Stochastic Population Dynamics Research Articles

Fifty-year population trajectory in a marginal American Pika (Ochotona princeps) population

Abstract The American Pika (Ochotona princeps) has been considered a species at risk due to warming temperatures associated with climate change. Many life-history attributes of pikas contribute to the sensitivity of pikas to warming temperatures. Repeated censuses of a marginal (warm, low-elevation) population of pikas at Bodie State Historic Park, California, conducted from 1972 to 2022, are presented to track the population trajectory of pikas for a time period predating recent awareness of global warming to the present day, thus giving a comprehensive portrayal of how American pikas may be responding to climate change. The northern constellation of anthropogenic habitat patches (mine ore dumps) showed no decline in percent of patches occupied or in total number of pika territories over time, suggesting that pikas in that area have not been at risk of extirpation resulting from climate change. In contrast, the pika population in the southern constellation of patches showed significant declines in percent of patches occupied and number of pika territories occupied. That area was unoccupied for about a decade beginning in 2006, but was recently recolonized from the northern constellation of patches. The most likely cause of the initial decline and transient extirpation in the south appears to result from fragmentation and stochastic population dynamics, independent of climatic factors that we investigated. Assessments of climatic impacts on American pikas should take into account the dynamics documented in the Bodie pika population and its resilience over time.

Feedback between stochastic gene networks and population dynamics enables cellular decision-making.

Phenotypic selection occurs when genetically identical cells are subject to different reproductive abilities due to cellular noise. Such noise arises from fluctuations in reactions synthesizing proteins and plays a crucial role in how cells make decisions and respond to stress or drugs. We propose a general stochastic agent-based model for growing populations capturing the feedback between gene expression and cell division dynamics. We devise a finite state projection approach to analyze gene expression and division distributions and infer selection from single-cell data in mother machines and lineage trees. We use the theory to quantify selection in multi-stable gene expression networks and elucidate that the trade-off between phenotypic switching and selection enables robust decision-making essential for synthetic circuits and developmental lineage decisions. Using live-cell data, we demonstrate that combining theory and inference provides quantitative insights into bet-hedging-like response to DNA damage and adaptation during antibiotic exposure in Escherichia coli.

The most probable transition pathway of a predator–prey system under noise

Stochastic population dynamics plays a crucial role in biological systems. The number and distribution of biological populations often affect the stability of ecosystems, and even lead to the transition of ecosystems from one steady-state to another. The transition between different states of a population has a significant impact on both population dynamics and biological evolution. In this paper, we examined a stochastic population model and used Onsager–Machlup action functional and neural shooting method to obtain the most probable transition pathway between different states. Moreover, we analyzed the influence of noise intensity, maximum harvest rate [Formula: see text] and time [Formula: see text] on most probable transition pathway in two different parameter scenarios. Our research may provide a better understanding of evolutionary process such as genetic variation in populations and the evolution of species adaptation, meanwhile, may provide a theoretical guidance for species conservation and management in real world.

Using early detection data to estimate the date of emergence of an epidemic outbreak.

While the first infection of an emerging disease is often unknown, information on early cases can be used to date it. In the context of the COVID-19 pandemic, previous studies have estimated dates of emergence (e.g., first human SARS-CoV-2 infection, emergence of the Alpha SARS-CoV-2 variant) using mainly genomic data. Another dating attempt used a stochastic population dynamics approach and the date of the first reported case. Here, we extend this approach to use a larger set of early reported cases to estimate the delay from first infection to the Nth case. We first validate our framework by running our model on simulated data. We then apply our model using data on Alpha variant infections in the UK, dating the first Alpha infection at (median) August 21, 2020 (95% interpercentile range across retained simulations (IPR): July 23-September 5, 2020). Next, we apply our model to data on COVID-19 cases with symptom onset before mid-January 2020. We date the first SARS-CoV-2 infection in Wuhan at (median) November 28, 2019 (95% IPR: November 2-December 9, 2019). Our results fall within ranges previously estimated by studies relying on genomic data. Our population dynamics-based modelling framework is generic and flexible, and thus can be applied to estimate the starting time of outbreaks in contexts other than COVID-19.

