Continuous approximations for the fixation probability of the Moran processes on star graphs

TL;DR

This work develops continuous, ODE-based approximations for fixation probabilities of Moran processes on star graphs under weak selection with general frequency-dependent birth and death fitnesses for both BD and DB updates. A rigorous O(1/N) error bound is established for the DB process, while the BD case yields a BD-complete-graph-like approximation with a structural adjustment due to the star geometry. The authors also analyze invasion probabilities, providing threshold results that delineate amplifier, suppressor, and preserver regimes depending on initial mutant placement and fitness functions, including linear fitness from 2-player games. Numerical experiments confirm high accuracy of the approximations for moderate to large N across diverse fitness landscapes, underscoring the method’s broad applicability to structured populations and non-linear fitness. The results illuminate how population structure and update rules jointly shape evolutionary outcomes, with potential implications for cancer evolution and other spatially constrained systems.

Abstract

We consider a generalized version of the birth-death (BD) and death-birth (DB) processes introduced by Kaveh, Komarova, and Kohandel (2015), in which two constant fitnesses, one for birth and the other for death, describe the selection mechanism of the population. Rather than constant fitnesses, in this paper we consider more general frequency-dependent fitness functions (allowing any smooth functions) under the weak-selection regime. A particular case arises in evolutionary games on graphs, where the fitness functions are linear combinations of the frequencies of types. For a large population structured as a star graph, we provide approximations for the fixation probability which are solutions of certain ODEs (or systems of ODEs). For the DB case, we prove that our approximation has an error of order $1/N$, where $N$ is the size of the population. The general BD and DB processes contain, as special cases, the BD-* and DB-* (where * can be either B or D) processes described in Hadjichrysanthou, Broom, and Rychtář (2011) -- this class includes many examples of update rules used in the literature. Our analysis shows how the star graph may act as an amplifier, suppressor, or remains isothermal depending on the scaling of the initial mutant placement. We identify an analytical threshold for this transition and illustrate it through applications to evolutionary games, which further highlight asymmetric structural effects across different game types. Numerical examples show that our fixation probability approximations remain accurate even for moderate population sizes and across a wide range of frequency-dependent fitness functions, extending well beyond previously studied linear cases derived from evolutionary games, or constant fitness scenarios.

Figures (11)

• Figure 1: The approximate fixation probability $f(x)$, where $x$ represents the proportion of individuals of type $B$ living at the leaves, is plotted for three distinct cases in the BD process for the population size $N=100$. In the first case (shown in red), the birth fitness function of $B$ is $\varphi_1(x) = 1 + \frac{\psi_1(x)}{N}$, where $\psi_1(x) = 10(x-0.5)$, and the death fitness function of $B$ is $\varphi_2(x) = 1 + \frac{\psi_2(x)}{N}$, where $\psi_2(x) = x+1$. In the second case (shown in blue), the birth fitness function of $B$ is $\tilde{\varphi}_1(x) = 1 -\frac{\psi_2(x)}{N}$, and the death fitness function of $B$ is $\tilde{\varphi}_2(x) = 1 - \frac{\psi_1(x)}{N}$. Finally, in the third case (shown in green), which is equivalent to the first case, the birth fitness function of type $B$ is $\bar{\varphi}_1(x) = 1 -\frac{1}{2} \frac{\psi_2(x)}{N}$, and the death fitness function of type $B$ is $\bar{\varphi}_2(x) = 1 -2 \frac{\psi_1(x)}{N}$.
• Figure 2: Approximations for the fixation probability in the star graph, in blue, and in the complete graph in orange. The fixation probability for the neutral case is given in green. On the left: $N=1000$, $\kappa^{-1}=10$, $\psi_1(x)=(x-1.5)$ and $\psi_2=0$; $A$ dominates. On the right: $N=1000$, $\kappa^{-1}=10$, $\psi_1(x)=(x+0.5)$ and $\psi_2=0$; $B$ dominates.
• Figure 3: Approximations for the fixation probability in the star graph, in blue, and in the complete graph in orange, for the BD process. The fixation probability for the neutral case is given in green. On the left: $N=1000$, $\kappa^{-1}=10$, $\psi_1(x)=(0.5-x)$ and $\psi_2=0$; we have the coexistence game. On the right: $N=1000$, $\kappa^{-1}=10$, $\psi_1(x)=(x-0.5)$ and $\psi_2=0$; we have the coordination game.
• Figure 4: Limit of the ratio of invasion probabilities $\beta=\lim_{\bar{z} \to 0}\phi_{2,\rho} / \phi_1$ in the DB process for $\psi_1(x) - \psi_2(x) = r$, with $\rho_N = \alpha / N$, over the parameter ranges $-3 \leq r \leq 3$ and $0 \leq \alpha \leq 5$. (a) 3D surface plot of the limit ratio $\beta$ in orange. (b) Contour plot of $\beta^\tau$, for $r\neq 0$, where the color scale indicates whether the star graph acts as an amplifier (red) or a suppressor (blue) relative to the complete graph. The dashed line correspond to $\beta=1$.
• Figure 5: Contour plot of $\beta^\tau$ for the DB process, where the fitness functions are derived from a two-player game with weak selection. We consider $\psi_1(x) = \gamma(x - x^*)$, $\psi_2(x) = 0$ and $\rho_N=1/N$, a parametrization that captures all classical game types (dominance, coexistence, and coordination) arising from $2 \times 2$ payoff matrices. The domain spans $-2 < \gamma < 2$ and $-2 < x^* < 2$, and the color scale indicates whether the star graph acts as an amplifier (red) and suppressor (blue) relative to the complete graph.

Theorems & Definitions (23)

• Theorem 1
• Remark 1
• Definition 1
• Proposition 1
• proof
• Remark 2
• Remark 3
• Theorem 2
• Remark 4
• Remark 5