Cultural traits, unlike genes, are acquired sequentially across and within individuals' lifetimes through repeated learning, innovation, and loss. While many tractable neutral models of cultural evolution assume one-shot acquisition, in which an individual's repertoire is fixed when it enters the population, we study a neutral model in which long-lived individuals in a fixed population repeatedly copy traits from one another and occasionally forget them, so that cultural turnover is driven by learning and forgetting within lifetimes rather than by births and deaths. The process admits a unique limiting distribution, with a stationary trait popularity spectrum in closed form. When the social learning efficiency is subcritical, the spectrum is monotone and total cultural diversity grows linearly with population size. Above the critical value the spectrum concentrates around an intermediate frequency and diversity grows exponentially with population size. Our main result concerns this supercritical regime: we prove that the spectrum is asymptotically Gaussian, with explicit mean, width, height and error bounds, and from this we derive the asymptotic behaviour of cultural diversity at the population and individual levels and of the time to stationarity. The same spectrum arises in the group-level model of Nakamura et al. (2020), so the shape of a spectrum does not by itself identify the mechanism behind it. Because the supercritical regime arises whenever individuals get more than one chance to acquire a trait, intermediate-peaked popularity distributions and abundant shared culture need not by themselves be read as evidence of biased or selective transmission.
Pairwise coalescence times are of fundamental importance in population genetic theory and inference. However, computing expectations and related quantities under the structured coalescent remains challenging. Here, we analyze two commonly used approximation strategies — the time-decomposition approach and the lineage-independence assumption — and characterize their underlying theoretical assumptions and computational properties. We also summarize recent algorithmic advances that relate the problem to the generalized Lyapunov equation. These methods enable exact computation with reduced complexity, and we further extend them to more general model settings. Finally, we discuss the potential role of iterative solvers in accelerating inference. Together, these insights clarify the computational landscape and provide a foundation for more flexible and accurate inference in structured populations.
Matrix projection models have long served as powerful tools for tracking populations and their structure over time. One of the main appeals of projection matrices is their simplicity, which arises from grouping individuals into classes that are assumed to share identical vital rates (e.g., mortality, birth rates). In natural populations, however, there are many traits that influence vital rates, resulting in a trade-off between accurately capturing this heterogeneity and minimizing model complexity. Here, we prove two theorems about when and how projection matrices for populations structured by both age and phenotypic classes can be collapsed to smaller and more easily evaluated matrices that generate, or allow recovery of, the original abundance dynamics after at most one maximum lifespan. We refer to such collapsed matrices as “dynamically sufficient” matrices. The proof identifies two kinds of redundant variables, one arising from phenotype-space directions that are not produced or are lost as cohorts age (“survival collapsing”) and one from directions that do not make independent contributions to future reproduction (“reproductive collapsing”), which can each be removed from projection matrices while maintaining dynamical sufficiency. We follow this formal proof with a biological example that illustrates how the two theorems can be applied to simplify projection matrices in ecological and evolutionary analyses. Finally, we warn that empirical studies averaging over hidden heterogeneity without collapsing using these theorems will typically produce inaccurate estimates of vital rates, and we quantify the proportional error in the predicted population growth rate. Our collapsing theorems can be used in future studies to incorporate individual heterogeneity within age classes while minimizing the numerical and empirical costs of added complexity.
Elucidating the mechanisms of species coexistence is not only a central problem in ecology but also provides fundamental insights for conservation biology and the management of biological resources. In general, coexistence of two similar species is considered difficult; however, examples are known in which spatiotemporal heterogeneity reduces niche overlap and promotes coexistence. Ontogenetic niche shifts occur in nearly all species, yet their effects are often overlooked in theoretical models. The present study examines the effects of ontogenetic niche shifts on coexistence conditions, both in the presence and absence of patchy environments. In coexistence theory, the Lotka-Volterra (LV) competition model plays a key role, and particular attention here is given to developing an ontogenetic niche shift (ONS) model that can be directly compared with the coexistence conditions of the LV framework. Specifically, starting from a nonlinear integro-partial differential equation model that can be regarded as a state-structured extension of the generalized LV model, a two-species competitive system with ontogenetic niche shift, consisting of immature and mature stages, is analyzed. The results show that coexistence can occur even when the interspecific competition coefficient for immature individuals exceeds the intraspecific competition coefficient. In addition, direct comparison between the ONS model and the corresponding classical LV model showed that the ONS model allowed coexistence in parameter regions where the classical LV model predicted exclusion. A patchy environment can also provide an alternative pathway to coexistence when two patches are weakly connected. These findings indicate that the predictions of the classical LV competition model are not universal and suggest that ontogenetic niche shifts may allow two ecologically similar species to coexist even when interspecific competition exceeds intraspecific competition, without invoking mechanisms such as the storage effect or higher-order interactions.