Universal abundance fluctuations across microbial communities, tropical forests, and urban populations.

The growth of complex populations, such as microbial communities, forests, and cities, occurs over vastly different spatial and temporal scales. Although research in different fields has developed detailed, system-specific models to understand each individual system, a unified analysis of different complex populations is lacking; such an analysis could deepen our understanding of each system and facilitate cross-pollination of tools and insights across fields. Here, we use a shared framework to analyze time-series data of the human gut microbiome, tropical forest, and urban employment. We demonstrate that a single, three-parameter model of stochastic population dynamics can reproduce the empirical distributions of population abundances and fluctuations in all three datasets. The three parameters characterizing a species measure its mean abundance, deterministic stability, and stochasticity. Our analysis reveals that, despite the vast differences in scale, all three systems occupy a similar region of parameter space when time is measured in generations. In other words, although the fluctuations observed in these systems may appear different, this difference is primarily due to the different physical timescales associated with each system. Further, we show that the distribution of temporal abundance fluctuations is described by just two parameters and derive a two-parameter functional form for abundance fluctuations to improve risk estimation and forecasting.

Solving the stochastic dynamics of population growth.

Population growth is a fundamental process in ecology and evolution. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not consider the variability of birth times, which is particularly important with small population sizes. We show that some moment-closure approximations describe the growth dynamics better than the deterministic prediction. However, they do not reduce the error satisfactorily and only apply to some population growth models. We explicitly solve the stochastic growth dynamics, and our solution applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community and population dynamics. It will allow the development of new tools for a more accurate analysis of experimental and empirical results, including the inference of important growth parameters.

An exploratory assessment of the Bering-Chukchi-Beaufort Seas stock of bowhead whales using a stochastic population dynamics model

The stochastic population dynamics model used by Aboriginal Whaling Management Procedure developers is revised to correct weaknesses related to uncertainty parameterisation and replacement yield estimation. Two variants of this model, along with the standard deterministic version, are used to assess the Bering-Chukchi-Beaufort Seas stock of bowhead whales. The variants differ with respect to the magnitude and complexity of the stochastic variation they introduce into natural mortality and birth/calf survival processes. An allowable catch statistic, E (Q0), is defined for appropriate use with stochastic model assessments. Using the same assessment methods, likelihood and priors as IWC (1999a), 5th percentiles of E (Q0) were found to be 117, 106 and 91 for the deterministic, simpler stochastic and extreme stochastic models, respectively. Bayes factor results show that there is no evidence suggesting that either stochastic model should be favoured over any simpler alternative, and the deterministic model yielded the best fit overall. The E (Q0) estimates confirm and strengthen past IWC Scientific Committee inference that under current bowhead subsistence hunting levels the stock should continue to increase towards stabilisation above its maximum sustainable yield level.

Confronting population models with experimental microcosm data: from trajectory matching to state‐space models

AbstractPopulation and community ecology traditionally has a very strong theoretical foundation with well‐known dynamical models, such as the logistic and its variations, and many modifications of the classical Lotka–Volterra predator–prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non‐)linear regressions, are biased, and whether more elaborate process error models or state‐space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro‐ and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ordinary differential equation (ODE) models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state‐space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state‐space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state‐space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R.

