Understanding the invader-driven replicator dynamics

TL;DR

The paper develops a comprehensive analysis of an invader-driven replicator system in which each species has a fixed invasion fitness against any resident. It derives explicit steady-state characterizations, proving a unique interior equilibrium when positive traits enable coexistence and identifying a threshold condition $\lambda_{k+1}<Q_k^*<\lambda_k$ that determines the surviving subset $E_k=\{1,...,k\}$. For random uniform traits, it provides exact and asymptotic results for the distribution and mean number of coexisting species, showing $\mathbb{E}[n]\sim\sqrt{2N}$ as the pool size $N$ grows, and analyzes sequential assembly and niche saturation with explicit invasion-outcome criteria. The work connects the invader-driven replicator to rank-one Lotka-Volterra models and to multi-strain SIS coinfection systems, and outlines empirical tests for the invader-driven hypothesis using cross-site serotype data, highlighting both the potential and the limitations of applying these mathematical principles to real ecosystems. Overall, the framework offers precise coexistence, invasibility, and assembly insights in a tractable, analytically solvable setting with broad ecological and epidemiological relevance.

Abstract

In this paper, we study a special case of the invasion fitness matrix in a replicator equation: the invader-driven case. In this replicator, each species is defined by its unique active invasiveness potential (initial growth rate when rare), upon invading any other species, independently of the partner. We derive explicit expressions and theorems to fully characterize the steady-states of this system, including its unique interior coexistence regime, reached for positive species traits, or alternative boundary exclusion states, reached for negative species traits. We study the internal stability of coexistence steady-states, and the system's stability to outsider invasion, relevant for system assembly. We provide detailed analytical results for critical diversity thresholds, and for the special case of random uniform species traits, we analytically compute the probability of stable $k-$species coexistence in a random pool of size $N$, and show that the mean number of co-existing species can be approximated as $\mathbb{E}[n] \sim \sqrt{2N}$. We also derive explicit mathematical conditions for invader traits and invasion outcomes (augmentation, rejection, and replacement), dependent on the history of system assembly. Finally, by outlining links of this replicator case with corresponding (rank-1) Lotka-Volterra ecological systems and specific epidemiological multi-strain SIS models with coinfection, we highlight the relevance of applying these mathematical principles to improve the theoretical and empirical understanding of multi-species coexistence.

Figures (11)

• Figure 1: Illustration of the invader-driven replicator dynamics. We present simulations in three panels for an invader-driven dynamics of 12 species. (a) Invasion fitness matrix $\Lambda = (\lambda_{i}^j)$, with $\lambda_{i}^j \sim \mathcal{U}\left[-1,1\right]$ for all $1\leq i,j \leq 12$ and $\lambda^i_i = 0$ for all $1\leq i\leq 12$. (b) Replicator dynamics from an equal initial condition for all species, showing convergence to a coexistence equilibrium with a subset of speciess persisting. (c) Linearity between $\lambda_i$ and $\left(1 - z_i^*\right)^{-1}$ with all persistent species $i$. The surviving species (here, species $10$, $11$, $12$$4$, $8$, $9$ and $5$) are marked by red with their identities.
• Figure 2: Invader-driven system assembly. Diagram of the ecological assembly mechanism of $k$ species in an invader-driven system, with $0<\lambda_i<\lambda_{i-1}<\dots<\lambda_1$ and $i=1..N$. As more species are added to the system, the mean invasion resistance $Q^*$ increases and a natural threshold emerges below which less fit species cannot coexist anymore in the invader-driven replicator.
• Figure 3: Coexistence subsets $E$ and global mean fitness $Q$ at invader-driven replicator equilibria. We examine all the feasible steady-states of $50$ different invader-driven systems with $\lambda_i\sim \mathcal{U}\,[0,1]$$\forall\,i\in S$. The top heatmap shows $Q$ for all the equilibria subsets $E$ (rows) in each system (columns), colored by magnitude and marked with a black star if the equilibrium is linearly stable (computed numerically). Subsets are ordered from bottom to top by increasing $k$. The bottom panels show, for one selected system (highlighted in the grid) with $\lambda_1 = 1 \ ,\ \lambda_2 = 0.674 \ ,\ \lambda_3 = 0.536\ ,\ \lambda_4= 0.342$, on the left a scatter plot of $Q$ for the different equilibria of the system, and on the right the time evolution of the frequencies $z_i(t)$, $i\in S$ and the mean fitness $Q(t)$. In the scatter plot, the unstable equilibria are red dots, and the stable one, which corresponds with the maximum value of $Q$ within the system, is green. Moreover, the green label $\{1,\,2,\,3\}$ indicates the result from the particular dynamics simulation.
• Figure 4: Distribution of the number of coexisting species in the invader-driven replicator, for different species pool size $N$. Summary of 1000 simulations for each $N$ between 2 and 10.
• Figure 5: Probability of $n-$ species coexistence in a pool of $N$ species (here $N=10$). We plot the probability mass function of the number of coexisting species at the unique stable equilibrium of the invader-driven replicator \ref{['eq:repli_invader']} with $N = 10$ and i.i.d. invader fitnesses $\lambda_i \sim \mathcal{U}[0,1]$. Blue bars: empirical frequencies from 10,000 independent ODE runs started in the simplex interior (survival counted when $z_i(T) > 10^{-4}$). Orange bars: probabilities obtained by Monte-Carlo evaluation of the Theorem \ref{['thm:prob']} integral formula, reported with 95$\%$ confidence intervals (as listed in the Supplementary material S2). The two estimates yield similar probabilities of the target events. For a similar figure for $N=20$ see Figure \ref{['fig:20strains_probs']}.

Theorems & Definitions (31)

• Proposition 1
• proof
• Remark 2
• Remark 3
• Proposition 4
• proof
• Theorem 5
• proof
• Remark 6
• Proposition 7