In a previous work, a complete methodology for monitoring the growth of a filamentous fungus was introduced, allowing the automated extraction of its graph structure and of its key statistics. In parallel, a stochastic growth-fragmentation model for the dynamics of such mycelial networks was introduced and studied. This simple model depends on three parameters: the elongation speed v of a single filament, the branching rate b1 of a filament at its open end, and the per unit length rate b2 at which a budding event happens and creates a new filament branching off from an existing one. Leaving aside the spatial structure of the mycelium and describing its structure essentially through the empirical measure of the lengths of its filament segments, the three parameters of the growth dynamics encoded in the model summarise the local balance between mass creation and regulation through non-linear mechanisms such as anastomosis or density-dependent growth modulation. In this work, we develop a generic statistical inference method based on the large-time behaviour of the stochastic model, and on the high-resolution pictures of the mycelial network obtained using the methodology for fungal growth monitoring, to reconstruct the effective growth parameters v, b1 and b2 from a single panorama of the filament network pictured after several hours of development and an empirical measurement of the exponential rate of increase of the number of branch points and apexes. We use this method to analyse the growth dynamics of Podospora anserina mycelia observed under standard conditions and when several forms of stress are applied, in order to quantify the effect of these stresses on the different mechanisms of fungal growth. By comparing the reconstructed effective parameters with the hyphal elongation speed and branching rates estimated from the much more complex dynamical tracking of individual filaments, we find that the effective branching rates we infer are in close agreement with the individual branching rates estimated from the dynamical tracking procedure. This suggests that, in this application at least, the effects of non-linear local regulation mechanisms on branching are well encoded by an effective Markovian rate during the exponential growth phase of the mycelium. By contrast, the effective elongation speed reconstructed with our model-based approach is approximately twice as low as the hyphal elongation speed estimated from the dynamical tracking procedure, a bias that may be explained by a mismatch between the definitions of "elongation speed" used in the two approaches. Nevertheless, our results show that the three parameters of our growth-fragmentation model, combined with experimental data that are now readily available, enable us to quantify different components of the exponential growth dynamics of an expanding mycelial network.
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.
Behavioral responses to infection risk can alter disease spread when isolation arises endogenously from epidemic conditions rather than from externally imposed rules. We study this mechanism in a behavior-coupled Susceptible-Exposed-Infected-Recovered (SEIR) model in which infectious individuals choose between isolation and non-isolation through an evolutionary game. The behavioral dynamics are governed by replicator equations derived from payoffs that depend on perceived infection risk, the social benefits of contact, and the cost of isolation, creating a two-way coupling between disease prevalence and behavioral change. We characterize the disease-free and endemic equilibria through the basic reproduction number and show that endogenous isolation qualitatively changes epidemic behavior relative to the classical SEIR system. In particular, prevalence-dependent behavioral adaptation generates time-varying adherence and recurrent infection waves even within an otherwise standard compartmental structure. Lower isolation cost or greater perceived risk shifts the evolutionary dynamics toward isolation, reducing transmission, lowering peak prevalence, and dampening later waves. We further study intervention in this coupled system by allowing time-dependent incentives to modify behavioral payoffs and promote isolation. The resulting optimal control problem is characterized using Pontryagin's Maximum Principle, showing that adaptive incentives can produce stronger and more sustained epidemic suppression than comparable static measures. Calibration to COVID-19 incidence data from Medan, Indonesia serves as an empirical illustration of the theoretical results. Overall, the analysis identifies behavior-disease coupling through endogenous isolation as a population-level mechanism that modifies both equilibrium structure and transient outbreak dynamics, while providing a general framework that links evolutionary behavioral adaptation and epidemic control.