Breeding Population Dynamics of Threatened Crawfish Frogs Inform Targets for Habitat Management

We used data from a five-year study at two focal breeding wetlands of a single Crawfish Frog (Rana areolata) population to demonstrate the relative importance and potential of tadpole survival as a target for population management. We concurrently estimated 11 vital rates in the field and used a females-only matrix projection model to estimate elasticities of demographic rates. We then simulated stochastic population dynamics at each wetland with and without immigration to estimate the intrinsic capacity for each focal breeding wetland to sustain a Crawfish Frog breeding population and the likelihood of breeding population persistence at each wetland. Elasticity of tadpole survival was second only to juvenile survival elasticity and 1.34–2.04 times greater than adult survival elasticity. Projections indicated that the Crawfish Frog population was not at risk of extinction but only one breeding site was capable of self-sustaining a breeding population. Because of low tadpole survival, the other breeding site was completely dependent on immigration to persist and was functioning as a population sink. Despite higher variability compared to terrestrial vital rates, larval survival did have a strong effect on population growth. Tadpole survival at the more productive breeding site was density dependent and likely related to wetland vegetation and predator and competitor abundance. Two additional findings were that annual survival of frogs following their first known breeding event was 48% lower compared to survival of frogs that had bred in two or more prior years, and adult temporary emigration from the breeding population was moderately high. Our study demonstrates the benefits of using population models that integrate density-dependent processes, temporary emigration from the breeding population, and state-specific adult survival, to identify larval habitats that function as population sinks and limit current population size and persistence probability. We contend that tadpole survival is an important and feasible habitat management target within broader conservation strategies for Crawfish Frogs and other amphibian species.

A transport process on graphs and its limiting distributions

AbstractGiven a finite strongly connected directed graph $G=(V, E)$ , we study a Markov chain taking values on the space of probability measures on V. The chain, motivated by biological applications in the context of stochastic population dynamics, is characterized by transitions between states that respect the structure superimposed by E: mass (probability) can only be moved between neighbors in G. We provide conditions for the ergodicity of the chain. In a simple, symmetric case, we fully characterize the invariant probability.

Stochastic population dynamics of cancer stemness and adaptive response to therapies.

Intratumoral heterogeneity can exist along multiple axes: Cancer stem cells (CSCs)/non-CSCs, drug-sensitive/drug-tolerant states, and a spectrum of epithelial-hybrid-mesenchymal phenotypes. Further, these diverse cell-states can switch reversibly among one another, thereby posing a major challenge to therapeutic efficacy. Therefore, understanding the origins of phenotypic plasticity and heterogeneity remains an active area of investigation. While genomic components (mutations, chromosomal instability) driving heterogeneity have been well-studied, recent reports highlight the role of non-genetic mechanisms in enabling both phenotypic plasticity and heterogeneity. Here, we discuss various processes underlying phenotypic plasticity such as stochastic gene expression, chromatin reprogramming, asymmetric cell division and the presence of multiple stable gene expression patterns ('attractors'). These processes can facilitate a dynamically evolving cell population such that a subpopulation of (drug-tolerant) cells can survive lethal drug exposure and recapitulate population heterogeneity on drug withdrawal, leading to relapse. These drug-tolerant cells can be both pre-existing and also induced by the drug itself through cell-state reprogramming. The dynamics of cell-state transitions both in absence and presence of the drug can be quantified through mathematical models. Such a dynamical systems approach to elucidating patterns of intratumoral heterogeneity by integrating longitudinal experimental data with mathematical models can help design effective combinatorial and/or sequential therapies for better clinical outcomes.

Stationary distribution of stochastic population dynamics with infinite delay

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C∞-smooth density.

Tracking the stochastic growth of bacterial populations in microfluidic droplets

Bacterial growth in microfluidic droplets is relevant in biotechnology, in microbial ecology, and in understanding stochastic population dynamics in small populations. However, it has proved challenging to automate measurement of absolute bacterial numbers within droplets, forcing the use of proxy measures for population size. Here we present a microfluidic device and imaging protocol that allows high-resolution imaging of thousands of droplets, such that individual bacteria stay in the focal plane and can be counted automatically. Using this approach, we track the stochastic growth of hundreds of replicate Escherichia coli populations within droplets. We find that, for early times, the statistics of the growth trajectories obey the predictions of the Bellman–Harris model, in which there is no inheritance of division time. Our approach should allow further testing of models for stochastic growth dynamics, as well as contributing to broader applications of droplet-based bacterial culture.

