Supercritical Branching Process with Immigration

Updated 24 August 2025

Supercritical branching process with immigration is defined by reproduction exceeding replacement and Poisson arrivals that continuously seed independent subpopulations.
Key methodologies, such as spine and backbone decompositions, rigorously analyze convergence properties and establish gamma limit laws and central limit theorems.
The process underpins applications in ecology and genetics by linking exponential population growth with species abundance models like the GEM distribution.

A supercritical branching process with immigration models populations where both intrinsic reproduction exceeds replacement (mean reproduction number per individual $>1$ ), and additional individuals (immigrants) enter from outside the system. In the supercritical regime, the population tends to grow exponentially, but immigration modifies convergence, extinction, and type structure in nontrivial ways. The mathematical foundations incorporate discrete Galton–Watson processes, age-dependent (Crump–Mode–Jagers, CMJ) branching, continuous-state branching processes (CSBP), and multi-type extensions.

1. Fundamental Construction and Growth

The canonical supercritical age-dependent branching process with immigration, as formalized in the Crump–Mode–Jagers framework, involves individuals with i.i.d. lifespans (distribution $\Lambda$ ), reproducing at constant rate $b > 0$ . Immigration occurs via a Poisson process (rate $\theta>0$ ); at each immigration time $T_i$ , a single individual founds an independent new family/subpopulation (splitting tree):

$X(t) = \sum_{x} 1\{\alpha_x \le t < \alpha_x + \zeta_x\}$

( $\alpha_x$ : birth time, $\zeta_x$ : lifetime).

For the total population with immigration: $I(t) = \sum_{T_i \le t} X_i(t-T_i)$ where $X_i$ is a CMJ process started by immigrant $i$ . Under supercritical conditions ( $m = E[\text{number of children}] > 1$ ), the rescaled process converges almost surely: $e^{-\eta t} I(t) \to I \;\; \text{a.s.}$ where $\eta$ solves the Malthusian equation: $\psi(\eta) = 0, \quad \psi(\lambda) = \lambda - \int_{(0,\infty)} [1 - e^{-\lambda r}] \Lambda(dr)$ The limiting random variable $I$ follows a gamma law with parameters $(\theta/b, c)$ ( $c = \psi'(\eta) > 0$ ), Laplace transform: $\mathbb{E}[e^{-aI}] = \left(\frac{c}{a + c}\right)^{\theta/b}$ The total population, conditional on non-extinction, exhibits exponential growth.

2. Immigration Mechanisms and Population Structure

Immigration generates multiple independent subfamilies. Each immigrant creates a "family" whose descendants form a subpopulation, and the aggregation over all families yields the total population. Considerations include:

Immigration as a Poisson process (constant rate $\theta$ ) (Richard, 2010).
Type structure: Immigrants can be of different types; types may be unique (Model I), chosen from a discrete set (Model II), or from a continuous spectrum (Model III).
In supercritical multi-type settings, both reproduction and immigration may vary by type; see discrete definitions in (Li et al., 21 Aug 2025, Rabehasaina et al., 2019).

Population composition is governed by the immigration-to-birth ratio ( $\theta/b$ ): in Model I (new unique type per immigrant), the vector of relative abundances $(P_1,P_2,\ldots)$ of surviving families converges almost surely to the GEM( $\theta/b$ ) distribution: $P_i = B_i \prod_{j=1}^{i-1}(1-B_j), \quad B_i \sim \text{Beta}(1, \theta/b)$ This limit law is robust: independent of details of the lifetime law, depending only on rates.

3. Advanced Methodologies: Spine and Backbone Decomposition

Limit theorems and convergence properties leverage refined probabilistic techniques:

Spine decomposition (Richard, 2010): Identifies an immortal lineage ("spine"), analyzes the process as contributions from the spine plus "grafted" subtrees (which are conditioned to become extinct). Facilitates uniform integrability and rigorous control of fluctuations.
Backbone decomposition (Ren, 2011): In continuous-state branching processes with immigration, the law coincides with a continuous-time Galton–Watson process with immigration, "dressed" by mass from conditioned extinct subcritical CSBP excursions. The process can be split into:
- A backbone (carrying the "prolific" genealogies).
- Poissonian dressing representing pathwise fluctuations about the backbone.
- Explicit generator formulas for backbone branching ( $F(r) = (1/\lambda^*)\psi(\lambda^*(1-r))$ ) and immigration ( $G(r)=\phi(\lambda^*)-\phi(\lambda^*(1-r))$ ).

