Dissimilarity measures for generalized Lotka-Volterra systems on networks

TL;DR

This work introduces a general framework to quantify dissimilarities between generalized Lotka–Volterra dynamics on networks, unifying comparisons across parameter changes, network topology, and even altered nonlinearities. It defines precise, computable measures—rooted in both transient and stationary behavior—such as $\mathcal{D}^{(\mathrm{LV})}(t)$ and $\mathcal{D}_{\mathrm{max}}$—to compare reference and modified systems. Through analyses of two-species LV, small three-node networks, and larger modular networks with cliques, the study demonstrates that small structural or nonlinear changes can produce substantial dynamical differences, including potential instabilities. The framework offers a practical tool for robustness assessment and structural sensitivity detection in ecological, microbiome, and neuroscience contexts, enabling systematic comparisons of nonlinear population dynamics on complex networks.

Abstract

In this paper, we introduce a general framework to quantify dissimilarities between generalized Lotka-Volterra dynamical processes, ranging from classical predator-prey systems to multispecies communities interacting on networks. The proposed measures capture both transient and stationary dynamics, allowing systematic comparisons across systems with varying interaction parameters, network weights, or topologies. Our analysis shows that even subtle structural changes can lead to markedly distinct outcomes: in two-species systems, interaction strength and initial conditions strongly affect divergence, while in small directed networks, differences that are invisible at the adjacency-matrix level produce divergent dynamics. In modular networks, the fraction and distribution of negative interactions control the transition from stable to unstable dynamics, with localized perturbations within cliques yielding different global outcomes than distributed ones. Beyond structural variations, the framework also applies when modified processes follow distinct nonlinear equations, demonstrating its versatility. Taken together, these results highlight that dynamical dissimilarity measures provide a powerful tool to analyze robustness, detect structural sensitivity, and predict instabilities in nonlinear systems. More broadly, this approach supports the comparative analysis of biological systems, where complex interaction networks and nonlinear dynamics are central to stability and resilience.

Figures (7)

• Figure 1: Dissimilarity in the dynamics of prey and predator populations using a reference system with $\alpha = 1$. (a) Functions $u(\tau)$ (original process) and $u^\star(\tau)$ (modified process with $\alpha^\star = 1.2$) associated to preys as a function of $\tau$, with $u_{0} = u_{0}^{\star} = 2$, panel (b) shows the same result for predators $v(\tau)$ and $v^\star(\tau)$. (c) Dissimilarity metric $\mathcal{D}^{(\mathrm{LV})}(\tau)$ in Eq. (\ref{['eq:D(t)']}) for the two systems in (a)–(b), $\{(u,v), (u^{\star}, v^{\star})\}$, as a function of time. (d) Maximum values $\mathcal{D}^{(\mathrm{LV})}_{\mathrm{max}}(\tau)$; the time $\tau = 12.741$ corresponds to the first moment when the two systems exhibit a $50\%$ difference with respect to the initial conditions. (e) Time $T_{50\%}$ required for two systems (reference–modified) to differ by $50\%$ as a function of $\alpha^\star$, for different values of $u_0 = u(0)$ (indicated by the colors of each line).
• Figure 2: Comparison of two gLV dynamical processes in a graph with three nodes. (a) Reference case: a complete graph with uniform interactions, $\Lambda_{ij} = 1$. (b) Modified graph: a single interaction is altered with $\Lambda^\star_{ab} = 1 - \beta$ for the link $a \to b$, where $\beta \geq 0$; all other entries $\Lambda^\star_{ij}$ remain as in the reference. (c)-(e) Temporal evolution of the dissimilarity $\mathcal{D}^{(\mathrm{LV})}(t)$ between the dynamics on the reference graph (a) and the modified graph for $\beta = 0.5$, $1.0$, and $1.5$, respectively. The curves correspond to 100 realizations with random initial conditions satisfying $\mathcal{N}_0 = 1$. (f) Probability density $\rho(\mathcal{D}_{\mathrm{max}})$ of the maximum dissimilarity $\mathcal{D}_{\mathrm{max}}=\mathcal{D}_{\mathrm{max}}(\mathbf{\Lambda}, \mathbf{\Lambda}^\star)$ between the two structures for different values of $\beta$ and values $\mathcal{N}_0$ for initial populations. The results in (f) correspond to $\mathcal{N}_0 = 1$, using random initial conditions and $10^4$ realizations of the process. The same analysis is repeated in (g) for $\mathcal{N}_0 = 3$, and in (h) for $\mathcal{N}_0 = 5$.
• Figure 3: Comparison of gLV dynamical processes for all non-isomorphic connected directed graphs of size $N = 3$. The thick continuous lines show the ensemble average $\left\langle \mathcal{D}^{(\mathrm{LV})}(t) \right\rangle$ as a function of time $t$, computed from $10^4$ realizations with random initial conditions fulfilling $\mathcal{N}_0 = 1$. Results are arranged in panels where row $\mu$ and column $\nu$ correspond to the comparison between the dynamics defined by the graph $\mathcal{G}_\mu$ and the graph $\mathcal{G}_\nu$. For each time $t$, the shaded region around the average represents the standard deviation of $\mathcal{D}^{(\mathrm{LV})}(t)$ obtained for the different realizations. The colors, encoded in the colorbar, represent the maximum value $\left\langle \mathcal{D}\right\rangle_\mathrm{max}$ of the average results.
• Figure 4: Statistical analysis of $\mathcal{D}_\infty$ for the comparison of gLV dynamical processes on graphs of size $N = 3$. The panels show the probability density $\rho(x)$ of the rescaled variable $x = \mathcal{N}_0 \mathcal{D}_\infty$, obtained from comparisons between the dynamics of pairs of graphs $\mathcal{G}_1, \mathcal{G}_2, \ldots, \mathcal{G}_5$. Results are obtained using $10^4$ realizations of random initial conditions for $\mathcal{N}_0 = 1$, $2$, $3$, and $4$.
• Figure 5: Network with three cliques with a total of $N = 12$ nodes. The connection weights $\Lambda_{ij}$ are chosen randomly with a uniform distribution taking values between 0 and 1 as indicated in panel (a). In this structure, the sign is randomly changed, where $p$ is the probability of maintaining a positive sign and $1-p$ of changing to negative. The cases explored are: (a) $p=1.0$, (b) $p=0.8$, (c) $p=0.5$, (d) $p=0.0$. (e) Dissimilarity $\langle \mathcal{D}_{\mathrm{max}}(\mathbf{\Lambda}, \mathbf{\Lambda}^\star) \rangle$ as a function of the probability $1-p$ of sign change. The ensemble average $\langle \mathcal{D}_{\mathrm{max}}(\mathbf{\Lambda}, \mathbf{\Lambda}^\star) \rangle$ is calculated using 200 realizations for each value of $p$ and represented with a circle. The dashed line is a moving average that serves as a guide to observe the trend of the obtained values. The inset shows the results in semilogarithmic scale and the error bars represent the standard deviation of the data.