Long-Term Conservation Effects of Protected Areas in Stochastic Population Dynamics

Terrestrial and marine protected areas are essential tools in mitigating anthropogenic impacts and promoting population persistence and resource sustainability. Adequately implemented protected areas (PAs) aim to promote conservation by increasing population size and reducing its variability. To resolve how these effects depend on PA features, I develop and analyze new models of stochastic processes that encompass the fluctuations generated by demographic or environmental stochasticity in PAs management. The stochastic model is built upon individual processes. In the model, density-independent mortality, migration between PAs and non-PAs, organism preference for PAs, and size characterize the features of the PA. The effect of PAs size is also examined. The long-term conservation effects are quantified using the coefficient of variation (CV) of population size in PAs, where a lower CV indicates higher robustness in stochastic variations. The results from this study demonstrate that sufficiently reduced density-independent mortality in PAs and high site preference for PAs and immigration rate into PAs are likely to decrease the CV. However, different types of stochasticity induce rather different consequences: under demographic stochasticity, the CV is always reduced because PAs increase the population size therein, but an increased population size by PAs does not always decrease the CV under environmental stochasticity. The deterministic dynamics of the model are investigated, facilitating effective management decisions.

Cavity and replica methods for the spectral density of sparse symmetric random matrices

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

Beyond the adiabatic limit in systems with fast environments: A τ-leaping algorithm.

We propose a τ-leaping simulation algorithm for stochastic systems subject to fast environmental changes. Similar to conventional τ-leaping the algorithm proceeds in discrete time steps, but as a principal addition it captures environmental noise beyond the adiabatic limit. The key idea is to treat the input rates for the τ-leaping as (clipped) Gaussian random variables with first and second moments constructed from the environmental process. In this way, each step of the algorithm retains environmental stochasticity to subleading order in the timescale separation between system and environment. We test the algorithm on several toy examples with discrete and continuous environmental states and find good performance in the regime of fast environmental dynamics. At the same time, the algorithm requires significantly less computing time than full simulations of the combined system and environment. In this context we also discuss several methods for the simulation of stochastic population dynamics in time-varying environments with continuous states.

Stochastic recruitment alters the frequencies of alternative life histories in age‐structured populations

AbstractAlternative reproductive tactics, where members of a population exhibit divergent mating approaches, are widespread in fishes. The maintenance of such polymorphisms is often explained using game‐theoretic concepts such as frequency‐dependent selection, which typically assume populations are at equilibrium. However, fish populations exhibit stochastic variability that is inconsistent with the assumption of equilibrium dynamics. We explored how stochastic population dynamics interact with sexual and harvest‐induced selection to shape the frequencies of alternative male phenotypes in Pacific salmon (Oncorhynchus spp., Salmonidae). Males of several salmon species are composed of typical “hooknoses” that compete on the spawning grounds for mating opportunities, and rarer “jacks” that mature younger and at smaller sizes to attempt reproduction by “sneaking.” Using stochastic simulations, we found that realistic levels of recruitment variation could increase the prevalence of jacks relative to equilibrium expectations. This result is attributable to the “cohort mismatch” phenomenon, whereby differences in age at maturity between male phenotypes cause jacks from strong recruitments to spawn in the same year as hooknoses from weaker cohorts, elevating the proportion of jacks among spawners. Simulations were supported by data from major populations of sockeye salmon (O. nerka, Salmonidae) throughout their North American range, which indicated a positive association between recruitment variability and jack prevalence in some regions. Moreover, we found that frequency‐dependent mating dampened population responses to fisheries‐induced selection. Our findings emphasize that stochastic ecological processes are important to the maintenance of alternative life histories and interact with selection to produce emergent dynamics that have not previously been recognized.

