λ-Mixed Moran Process

• λ-Mixed Moran process is a hybrid evolutionary model that interpolates between different updating mechanisms to capture dynamics in varied and structured environments.
• On graphs, it combines Birth-death and death-Birth steps via a convex mixture to yield explicit fixation probabilities and polynomial absorption times.
• Extensions include environmental switching, heterogeneous resampling, and jump-diffusion limits, offering key insights into evolutionary game theory and population genetics.

The $\lambda$-mixed Moran process is an umbrella term for a spectrum of evolutionary models that interpolate between different Moran process variants using a mixing parameter $\lambda$ to blend updating mechanisms or environmental regimes. These models capture diverse phenomena in structured, heterogeneous, or fluctuating environments, and have substantial implications in evolutionary game theory, population genetics, and ecological persistence. Contemporary research has formalized several mathematically distinct but conceptually related processes under this terminology, covering stochastic processes on graphs, random environments, heterogeneous resampling rates, and models bridging standard Wright-Fisher and Bolthausen–Sznitman genealogies.

1. $\lambda$-Mixed Moran Process on Graphs

A rigorous formulation of the $\lambda$-mixed Moran process for evolutionary dynamics on undirected graphs was introduced as a convex combination of the classical Birth-death (Bd) and death-Birth (dB) Moran steps, mediated by parameter $\lambda\in[0,1]$ (Brewster et al., 23 Nov 2025). Let $G=(V,E)$ be an undirected connected graph with $n=|V|$ vertices. Each vertex harbors an individual of either resident (fitness 1) or mutant (fitness $r>0$) type; $S\subseteq V$ denotes the mutant set.

At each discrete time step:

• With probability $\lambda$, a Bd move: select a reproducer $u$ with fitness-weighted probability, then select a neighbor $v$ of $u$ uniformly to be replaced.
• With probability $1-\lambda$, a dB move: select a death site $v$ uniformly, then select a reproducing neighbor $u$ (of $v$) with fitness-weighted probability.

The transition kernel $P_\lambda(S\to S')$ is a linear combination of the respective Bd and dB transition rates, yielding a finite absorbing Markov chain on $2^{V}$. Absorbing states correspond to fixation ($S=V$) or extinction ($S=\emptyset$).

Key results include:

• For $\lambda=1/2$ (uniform mixture), in the neutral case ($r=1$), the fixation probability from any initial $S_0$ is $|S_0|/n$, mirroring the symmetry of the process.
• The expected fixation (absorption) time satisfies $E[T]=O_r(n^4)$, with polynomial-time approximation algorithms available for key statistics in almost-regular, random, and bidegreed graphs.

Specializations to random graphs $G(n,p)$ and explicit formulas for cycles and stars were developed, illustrating the analytical tractability of the mixed process in regimes that classic Moran models do not directly address (Brewster et al., 23 Nov 2025).

2. Moran Process with Environmental Switching: $\lambda$ as Markov Jump Rate

Another $\lambda$-mixed formulation considers a population of $N$ individuals, $K+1$ types, and a finite set of environmental regimes indexed by a finite set $E$, each specifying a fitness vector (Guillin et al., 2019). The environment process $(s^{(N)}_n)$ is a Markov chain on $E$ with scaled transition rates such that

$$\lambda_{ij} = \lim_{N\to\infty} N P^N_{i,j}\,, \quad i\neq j.$$

At each event, a standard Moran step is performed, but the replacement probabilities depend on the current environmental fitness vector $s\in E$. In the large-$N$ limit, the process converges to a Piecewise Deterministic Markov Process (PDMP) on type frequencies and environmental state, with deterministic ODE segments per environment and random jumps (with rates $\lambda_{ij}$) between environmental states. The generator for smooth $f$ is:

$$L f(x,i) = \nabla_x f(x,i) \cdot b^{(s_i)}(x) + \sum_{j\neq i} \lambda_{ij} [f(x,j) - f(x,i)],$$

where $b^{(s_i)}(x)$ encodes the replicator-like vector field based on environmental fitnesses.

Persistence theory applies: a system can exhibit stochastic persistence (coexistence of all types) if the weighted sum of invasion rates

$\sum_{i=1}^{K+1} c^i \lambda^i(\mu) > 0$

for suitable weights $c^i>0$ and any ergodic measure $\mu$ on the extinction set (Guillin et al., 2019). This framework yields explicit coexistence or extinction phase diagrams in low-dimensional cases and connects to stochastic Lyapunov method arguments.

