arXiv subjects

Alan Hastings

Publications and source records attributed to Alan Hastings.

At least 19 recordsLinked to original sources

arXiv2026

Towards a unified framework for multiple stable states in ecological systems

Multiple stable states - the coexistence of two or more distinct ecological configurations under identical environmental conditions - have attracted sustained interest in ecology, yet the field still lacks a unified framework connecting ecological mechanisms to dynamical models. Here, we review empirical and theoretical approaches to multiple stable states, synthesising perspectives on stability, tipping, hysteresis, and transient dynamics, and contextualise these within a common mathematical framework. Drawing on examples of well-known ecosystem models, we highlight the central and necessary role of positive feedback loops and identify other common, unifying features of ecological systems that exhibit multiple stable states. We further discuss the relationship between stable and transient dynamics, the roles of spatial and temporal scales in feedback identification, and the implications for ecological restoration and management. We conclude with open questions and challenges for the field, including extending multistability theory to persistent-transient frameworks and harnessing emerging data-collection technologies to sharpen empirical inference.

q-bio.PE↗

arXiv2026

Tipping points in complex ecological systems

Tipping points are one of the hot topics in modern physics of complex systems. But what is a tipping point? A generic definition declares it as ``a state of the system where a small change in its parameters can lead to a significant change in its properties''. Additional ingredients that often enter the definition of tipping process are the abruptness of the resulting change and its irreversibility, i.e. it is impossible to recover the initial state if one reverses the protocol of change of the parameters. However, there exists a number of different mathematical structures that can show this behavior, the one that was originally suggested as a tipping point (nowadays usually referred to as bifurcation induced tipping) is just one of many. Different preconditions and/or different level of details included into the model, reflecting also different environmental forcing, can lead to a variety of tipping mechanisms. Furthermore, in a spatially extended system and/or a system with multiple scales, different parts can react to a change in environmental conditions differently or at a different time, interacting with each other to create a tipping cascade. In this paper, using ecosystems as a paradigm of complex nonlinear open systems, we provide a critical overview of the progress made in tipping point science over the last 15 years. We highlight the main findings, identify gaps in our knowledge, and outline a roadmap for further progress.

q-bio.PE↗

arXiv2025

Long-range dispersal promotes spatial synchrony but reduces the length and time scales of synchronous fluctuations

Synchronous oscillations of spatially disjunct populations are widely observed in ecology. Even in the absence of spatially synchronized exogenous forces, metapopulations may synchronize via dispersal. For many species, most dispersal is local, but rare long-distance dispersal events also occur. While even small amounts of long-range dispersal are known to be important for processes like invasion and spatial spread rates, their potential influence on population synchrony is often overlooked, since local dispersal on its own can be strongly synchronizing. In this work, we investigate the effect of random, rare, long-range dispersal on the spatial synchrony of a metapopulation and find profound effects not only on synchrony but also on properties of the resulting spatial patterns. While controlling for the overall amount of emigration from each local subpopulation, we vary the fraction of dispersal that occurs locally (to nearest neighbors) versus globally (to random locations, irrespective of distance). Using a metric that measures the instantaneous level of global synchrony, we show that this form of long-range dispersal significantly favors the spatially synchronous state and homogenizes the population by decreasing the size of clusters of subpopulations that are out of phase with the rest of the metapopulation. Moreover, the addition of non-local dispersal significantly decreases the equilibration time of the metapopulation.

q-bio.PE↗

arXiv2025

Deciphering culprits for cyanobacterial blooms and lake vulnerability in north-temperate lakes

Harmful cyanobacterial blooms (CBs) are increasingly prevalent worldwide, posing significant environmental and health concerns. We derive a stoichiometric model describing the population dynamics and toxicity of cyanobacteria in north-temperate freshwater ecosystems. Our model quantifies the hypoxic effects of CBs on fish mortality and evaluates the impact of microcystin-LR (MC-LR) on aquatic macro-invertebrates, phytoplankton, and fish species. Analyzing data from diverse north-temperate lakes with varying physical characteristics, we identify eutrophication as a pivotal catalyst in bloom proliferation. Under predicted warming scenarios coupled with increased eutrophication, peak MC-LR concentrations will surge dramatically, and blooms will occur earlier in the year. We uncover severe bioaccumulation of MC-LR in higher trophic species; the response to CBs among fish at intermediate trophic levels was heterogeneous across lakes. We compare our model against observations from several north-temperate lakes, demonstrating its robustness and applicability. Our insights are critical for informing targeted interventions to mitigate CBs.

math.DS↗

arXiv2025

Behavior of Ising spins and ecological oscillators on dynamically rewired small-world networks