The Kingman coalescent process models the genealogy of a sample taken from a large population of individuals who reproduce and die according to models such as the Moran or Wright-Fisher processes. The occurrence and spread of neutral mutations in the sample can be modelled by a Poisson process over sample phylogenies from the Kingman coalescent. We study the joint probability distribution of frequencies for multiple mutations occurring on the same tree. We call this the Joint Spectrum over Trees (JST). We derive a closed-form solution for this joint distribution in the case of two mutations with varying population size. We specialise the result for specific population histories, including constant population size. In the process, we highlight how different averaging procedures can lead to different distributions for the frequency of mutations, even when considering only the frequency of a single mutation. We provide a systematic approximation scheme for the Joint Spectrum over Trees under constant population when the number of samples is large. The exact form of the Joint Spectrum over Trees has implications for genealogical inference with the Kingman coalescent, specifically for the characterisation of tree structure near the root and in parameter inference when the underlying tree structures are unknown. To this end, we also comment on the validity of the independence approximation to the true joint distribution under different population histories.
In the Admixture Model, the probability that an individual carries a certain allele at a specific marker depends on the allele frequencies in K ancestral populations and the proportion of the individual's genome originating from these populations. The markers are assumed to be independent. The Linkage Model is a Hidden Markov Model that extends the Admixture Model by incorporating linkage between neighboring loci. We prove consistency and asymptotic normality of maximum likelihood estimators for the ancestry of individuals in the Linkage Model, complementing earlier results by (Pfaff et al., 2004; Pfaffelhuber and Rohde, 2022; Heinzel, 2025) for the Admixture Model. These results are used to prove that a statistical test that allows for model selection between the Admixture Model and the Linkage Model is an asymptotic level-α-test. Finally, we demonstrate the practical relevance of our results by applying the test to real-world data from The 1000 Genomes Project Consortium (2015).
Threshold ages - the ages at which mortality reductions shift from compressing to expanding lifespan variation - are fundamental features of age-at-death distributions. Classical continuous formulations typically derive threshold conditions on the half-line [0,∞); empirical life tables and computational implementations, however, operate on a finite, maximum observed age ω that can expand over time. We develop a unified framework for threshold dynamics under expanding support. First, we show that κ-homogeneous spread measures admit a scale-shape decomposition, separating the mechanical effect of domain size from distributional shape. Second, differentiating this decomposition yields a sensitivity decomposition into a shape channel and a domain-scaling channel induced by changes in ω. The latter shifts thresholds toward younger ages for all κ>0. Third, we prove existence, uniqueness, and a general threshold-shift formula, expressed through the normalized threshold y∗=x∗/ω. Applications to variance, the Gini coefficient, entropy, and higher moments, together with simulations and Human Mortality Database data, confirm the theoretical predictions. In protected, predominantly senescent populations, normalized thresholds cluster in narrow ranges, but historical analysis shows that y∗ can evolve substantially during demographic transitions (e.g., for Swedish females, the variance threshold shifts from yV∗≈0.45 in 1751 to yV∗≈0.75 in 2020).
In this manuscript, we construct an intraguild predation system based on optimal foraging theory in the presence of harvesting. The basal resource is considered as more profitable prey for the super predator (intraguild (IG)-predator), and the attacking probabilities of the IG-predator to the prey species are taken as the foraging strategy of the super predator. The underlying conditions for switching the foraging strategy depending on prey densities are determined with the aid of fitness function. The resulting dynamics of the harvesting intraguild system are compared from two perspectives: one is with fixed foraging, and the other is with optimal foraging strategy. The effects of system parameters in the occurrence of Hopf bifurcation are depicted numerically by adopting some fixed foraging strategies. Some numerical simulations have been used to describe the fascination of the system's dynamics, including the occurrence of partial preference, periodic coexistence, and extinction of intermediate prey with optimal foraging strategy.
Assortative mating for a given phenotype is the phenomenon by which mates select each other based on their phenotypic similarity. Other phenomena can create positive correlation between the parents' and the offspring's environment: vertical cultural transmission, or dynastic effects. When these phenomena occur together, they induce a gene-environment correlation at the population scale. It will impact genetic measures of associations such as SNP effect size and SNP-heritability estimates. In this paper, we provide a complete mathematical modelling of both assortative mating and vertical cultural transmission in the classical framework of the polygenic additive model. We establish for the first time the theoretical evolution and equilibrium values of all quantities of interest involved. We then derive its consequences on typical genetic epidemiology analyses; including both population and family-based study designs. We show the consequences on heritability estimation, Genome-Wide Association Studies (GWAS), and the variance explained by polygenic scores. We validate our calculations through simple forward-time simulations.