3. Moran Model with Random Heterogeneous Resampling Rates

The $\lambda$-mixed designation is also used in the context of the Moran model with random resampling rates (Athreya et al., 2024). Here, each individual $i$ is assigned a fixed Poisson resampling rate $R_i\sim \lambda$ (where $\lambda$ is a discrete probability law with countable support), and at each event, type resampling proceeds according to these heterogeneous rates. Denoting $Y^N(t)$ as the empirical measure over resampling rates among type-1 individuals at (rescaled) time $Nt$, it is shown that, under suitable conditions on $\lambda$, $Y^N$ converges to a process $Y(t)=S(t)\lambda$ where $S(t)$ is a scalar Fisher-Wright diffusion:

$$dS(t) = \sqrt{D\,S(t)(1-S(t))}\,dW(t), \quad D^{-1} = \int \frac{1}{r} \lambda(dr).$$

The only effect of rate-heterogeneity in the large-$N$ limit is to alter the genetic drift timescale via the effective diffusion constant $D$.

4. Strong Selection, $\Lambda$-Coalescents, and Jump-Diffusion Limits

Under regimes of strong selection, a $\lambda$-mixed Moran process can interpolate between the classical Wright–Fisher diffusion (Kingman coalescent limit) and jump-driven $\Lambda$-Wright–Fisher limits (Bolthausen–Sznitman coalescent) (Ged, 2020). In these models, type $j$ (fitness advantage $s_N(j-M(t))$) accrues beneficial mutations and undergoes rare, massive sweeps, while a weak selection parameter $\alpha$ operates on an overlaid X/Y-allele label.

The scaling limit for allele frequency $V_N$ of the disadvantaged group (Y) converges to the SDE

$$dV_t = -\alpha V_t(1-V_t)\,dt + \int_0^1\int_0^1 z\big[1_{\{u\le V_{t-}\}}-V_{t-}\big]\widetilde{M}(dt,dz,du),$$

with $\widetilde{M}$ a compensated Poisson random measure of intensity $dt\,dz\,du/z^2$ (i.e., $\Lambda(dz)=dz$ on $(0,1]$). This process bridges processes dominated by diffusive noise (Kingman, $\Lambda=\delta_0$) and those dominated by large, simultaneous mergers (Bolthausen–Sznitman, $\Lambda$ uniform).

5. Analytical and Computational Results

The $\lambda$-mixed Moran process on graphs admits explicit linear systems for fixation probabilities, with boundary conditions $\varphi(\emptyset)=0$, $\varphi(V)=1$ (Brewster et al., 23 Nov 2025). For bidegreed graphs, fixation probabilities admit closed-form formulas as degree-weighted averages, and expected absorption times scale as $O_r(n^4\alpha^2)$ with $\alpha=d_2/d_1$. For cycles and stars, the process reduces to tractable birth–death chains or low-dimensional recursions, and for $\lambda=1/2$ and neutral fitness ($r=1$), the process is exactly symmetric, guaranteeing fixation probability $1/n$ for any starting vertex. Polynomial-time (FPRAS) algorithms are available for large classes of almost-regular and random graphs.

For the PDMP setting with environmental switching, invasion rates and Lyapunov function techniques yield explicit criteria for long-term coexistence versus extinction, with phase diagrams accessible in low-dimensional settings. Numerical and analytical methods provide explicit stationary distributions in special cases (Guillin et al., 2019).

6. Biological and Mathematical Implications

The $\lambda$-mixed Moran process provides unified frameworks to interpolate between classic evolutionary assumptions:

• In graph settings, it formalizes mixtures of local replacement rules (Bd vs dB), capturing more general evolutionary update protocols and robustness of fixation results (Brewster et al., 23 Nov 2025).
• In random environments, it models environmental stochasticity and demonstrates how frequent switching can facilitate coexistence even in the absence of a consistently favored type (Guillin et al., 2019).
• With rate heterogeneity, it quantifies the effects of individual-level disorder on population-level genetic drift, showing that heterogeneous rates slow drift relative to homogeneous populations (Athreya et al., 2024).
• The combination of strong selection, mutation-accumulation, and weak selection yields genealogical structures that deviate sharply from the Kingman paradigm, manifesting in coalescent processes with multiple mergers and jump-diffusions (Ged, 2020).

This breadth of modeling capacity positions the $\lambda$-mixed Moran process as a central construct in stochastic evolutionary theory, with direct applications to ecology, population genetics, evolutionary graph theory, and beyond.