical regimes, often distinguishing between quenched (environment fixed) and annealed (averaged over environments) perspectives. Limit theorems characterise normalised population sizes, while large‐deviation principles quantify rare events such as unexpectedly rapid growth or precipitous decline. Connections to random walks arise through logarithmic means of offspring distributions, yielding criteria for almost sure convergence and rates of decay. Applications span ecology, epidemiology and information transmission, where environmental randomness modulates proliferative behaviour. Recent advances further probe multitype interactions, migration, heavy‐tailed effects and conditioned processes, revealing rich asymptotic structures and shedding light on global significance in natural and engineered systems.

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Recent work on multitype BPREs elucidates the survival probability in weakly subcritical regimes. In an independent and identically distributed random environment, the asymptotic decay of survival is characterised by the associated random‐walk drift of the Perron roots of reproduction matrices. This yields precise exponential rates for extinction probabilities at large generations, refining classical subcritical estimates. A separate line of inquiry introduces a sparse random environment model, intermediate between Galton–Watson and BPRE. Under critical conditions, new Yaglom‐type limit theorems capture the population size conditioned on non‐extinction, revealing universal scaling laws despite environmental sparsity. Another study examines branching processes with conditioned geometric offspring laws and single immigration in long‐tailed environments. It demonstrates that occasional extreme environmental values dominate the tail behaviour of generation sizes, leading to explicit asymptotic equivalence for large deviations. The “principle of a single atypical environment” identifies rare but decisive environmental shocks as the primary driver of exceptionally large populations, with implications for risk assessment in ecological and networked systems.

Branching Processes in Random Environments publication trend

The graph below shows the total number of articles in branching processes in random environments across all publications each year (not limited to Nature Index journals).

Technical terms

Branching process: A stochastic model in which individuals reproduce independently, generating a random offspring count each generation.

Random environment: A sequence of random variables that determine the reproduction law at each generation.

Quenched perspective: Analysis conditioned on a fixed realisation of the environment.

Annealed perspective: Analysis averaging both population randomness and environmental randomness.

Supercritical/subcritical/critical: Regimes determined by whether the average reproduction rate exceeds, falls below, or equals unity, governing long‐term survival.

Yaglom limit: A conditional limit theorem describing the distribution of population size given non‐extinction in critical processes.

Perron root: The leading eigenvalue of a mean reproduction matrix in multitype models, governing exponential growth rates.

Heavy‐tailed distribution: A probability distribution whose tail decays slower than an exponential, often yielding large‐deviation behaviour dominated by rare events.

References

1. The survival probability of a weakly subcritical multitype branching process in iid random environment. Electronic Journal of Probability (2024).
2. Critical branching processes in a sparse random environment. Modern Stochastics Theory and Applications (2023).
3. Branching processes with immigration in atypical random environment. Extremes (2021).
4. Upper large deviations for Branching Processes in Random Environment with heavy tails. Electronic Journal of Probability (2011).
5. Branching processes in correlated random environment. Electronic Communications in Probability (2019).