These decompositions are central for relaxing moment conditions and yielding almost sure convergence results in general settings, including non-exponential lifetime distributions.

4. Limit Theorems: Scaling and Deviations

Various scaling results and deviation principles are established:

Law of large numbers: Appropriately normalized populations (e.g., $e^{-\eta t} I(t)$ ) converge to a nondegenerate limit (gamma or exponential) under supercriticality with immigration (1007.54282009.12564Barczy et al., 2018).
Central limit theorems: Fluctuations around the deterministic exponential growth are asymptotically Gaussian (normal), subject to second moment bounds (Wang, 2016 Barczy et al., 2018). In multi-type CBI processes, $e^{-s(B)t}(u,X_t)\to W_{u,X_0}$ in $L_1$ , and CLT results describe the fluctuations along left eigenvectors.
Large deviation rates: For supercritical multitype processes, deviation probabilities for normalized projections decay geometrically, and, under conditioning or stronger moment assumptions, supergeometrically. Quantitative bounds for probabilities of the form

$P_i\left(\left|\frac{l\cdot X_n}{\mathbf{1}\cdot X_n}-\frac{l\cdot v}{\mathbf{1}\cdot v}\right|>\varepsilon\right)$

are given in (Li et al., 21 Aug 2025). The martingale $Y_n$ converges almost surely, with deviation rates for $|Y_n-Y|$ bounded by supergeometric exponentials.

Small value probabilities: Exact left-tail asymptotics (probabilities of very small normalized population size) depend on extinction probabilities, "minimal" offspring and immigration events (Chu et al., 2013). Four cases are distinguished depending on the offspring and immigration distributions.

5. Structured Populations and Species Diversity

Immigration modifies the long-term species/type diversity, especially the limiting abundance distribution.

In Model I (unique type per immigrant), the rescaled family sizes $(Z^{(i)}(t)/I(t))$ converge to the GEM( $\theta/b$ ) law (Richard, 2010). This result links branching processes to neutral models in ecology and genetics (such as Kingman coalescent with mutation).
Multi-type supercritical models yield convergence of relative frequencies: For $p$ types, $(e_i,X_t)/\sum_{k=1}^p (e_k,X_t)$ converges almost surely to the component $(e_i,u)$ of the left Perron–Frobenius eigenvector $u$ (Barczy et al., 2018).
In settings with varying environments and generation-dependent immigration, normalization by a scale $a_n$ yields asymptotic gamma distributions for population size (González et al., 28 Jan 2024).

6. Extensions: Random Environments, Competition, Controlled Immigration

Random environments: Branching and immigration laws can vary randomly per generation (Zhao et al., 23 Feb 2024). Exponential decay rates for small positive values and limit theorems (CLT and Edgeworth expansions for $\log Z_n$ ) generalize previous homogeneous results.
Competition mechanisms: Adding nonlinear drift terms (modeling competition) achieves exponential ergodicity even in supercritical CB-processes with immigration (Li et al., 2022). The process converges exponentially fast in a weighted total variation distance to a unique invariant law, controlled via coupling and Lyapunov techniques.
Controlled immigration: Immigration triggered when the catalyst population falls below a threshold is analyzed via reflected SDEs and averaging principles (Budhiraja et al., 2012). When the catalyst changes rapidly, averaging yields simplified limits for the reactant population.

7. Statistical Inference and Stable Estimation

Parameter estimation in supercritical CBI processes with immigration is complicated by mixed normal limits and nondegeneracy due to randomness of population growth. Conditional least squares estimators for drift parameters converge stably (in the sense of mixing convergence); normalization rates and mixed normal limits are characterized in detail (Barczy, 2022). This enables reliable inference for model parameters in both discrete and continuous time, despite pathwise randomness and long-term growth.

The supercritical branching process with immigration exhibits exponential growth modulated by incoming immigrants, with limit laws strongly affected by the immigration-to-birth ratio ( $\theta/b$ ), the spectral properties of the mean matrix in multi-type settings, and the precise structure of reproduction and immigration mechanisms. Spine decomposition and backbone techniques allow generalization to age-dependent lifespans and continuous-state models, accommodating minimal moment assumptions and non-Markovian dynamics. Limit theorems—laws of large numbers, central limit, large deviation rates, and small value asymptotics—provide a comprehensive framework to predict both total population behavior and fine-scale composition, with universal links to species abundance models, population genetics, and ecological statistics. Applications span branching models in ecology, genetics, epidemiology, and queuing theory, leveraging theoretical results for both predictive modeling and statistical estimation.