Branching & Immigration-Birth Models

• Branching and immigration-birth representation is a framework that decomposes population evolution into internal reproduction (branching) and external arrivals (immigration) across various stochastic models.
• The representation underpins rigorous genealogical analysis by linking discrete Galton–Watson and continuous-state branching processes via explicit generating functions and Laplace transforms.
• It enables analytical, statistical, and simulation studies by providing formulas for moments, generating functions, and limit theorems across diverse population structures.

Branching and Immigration-Birth Representation

A branching and immigration-birth representation refers to a structural decomposition of stochastic population models in which the population's evolution is driven both by internal reproduction (branching) and by arrival of new individuals from outside the system (immigration). These representations arise in discrete (Galton–Watson) and continuous-state (CSBP, CBI, superprocess) contexts, generalize to age-structured and multitype populations, and underpin genealogical analyses, limit theorems, and simulation schemes in stochastic population theory.

1. Fundamental Models and General Structure

The canonical models consist of either discrete or continuous-state branching processes with immigration. In discrete time, the Galton–Watson process with immigration (GWI) is defined recursively by

$Z_{n+1} = \sum_{i=1}^{Z_n} X_i^{(n)} + Y_n$

where $X_i^{(n)}$ are i.i.d. with offspring distribution, and %%%%1%%%% are i.i.d. immigration counts (Han et al., 2018). In continuous time, a continuous-state branching process with immigration (CBI) has the Laplace semigroup

$$E_x \left[ e^{-\lambda Y_t} \right] = \exp\left( - x v_t(\lambda) - \int_0^t \phi(v_s(\lambda)) ds \right)$$

where $v_t$ satisfies $\partial_t v_t = -\psi(v_t)$ (Foucart et al., 2012).

The total population at any time can be represented as the superposition of the descendants of all immigrants and initial ancestors. For a GWI,

$$Z_n = \sum_{\ell=0}^{n} \sum_{j=1}^{Y_{\ell}} \xi_{\ell, j}(n-\ell)$$

where $\xi_{\ell, j}(n-\ell)$ is the size at generation $n$ of the $j$-th immigrant arriving at time $\ell$ (Han et al., 2018). For CBI, the backbone decomposition is analogous, often framed via Poisson random measures governing immigration epochs (Bi et al., 2013).

2. Branching, Immigration, and Genealogy in CSBP/CBI

CBI processes are characterized by Lévy–Khintchine branching and immigration mechanisms: $$\psi(\lambda) = b \lambda + c \lambda^2 + \int_{0}^{\infty} [e^{-\lambda x} - 1 + \lambda x \mathbf{1}_{x<1}] \Pi(dx)$$

$$\phi(\lambda) = \beta \lambda + \int_{0}^{\infty} [1 - e^{-\lambda x}] \nu(dx)$$

where $\psi$ governs offspring reproduction and $\phi$ the arrival of immigrants (Foucart et al., 2012). The genealogical tree of stationary CSBP with immigration under a subcritical stable mechanism ($\psi(\lambda) = \alpha \lambda + \gamma \lambda^b$, $1 < b \leq 2$) is, up to deterministic time-change, distributed as a continuous-time Galton–Watson process with immigration (GWI) (Abraham et al., 2020).

Backward-in-time, the process counting ancestors is a time-inhomogeneous pure-death Markov chain with explicit transition rates derived via the derivatives of $\psi$ and the extinction probability $c(t)$: $$q_{n,m}^{\mathrm{d}}(t) = \binom{n+1}{m} \frac{|\bar u^{(m)}(c(t))|}{|\bar u^{(n)}(c(t))|} |\psi^{(n-m+1)}(c(t))|$$ and similarly for the forward-in-time pure-birth chain (Abraham et al., 2020).

For general subcritical $\psi$, the pure birth/death chain structure persists, underpinning the full ancestral reconstruction.