Many ecological populations are known to display a cyclic behavior with period 2. Previous work has shown that when a metapopulation (group of coupled populations) with such dynamics is allowed to interact via nearest neighbor dispersal in two dimensions, it undergoes a phase transition from disordered (spatially asynchronous) to ordered (spatially synchronous) that falls under the 2-D Ising universality class. While nearest neighbor dispersal may satisfactorily describe how most individuals migrate between habitats, we should expect a small fraction of individuals to venture on a journey to further locations. We model this behavior by considering ecological oscillators on dynamically rewired small-world networks, in which at each time step a fraction $p$ of the nearest neighbor interactions is replaced by a new interaction with a random node on the network. We measure how this connectivity change affects the critical point for synchronizing ecological oscillators. Our results indicate that increasing the amount of long-range interaction (increasing $p$) favors the ordered regime, but the presence of memory in ecological oscillators leads to quantitative differences in how much long-range dispersal is needed to order the network, relative to an analogous network of Ising spins. We also show that, even for very small values of $p$, the phase transition falls into the mean-field universality class, and argue that ecosystems where dispersal can occasionally happen across the system's length scale will display a phase transition in the mean-field universality class.

cond-mat.stat-mech↗

arXiv2024

Determinism vs. stochasticity in competitive flour beetle communities

As ecologists increasingly adopt stochastic models over deterministic ones, the question arises: when is this a positive development and when is this an unnecessary complication? While deterministic models -- like the Lotka-Volterra model -- provide straightforward predictions about competitive outcomes, they are often unrealistic. Stochastic models are more realistic, but their complexity can limit their usefulness in explaining coexistence. Here, we investigate the relative importance of deterministic and stochastic processes in competition between two flour beetle species, Tribolium castaneum and Tribolium confusum. Specifically, we use highly-replicated one-generation experiments (784 microcosms) to parameterize a mechanistic model. Both the full stochastic model and the underlying deterministic skeleton exhibit priority effects, where one species excludes the other, but the identity of the winning species depends on initial abundances. Stochasticity makes the identity of the winner less predictable, but deterministic dynamics still make reliable predictions (94% accuracy across a range of reasonable initial abundances). We conclude that deterministic population dynamics are sufficient to account for patterns of coexistence (or lack thereof), a potentially general finding that is supported by recent field studies. Additionally, we resolve longstanding issues in flour beetle research by identifying selective egg predation as the mechanism for priority effects, demonstrating the primacy of demographic stochasticity (compared to environmental stochasticity), and reinterpreting classic competition experiments to show that apparent coexistence often represents long-term transient dynamics.

q-bio.PE↗

arXiv2023

Rate-induced tipping in complex high-dimensional ecological networks

In an ecosystem, environmental changes as a result of natural and human processes can cause some key parameters of the system to change with time. Depending on how fast such a parameter changes, a tipping point can occur. Existing works on rate-induced tipping, or R-tipping, offered a theoretical way to study this phenomenon but from a local dynamical point of view, revealing, e.g., the existence of a critical rate for some specific initial condition above which a tipping point will occur. As ecosystems are subject to constant disturbances and can drift away from their equilibrium point, it is necessary to study R-tipping from a global perspective in terms of the initial conditions in the entire relevant phase space region. In particular, we introduce the notion of the probability of R-tipping defined for initial conditions taken from the whole relevant phase space. Using a number of real-world, complex mutualistic networks as a paradigm, we discover a scaling law between this probability and the rate of parameter change and provide a geometric theory to explain the law. The real-world implication is that even a slow parameter change can lead to a system collapse with catastrophic consequences. In fact, to mitigate the environmental changes by merely slowing down the parameter drift may not always be effective: only when the rate of parameter change is reduced to practically zero would the tipping be avoided. Our global dynamics approach offers a more complete and physically meaningful way to understand the important phenomenon of R-tipping.

q-bio.PE↗

arXiv2023

Consensus Formation Among Mobile Agents in Networks of Heterogeneous Interaction Venues

Exploring the collective behavior of interacting entities is of great interest and importance. Rather than focusing on static and uniform connections, we examine the co-evolution of diverse mobile agents experiencing varying interactions across both space and time. Analogous to the social dynamics of intrinsically diverse individuals who navigate between and interact within various physical or digital locations, agents in our model traverse a complex network of heterogeneous environments and engage with everyone they encounter. The precise nature of agents internal dynamics and the various interactions that nodes induce are left unspecified and can be tailored to suit the requirements of individual applications. We derive effective dynamical equations for agent states which are instrumental in investigating thresholds of consensus, devising effective attack strategies to hinder coherence, and designing optimal network structures with inherent node variations in mind. We demonstrate that agent cohesion can be promoted by increasing agent density, introducing network heterogeneity, and intelligently designing the network structure, aligning node degrees with the corresponding interaction strengths they facilitate. Our findings are applied to two distinct scenarios: the synchronization of brain activities between interacting individuals, as observed in recent collective MRI scans, and the emergence of consensus in a cusp catastrophe model of opinion dynamics.