Reservoir ecological operation by quantifying outflow disturbance to aquatic community dynamics

Reservoir operation causes spatiotemporal variations in outflow, which influence the dynamics of downstream aquatic communities. However, empirical evidence of community responses to hydrological alteration remains limited for dam-regulated rivers. This study focused on quantifying the streamflow disturbance to multi-population dynamics in downstream of the China’s Danjiangkou Reservoir. First, the stochastic population dynamics model (PDM) was used to simulate aquatic community dynamics. Then, the flow–ecology relationship was established to identify community response to reservoir outflow. Third, two novel ecological indicators, stable time (ST) and coefficient of variation at stable time (CVST), were proposed to evaluate the resilience and resistance of multi-population systems, respectively. Finally, the reservoir operating rule curves were optimized by considering tradeoffs between socioeconomic and ecological objectives. The coevolution processes of multi-population systems (fish, phytoplankton, zooplankton, zoobenthos, and macrophytes) were simulated by stochastic PDMs. The population densities of stable states showed continuous downward trends with increasing degree of hydrological alteration for multi-population systems, and aquatic community systems could be destroyed when alteration reached its acceptable maximum. The greater the degree of hydrological alteration, the longer the recovery time from an unstable to a stable state, and the weaker resistance for each population system. The resilience and resistance of downstream multi-population systems were enhanced by optimizing reservoir outflow. The optimization results illustrated that the performances of the multiple objectives of water supply, hydropower generation, and ST were improved by 2.37%, 2.40%, and 2.67%, respectively, whereas the performance of CVST was the same as the conventional operation. The flow–ecology relationship provided an approach to quantify the impacts of reservoir outflow on an aquatic community, which is helpful in guiding ecological flow strategies.

Finite-size effects, demographic noise, and ecosystem dynamics

Strong positive feedback is considered a necessary condition to observe abrupt shifts in ecosystems. A few previous studies have shown that demographic noise—arising from the probabilistic and discrete nature of birth and death processes in finite systems—makes the transitions gradual. In this paper, we investigate the impact of demographic noise on finite ecological systems. We use a simple cellular automaton model with births and deaths influenced by positive feedback processes. We present our methods in a tutorial like format. Using the approach of van Kampen’s system-size expansion, we derive a stochastic differential equation that describes how local probabilistic rules scale to stochastic population dynamics in finite systems. We illustrate that as a consequence of enhanced demographic noise, finite-sized ecological systems can show an ‘effective abrupt transition’ even with weak positive interactions. Numerical simulations of our spatially explicit model confirm this analytical expectation. Thus, we predict that small-sized populations and ecosystems, in response to environmental drivers, are prone to abrupt collapse while larger systems—with the same microscopic interactions—show a smooth response.

Hunting quotas, selectivity and stochastic population dynamics challenge the management of wild reindeer

Wildlife managers adjust harvest quotas based on population changes and specific management goals, like controlling population size and demography or minimizing human-�wildlife interactions. However, establishing quotas that best meet these goals can be challenging due to e.g. population fluctuations, climate change, and bias or variation in harvest effort. High-Arctic Svalbard reindeer Rangifer tarandus platyrhynchus experience strong interannual variation in population size and demographic structure due to environmental stochasticity. Here, we analyzed the demographic and spatial bias of reindeer harvest in relation to quota regulations, hunter preferences, and population dynamics. Despite the protective management goal to avoid demographic impacts, 30 yr of data revealed that the harvest was consistently biased towards yearlings and male adults. This hunting selectivity resulted from both hunter preferences and the coarse license categories separating ‘calf’, ‘yearling or female adult’, and ‘free choice’ licenses. We developed Bayesian multinomial likelihood models to account for hunting selectivity and optimize annual quota distributions among license categories using population monitoring data. Optimized annual quota distributions varied strongly due to demographic fluctuations associated with strong climate variability, whereas simulated harvest offtakes showed that the protective management goal is inherently challenged by the coarse license categories. Although on average, only 7% of the hunted population was harvested annually, we found strong spatial variation in harvest pressure, with potential implications for spatial population dynamics. Our adaptive management approach accounting for hunting selectivity and demographic fluctuations can be of general relevance for harvested species in stochastic environments.