arXane › Quantitative Biology › Quantitative Biology
Wednesday, 4 November 2026: 2 new papers, 1 cross-list and 2 replacements in Populations and Evolution (q-bio.PE), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Reticulate processes such as hybridization, introgression, and horizontal gene transfer cannot be fully represented by a bifurcating tree. Enter phylogenetic networks: first as split graphs to visualize tree discordance, then as explicit probabilistic models that capture biological phenomena. Here, we describe the broad taxonomy of network representations, distinguishing the principal classes of explicit networks, their biological interpretability and our ability to accurately estimate them from empirical data. We also trace the evolution of the main network inferential methods from hybrid detection tests, distance- and subgraph-based amalgamation methods, probabilistic approaches under the multispecies network coalescent, composite-likelihood and divide-and-conquer frameworks, while highlighting the selective pressures of statistical identifiability and computational scalability that have shaped this evolution. As we move towards a network thinking paradigm, previously isolated methodological lineages from population genetics, phylogenomics, and mathematical network theory are now introgressing, uniting diverse network models into a shared framework that can integrate sequence- and species-level reticulate processes, increase robustness to systematic errors, and refine algorithms for genome-scale data, expanding the tree of life into a richer, more entangled yet clearer picture of evolution.
The basic reproductive number, $\mathcal{R}_0$, is widely used to determine whether an infectious disease can persist in a population. However, its interpretation may become unclear when the underlying disease-free demographic dynamics exhibit multiple types of stable attractors. In this study, we investigate how demographic bistability between fixed points and periodic cycles affects the calculation and interpretation of $\mathcal{R}_0$. We consider two discrete-time host-parasite models with infection: a tick-host model motivated by Borrelia burgdorferi transmission and a Nicholson-Bailey host-parasitoid model with intraspecific competition and infection. For each model, we calculate reproductive numbers associated with different disease-free attractors using the next-generation operator method and compare their invasion thresholds. In the tick-host model, the reproduction number calculated on the stable demographic cycle differs from that calculated at the positive coexistence fixed point, resulting in parameter regions where infection can invade at one demographic attractor but not the other. In contrast, in the host-parasitoid model, the host-driven invasion threshold is necessarily the same at the positive coexistence fixed point and on the host-only periodic cycles. These contrasting results demonstrate that demographic bistability alone does not necessarily produce attractor-dependent invasion thresholds. Rather, the effect of demographic dynamics on invasion depends on the structure of pathogen transmission. In particular, a complete host--vector--host transmission pathway can allow differences between demographic attractors to alter the invasion threshold. Our results emphasize the importance of considering both demographic dynamics and transmission structure when interpreting $\mathcal{R}_0$ in populations with multiple stable disease-free states.
Computing phylogenetic likelihoods becomes challenging when the continuous-time Markov chain has a large state space. At each branch, pruning algorithms require terms of the form $e^{Q t}\cdot B$, where $Q$ is an $S \times S$ rate matrix (e.g., for $S$ states), $t$ is a branch length, and the columns of matrix $B$ (e.g., $B \in \mathbb{R}_{+}^{S \times L}$) are partial-likelihood vectors for $L$ site patterns. Many implementations calculate the dense transition matrix $P(t)=e^{Q t}$ via eigendecomposition and multiply it by $B$. This strategy is efficient for small state spaces, but it can be inefficient for large sparse rate matrices because forming $P(t)$ or its eigen vector matrices destroys sparsity and its cost may not be well amortized when $L$ is small. We implement an alternative likelihood kernel based on the Al-Mohy and Higham (2011) algorithm for computing the action of the matrix exponential directly, avoiding explicit construction of $P(t)$ or its eigen vector matrices. The resulting method takes full advantage of the sparsity of $Q$. We implement this method as a CPU-based extension to the BEAGLE library and evaluate it on two large-state molecular models: a Markov-modulated codon model of heterotachy with $M$ selective regimes and $61\cdot M$ possible states, and a codon-pair model of interlocus gene conversion between two paralogs with $61^2=3\,721$ possible states. Our results characterize when action-based likelihood computation improves performance over calculating $e^{Q t}$ directly. We show that this method enables fast, scalable inference for complex models, removing a key computational barrier to the study of evolution with large state space models.