3. Lamperti-Type Representations and Scaling Limits

The Lamperti representation gives a construction of CBI processes by time-changing a pair of independent Lévy processes for branching and immigration: $Z_t = x + X_{ \int_0^t Z_s\,ds } + Y_t$ where $X$ is spectrally positive Lévy ($\psi$), $Y$ is a subordinator ($\phi$) (Caballero et al., 2010). This representation provides unique pathwise solutions, generator formulas, and scaling limits, showing that rescaled Galton–Watson processes with immigration converge in law to CBI processes as population size diverges.

Pitman and others extended this to critical processes conditioned on total population size, resulting in invariance principles for breadth-first walks and stable bridges (Caballero et al., 2010).

4. Cluster Structure, Superposition, and Coalescent Connections

The immigration-birth representation is central in understanding the cluster (family) structure in models with immigration. Each immigrant results in an independent sub-family (cluster), and the full process is the superposition of these clusters.

In age-dependent or multi-type branching processes (e.g., CMJ processes with Poissonian immigration), the total population is the sum over all immigrant-founded families (Richard, 2010). Structured types, via discrete or continuous spectra, permit limit theorems for the relative abundances of surviving families, including convergence to Dirichlet or GEM distributions (Richard, 2010). The backbone approach directly generalizes to stable CSBP genealogy and links to Beta-coalescents through time-change and measure-valued flows (Foucart et al., 2012).

Superprocess and nonlocal branching constructions embed these flows into measure-valued or path-valued objects that generalize Aldous–Pitman or Abraham–Delmas tree-valued dynamics (Li, 2012).

5. Markovian, Spatial, and Random Environment Extensions

Branching and immigration-birth representations extend to spatial processes, random environments, and more general Markovian settings. In spatial branching random walks with immigration, the total field at any site is decomposed into the sum of families from initial configuration and those stemming from immigrants arriving at each site, preserving independent family structure (2002.03999).

In random environments, one-immigrant-per-generation processes permit the analysis of the probability that all descendants at time $n$ stem from a single immigrant (the MRCA), with asymptotic probabilities determined by conditioned random walk path fluctuations (Smadi et al., 2021, Smadi et al., 2019).

Age-structured and resource-dependent models further formulate the process via stochastic equations, measure-valued SDEs, or combinatorial selection schemes, while maintaining backbone (immigration–birth cluster) structure (Ji et al., 2020, Bruss, 2018).

6. Analytical, Statistical, and Simulation Implications

The branching and immigration-birth viewpoint yields explicit formulas for generating functions, moments, and limiting distributions of population size and family sizes. It underpins simulation schemes, such as Euler Approximation for continuous-state processes (Caballero et al., 2010), and inference of invariants such as TMRCA, bottleneck effects, and genealogical properties (Bi et al., 2013, Foucart et al., 2012).

Statistical applications encompass species abundance, population sampling, and multi-sample problems (e.g., Fisher’s variance calculation), facilitated by Poisson marking and continuous-time embedding, and linked to classical results such as Ewens Sampling Formula and the Chinese Restaurant Process (Silva et al., 2022).

7. Connections, Limitations, and Theoretical Significance

The backbone representation is pervasive across models—from simple GWI to stable CSBP, spatial, age- or type-structured, superprocess, and random-environment models. Limit theorems, equilibrium calculations, duality formulas, and tree-structural links derive from this representation.

The existence and tractability of the framework depend on subcriticality, Grey’s condition, and parameter constraints (e.g., finite immigration rate, admissibility of offspring distributions), and certain global properties (ergodicity, stationarity) may break down in supercritical or infinite population scenarios (Abraham et al., 2020, Foucart et al., 2012). In critical and supercritical regimes, restriction to surviving clusters or immigration windows may be required.

The branching and immigration-birth representation provides a unifying lens for understanding the stochastic microstructure, genealogical dynamics, and macroscopic laws of populations under reproduction and external influx. Its analytical formalism enables rigorous treatment across a spectrum of mathematical biology, probability theory, and stochastic processes.