The impact of human health-related behavior on the spread of the disease continues to be prominent due to the proliferation of sentiment against public health-promoting behaviors. The recent SARS-COV-2 pandemic brought these effects into sharp focus. In the process, we have learned that our current understanding of how human behavior affects disease spread is insufficient in scope and intricacy. In particular, how different health-related behaviors themselves interact with each other, and how those interactions affect disease spread, is largely unknown. In this paper, we use a coupled-contagion model to study the interaction of two different health behaviors - vaccination and mask-wearing. We study how a positive or negative association between these behaviors - e.g. a pro-vaccine person may be more or less likely to be pro-mask-wearing - affects disease dynamics. We find that the strength of such an association alone can determine the outcome of disease spread in the short term and the long term. In addition to studying human health-related behaviors separately, it is vitally important to understand how these behaviors interact with each other and how these associations affect the outcomes of epidemics.
Mutation accumulation (MA) experiments are crucial for understanding evolution. In microbial populations, these experiments typically involve periods of population growth, where a single individual forms a visible colony, followed by severe bottlenecks. Studies on the effects of positive and negative selection in MA experiments have shown that, even with as few as ten generations of growth between bottlenecks, beneficial mutations will be substantially over-represented (Wahl and Agashe, 2022); this effect is known as “selection bias”. In previous work, we developed a discrete-time model and demonstrated that when stochastic offspring distributions are considered, selection bias is even stronger than previously predicted (Ezadian and Wahl, 2025). Since bacterial division is unlikely to remain synchronized over 15 or more generations, this study extends the discrete-time model to a continuous-time framework. Since lineages that start reproducing early accrue a compounded advantage, a continuous-time model offers an even more accurate correction for selection in MA experiments. In addition, we develop techniques for estimating the selection bias when only colonies above a fixed threshold size (e.g. visible colonies) are selected at the bottleneck, an experimental protocol that further exacerbates selection bias. We develop and describe computationally efficient techniques for correcting experimental MA data whether all colonies or only visible colonies are considered.
Characterizing patterns of biological diversity is a central goal of evolutionary biology. This requires understanding expectations for clade size-the relationship between the number of species in a clade and its age. Such expectations are key for identifying diversity outliers (e.g., specious or depauperate clades in macroevolution, or unusually large or small transmission clusters in epidemiology) and for testing alternative hypotheses about diversification. Here, we develop a general method for deriving closed-form expressions for the (joint) distribution of clade sizes under a given diversification model. We apply this approach to the constant- and variable-size coalescent model as well as the Yule model. Our results reveal that while the coalescent and Yule models produce qualitatively similar clade size patterns, they exhibit quantitative differences. Leveraging the flexibility of the coalescent framework, we further examine how transmission cluster size distributions differ between rapidly and slowly growing epidemics, finding - counterintuitively - that slowly growing epidemics are more likely to generate large clusters, a pattern often attributed to increased transmission.
The Feller diffusion is studied as the limit of a coalescent point process in which the density of the node height distribution is skewed towards zero. Using a unified approach, a number of recent results pertaining to scaling limits of branching processes are reviewed and reinterpreted as properties of the Feller diffusion arising from this limit. The notion of Bernoulli sampling of a finite population is extended to the diffusion limit to cover finite Poisson-distributed samples drawn from infinite continuum populations. We show that the coalescent tree of a Poisson-sampled Feller diffusion corresponds to a coalescent point process with a node height distribution taking the same algebraic form as that of a Bernoulli-sampled birth-death process. By adapting methods for analysing k-sampled birth-death processes, in which the sample size is pre-specified, we develop methods for studying the coalescent properties of the k-sampled Feller diffusion.