physics.soc-ph↗

arXiv2023

Stability and Fluctuations in Complex Ecological Systems

From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. This paper outlines eight areas including (1) improving our understanding of the effect of scale, both temporal and spatial, for both deterministic and stochastic problems; (2) clarifying the different terminologies and definitions used in different scientific fields; (3) developing a comprehensive set of data analysis techniques arising from different fields but which can be used together to improve our understanding of existing data sets; (4) having theoreticians/computational scientists collaborate closely with empirical ecologists to determine what new data should be collected; (5) improving our knowledge of how to protect and/or restore ecosystems; (6) incorporating socio-economic effects into models of ecosystems; (7) improving our understanding of the role of deterministic and stochastic fluctuations; (8) studying the current state of biodiversity at the functional level, taxa level and genome level.

q-bio.PE↗

arXiv2023

The storage effect is not about bet-hedging or population stage-structure

The storage effect is a well-known explanation for coexistence in temporally varying environments. Like many complex ecological theories, the storage effect is often used as an explanation for observed coexistence on the basis of heuristic understanding, rather than careful application of a detailed model. One interpretation states that species coexist by specializing on specific environmental states, and therefore must have a robust life-stage (e.g., long-lived adults, a seed-bank) in order to "wait it out" for favorable conditions. Here we show that this widely employed interpretation can be misleading. Multiple models show that stage-structure, long lifespans, and overlapping generations are neither necessary nor sufficient for the storage effect. In models where a robust life-stage does engender a storage effect, it does not do so by preventing stochastic extinction or by improving relative bet-hedging. A robust life-stage is best understood as one of many ways to fulfill an abstract condition for the storage effect: an interaction effect of environment and competition on per capita growth rates. Using a dataset of annual plants from a Mediterranean grassland in Spain, we show that such interaction effects occur between water availability and the number of germinant competitors, leading to storage in the absence of a persistent seed bank. Empiricists hoping to uncover the storage effect should look for interaction effects between environmental conditions and competition -- easily identifiable with multiple regression -- at all stages of a species' life-cycle.

q-bio.PE↗

arXiv2022

Resolving conceptual issues in Modern Coexistence Theory

In this paper, we discuss the conceptual underpinnings of Modern Coexistence Theory (MCT), a quantitative framework for understanding ecological coexistence. In order to use MCT to infer how species are coexisting, one must relate a complex model (which simulates coexistence in the real world) to simple models in which previously proposed explanations for coexistence have been codified. This can be accomplished in three steps: 1) relating the construct of coexistence to invasion growth rates, 2) mathematically partitioning the invasion growth rates into coexistence mechanisms (i.e., classes of explanations for coexistence), and 3) relating coexistence mechanisms to simple explanations for coexistence. Previous research has primarily focused on step 2. Here, we discuss the other crucial steps and their implications for inferring the mechanisms of coexistence in real communities. Our discussion of step 3 -- relating coexistence mechanisms to simple explanations for coexistence -- serves a heuristic guide for hypothesizing about the causes of coexistence in new models; but also addresses misconceptions about coexistence mechanisms. For example, the storage effect has little to do with bet-hedging or "storage" via a robust life-history stage; relative nonlinearity is more likely to promote coexistence than originally thought; and fitness-density covariance is an amalgam of a large number of previously proposed explanations for coexistence (e.g., the competition-colonization trade-off, heteromyopia, spatially-varying resource supply ratios). Additionally, we review a number of topics in MCT, including the role of "scaling factors"; whether coexistence mechanisms are approximations; whether the magnitude or sign of invasion growth rates matters more; whether Hutchinson solved the paradox of the plankton; the scale-dependence of coexistence mechanisms; and much more.

q-bio.PE↗

arXiv2022

Coexistence in spatiotemporally fluctuating environments

Ecologists have put forward many explanations for coexistence, but these are only partial explanations; nature is complex, so it is reasonable to assume that in any given ecological community, multiple mechanisms of coexistence are operating at the same time. Here, we present a methodology for quantifying the relative importance of different explanations for coexistence, based on an extension of Modern Coexistence Theory. Current versions of Modern Coexistence Theory only allow for the analysis of communities that are affected by spatial or temporal environmental variation, but not both. We show how to analyze communities with spatiotemporal fluctuations, how to parse the importance of spatial variation and temporal variation, and how to measure everything with either mathematical expressions or simulation experiments. Our extension of Modern Coexistence Theory allows empiricists to use realistic models and more data to better infer the mechanisms of coexistence in real communities.