Among the mathematical approaches used to model population dynamics, Hidden Markov Models (HMM) are well adapted to the case where the species of interest is difficult to observe. For a broader application of HMM in ecology, two limits need to be overcome. While HMMs can deal with detection errors, another important situation is when only some life stages of the population can be observed while the others remain hidden. The metapopulation level, rather than a single population, makes it possible to incorporate dispersal processes, often linked to hidden life stages. Therefore, there is a need to extend the HMM framework to the case of several couples of hidden and observed life stages interacting via dispersal. We propose a conceptual guide to model and estimate such dynamics using the framework of interacting Partially Observable Dynamic Bayesian Networks (PO-DBN). We show that only four interaction structures are needed to describe the main metapopulation models. We illustrate them on concrete examples. Well known computational challenges apply to inference in metapopulation models with partial observation, due to the problem dimension. We discuss parameter estimation using the EM algorithm and we establish that for two structures the complexity of EM is actually linear in the number of patches, which means that estimation is easily accessible for the associated metapopulations. For the two other structures, the EM complexity is exponential and we discuss methods for approximate inference. This study provides the practical foundations for modelling and estimating the dynamics of a metapopulation with hidden life stages.
Bacterial lineages are relatively short-lived on geological timescales, according to phylogenetic analyses, implying that bacterial extinction occurs at high rates. Since the vast majority of bacteria live in large populations in oceans and soils, many well-studied extinction mechanisms, such as demographic or environmental stochasticity, seem unlikely to drive this pattern. We outline mechanisms for the extinction of large bacterial populations, and discuss the emergence of a new virus as a possible cause of extinction. We use deterministic and stochastic models to characterise the persistence of a bacterial population, demonstrating that when resistance to a new virus does not emerge, large populations are more likely to go extinct than small populations, which contrasts with classically studied extinction mechanisms. When they go extinct, large populations also reach extinction more quickly. When phage-resistant bacteria appear, extinction is rare but its probability increases with population size in some parameter regimes. We also quantify bacterial extinction in spatially distinct subpopulations. We conclude that large bacterial populations are robust to many extinction mechanisms, and typically evolve resistance to new phages, as observed empirically. For bacterial lineages that have gone extinct, however, the failure to evolve resistance to a novel phage is a likely underlying mechanism.
The great population geneticist Sewall Wright had a lifelong interest in animal breeding programs. He was concerned that, in these programs, genetic variation would tend to be lost (as measured by him in a tendency for genetic heterozygosity to decrease) when a small number of males is mated to a comparatively large number of females. The calculations in his great 1931 paper were designed to quantify this tendency. Decades later other authors noted further cases when population genetic heterozygosity tended to be lost, a classic example being that of a population whose size changes cyclically with some small population sizes arising during the cycle. These observations led to several definitions of the effective population size. Perhaps the two most frequently discussed in the literature are the inbreeding effective population size and the variance effective population size. The first aim of this note is to show that Wright's (1931) calculations for two-sex populations relate to neither of these, and instead to the far less discussed eigenvalue effective population size. The second aim is to critically assess all three effective population sizes and put two-sex populations into a wider concept of subdivided populations. This analysis reveals some new results, and for two-sex populations without other types of subdivision, we conclude that the eigenvalue effective size is the most relevant type of effective size.
Avoidance behaviors can have a substantial influence on both the spread of an infectious disease and on the long-term dynamics of substance use disorder. However, this behavior is typically reactive based on delayed information related to personal risk as well as both the length of exposure to and retention of that information. This suggests that in epidemic models, feedback delays should be strong - that is, not exponentially distributed (Erlang-1) but instead peaking at a point strictly in the past (Erlang-2 or greater). However, almost all studies of infectivity feedback delays are exponential with qualitatively different results than are seen with strong delays. To address this gap, we analyze two compartmental models for infectious disease epidemiology. Our results demonstrate that sustained oscillations in the total number of active cases may appear even as early as the first two years of an outbreak, suggesting that human behavior may be an explanatory factor for periodic fluctuations evident in recent pandemic time-series data (e.g. COVID-19). We then extend our analysis to a study of the role of information feedback on substance use disorder epidemiology, with a focus on both the transient and asymptotic dynamics of drug waves. We show that under certain conditions, oscillations in substance use disorder can become sustained and that models without strong feedback delays can fail to produce important qualitative transient behavior in substance use incidence rates. To our knowledge, this work represents the first mathematical model exhibiting oscillations with non-contact (linear) pathways to substance use disorder.