q-bio.PE↗

arXiv2022

Towards a heuristic understanding of the storage effect

The storage effect is a general explanation for coexistence in a variable environment. The generality of the storage effect is both a strength - it can be quantified in many systems - and a challenge - there is not a clear relationship between the abstract conditions for storage effect and species' life-history traits (e.g., dormancy, stage-structure, non-overlapping generations), thus precluding a simple ecological interpretation of the storage effect. Our goal here is to provide a clearer understanding of the conditions for the storage effect as a step towards a better general explanation for coexistence in a variable environment. Our approach focuses on dividing one of the key conditions for the storage effect, covariance between environment and competition, into two pieces, namely that there must be a causal relationship between environment and competition, and that the effects of the environment do not change too quickly. This finer-grained definition can explain a number of previous results, including 1) that the storage effect promotes annual plant coexistence when the germination rate fluctuates, but not when the seed yield fluctuates, 2) that the storage effect is more likely to be induced by resource competition than apparent competition, and 3) that the spatial storage effect is more probable than the temporal storage effect. Additionally, our expanded definition suggests two novel mechanisms by which the temporal storage effect can arise: transgenerational plasticity, and causal chains of environmental variables. These mechanisms produce coexistence via the storage effect without any need for stage structure or a temporally autocorrelated environment.

q-bio.PE↗

arXiv2022

Methods for calculating coexistence mechanisms: Beyond scaling factors

How do species coexist? A framework known as Modern Coexistence Theory measures mechanisms of coexistence by comparing a species perturbed to low density (the invader) to other species that remain at their typical densities (the residents); this invader-resident comparison measures a rare-species advantage that results from specialization. However, there are several reasonable ways (i.e., methods) to compare invaders and residents, each differing in practicality and biological interpretation. Here, using theoretical arguments and case studies, we compare four such methods for calculating coexistence mechanisms: 1) Scaling factors, the traditional approach where resident growth rates are scaled by a measure of relative sensitivity to competition, obtained by solving a system of linear equations; 2) The simple comparison, which gives equal weight to all resident species; 3) Speed conversion factors, a novel method in which resident growth rates are scaled by a ratio of generation times, and; 4) The invader-invader comparison, another novel method in which a focal species is compared to itself at high vs. low density. We conclude that the conventional scaling factors can be useful in some theoretical research, but are not recommended for empirical applications, i.e., determining the mechanisms of coexistence in real communities. Instead, we recommend the simple comparison and speed conversion factor methods. The speed conversion factors are most useful when comparing species with dissimilar generation times. However, ecologists often study coexistence in guilds of species with similar life-histories, and therefore, similar generation times. In such scenarios, the easier-to-use simple comparison method is reasonable.

q-bio.PE↗

arXiv2022

Characterizing long transients in consumer-resource systems with group defense and nonreproductive stages

During recent years, the study of long transients has been expanded in ecological theory to account for shifts in long-term behavior of ecological systems. These long transients may lead to regime shifts between alternative states that resemble the dynamics of alternative stable states for a prolonged period of time. One dynamic that potentially leads to long transients is the group defense of a resource in a consumer-resource interaction. Furthermore, time lags in the population caused by discrete reproductive pulses have the potential to produce long transients, either independently or in conjunction to the transients caused by the group defense. In this work, we analyze the potential for long transients in a model for a consumer-resource system in which the resource exhibits group defense and reproduces in discrete reproductive pulses. We develop this discrete-time model by discretizing a pulse differential equation. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource. In addition, we identify a transcritical bifurcation in our system, under which a ghost limit cycle appears. These transients resemble stable states for a prolonged transient time period. We estimate the transient time of our system from these transients using perturbation theory. This work advances an understanding of how systems shift between alternate states and their duration of staying in a given regime and what ecological dynamics may lead to long transients.

q-bio.PE↗

arXiv2020

A Low-Dimensional Network Model for an SIS Epidemic: Analysis of the Super Compact Pairwise Model

Network-based models of epidemic spread have become increasingly popular in recent decades. Despite a rich foundation of such models, few low-dimensional systems for modeling SIS-type diseases have been proposed that manage to capture the complex dynamics induced by the network structure. We analyze one recently introduced model and derive important epidemiological quantities for the system. We derive the epidemic threshold and analyze the bifurcation that occurs, and we use asymptotic techniques to derive an approximation for the endemic equilibrium when it exists. We consider the sensitivity of this approximation to network parameters, and the implications for disease control measures are found to be in line with the results of existing studies.

math.DS↗

arXiv2020

Occupancy times for time-dependent stage-structured models

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick--von F{ö}rster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. We apply our approach to study a time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.

math.PR↗

arXiv2020

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation, and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator-prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e., synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry - henceforth the phenomenon of "synchronization within synchronization." Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

q-bio.PE↗