Cultural diversity is a fundamental aspect of human life; however, it is still challenging to understand mechanisms that promote cultural diversity. While previous studies tended to focus on how innovation generates cultural diversity, we investigated whether the invasion criterion, originally developed for understanding species coexistence in community ecology, can predict the maintenance of cultural diversity in a finite population under various transmission biases. Using mathematical and individual-based models, we show that the invasion criterion reliably predicts the duration of coexistence of multiple cultural traits in stochastic dynamics under content-, conformity-, and anticonformity-biased social learning. The invasion criterion also predicted cultural coexistence under prestige- and success-bias social learning, but the predictability was lower than in the above cases. Similarity-biased social learning, where people tend to learn from those with the same "tag" traits (e.g., gender), lowered the predictive power because the cultural traits were associated with the tag traits with a certain probability, violating the rarity assumption. Our results, therefore, demonstrate that the invasion criterion offers valuable tools for quantitatively predicting cultural coexistence, but predictability varied across transmission biases, suggesting that modifications are needed to improve the predictability in cultural dynamics. Our model also showed that increasing the number of cultural models, from which an individual copies a cultural trait, can prolong cultural coexistence under certain biases, with implications for fostering innovation and preserving culture. These findings suggest that the invasion criterion can be useful for understanding the maintenance of cultural diversity, as it is for species diversity.
Underlying many important statistical properties of a sample of DNA sequences is the total branch length L of the sample's genealogy. A classic example is the number K of mutations on the genealogy, whose expectation and variance are simple functions of the expectation and variance of L. However, higher moments of L and related quantities have received relatively little attention despite their potential utility. This paper systematically investigates the properties of L, its relationship with the total branch length of a certain number of descendants, and demonstrates their usefulness through several applications under the constant-in-state model, which is an extension of the Wright-Fisher model with constant effective population size. Specifically, the recurrence relations for the power moments of L as well as for the expectations of products between powers of L and branch length of specified sizes are derived. A closed-form expression for the power moments of L is also obtained. These results allow examination of the large-sample behavior of L through its skewness and kurtosis, revealing that L does not satisfy asymptotic normality under the constant-size Wright-Fisher model, although approximate normality emerges in rapidly growing populations. Moreover, the power moments of L provide a more straightforward route to deriving higher moments of K and yield a novel approach for computing the distribution of K. The associated mixed moments similarly lead to a novel method for calculating the probability of having a single mutation of a specified number of descendants.
Evolutionary analyses of large populations commonly incorporate stochasticity through temporal variation in selection while treating genetic transmission parameters as fixed. Much less attention has been given to stochasticity in transmission itself. We study a selected locus with alleles A and a under constant selection, linked to a neutral modifier locus whose alleles M1 and M2 control the mutation rate from A to a. Under constant transmission, the Reduction Principle applies: near a mutation-selection balance where M1 is fixed with mutation rate u1, a rare allele M2 invades if its associated rate u2u1. This result holds for both haploid and diploid populations and is independent of other parameters such as recombination, which affects only the rate, not the direction, of change. We extend this framework by allowing the mutation rate associated with the invading modifier allele M2 to fluctuate randomly across generations. In this stochastic setting, invasion is no longer determined by mean mutation rates alone. Instead, it depends on the temporal distribution of mutation rates, the strength of selection at the focal locus, and the recombination rate between modifier and target loci. Stochastic transmission therefore does not merely rescale deterministic predictions based on the Reduction Principle; it can alter the direction of selection acting on modifier alleles.
