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Harmful cyanobacterial blooms (CBs) have a growing global prevalence, emerging as a significant environmental concern due to their potential toxicity. Understanding how the different mechanisms affect CBs is crucial to develop actionable management strategies. For this, we derive a stoichiometric dynamical system that describes the qualitative population dynamics of cyanobacteria and their toxicity in north-temperate freshwater ecosystems. Our model quantifies the hypoxic effects of CBs on fish mortality and the effect of microcystin-LR (MC-LR), a potent toxin produced by cyanobacteria, on aquatic macro-invertebrates, phytoplankton, and fish species. By fitting the model to lakes with varying physical characteristics, eutrophic conditions, and water temperature, we can delineate and understand the driving components of CBs. We show that decreases in water exchange rate, depth of epilimnion, or light attenuation increases bloom intensity and duration. Furthermore, our models concur that eutrophication and increasing water temperatures exacerbate the intensity of CBs. We observe a severe bioaccumulative effect of MC-LR in aquatic species, stressing the potential impact on humans and other terrestrial animals. We validate our model with field measurements demonstrating its applicability to several realistic lake conditions. These insights are essential for informing targeted interventions to reduce CBs and their ecological impacts., Comment: Main Document: 11 pages

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., Comment: 22 pages, 6 figures, 1 table

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., Comment: 8 pages, 5 figures

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., Comment: 18 pages, 10 figures

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., Comment: 22 pages

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., Comment: 40 pages, 1 figure, 2 tables

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., Comment: 48 pages, 9 figures, 6 tables

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., Comment: 8 pages, 2 figures

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., Comment: 20 pages, 8 figures

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., Comment: 26 pages, 1 figure, 4 tables

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.

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.

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{\"o}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., Comment: 16 pages, 10 figures

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., Comment: 17 pages, 7 figures

Tipping points have been shown to be ubiquitous, both in models and empirically in a range of physical and biological systems. The question of how tipping points cascade through systems has been less well studied and is an important one. A study of noise-induced tipping, in particular, could provide key insights into tipping cascades. Here, we consider a specific example of a simple model system that could have cascading tipping points. This model consists of two interacting populations with underlying Allee effects and stochastic dynamics, in separate patches connected by dispersal, which can generate bistability. From an ecological standpoint, we look for rescue effects whereby one population can prevent the collapse of a second population. As a way to investigate the stochastic dynamics, we use an individual-based modeling approach rooted in chemical reaction network theory. Then, using continuous-time Markov chains and the theory of first passage times, we essentially approximate, or emulate, the original high-dimensional model by a Markov chain with just four states, where each state corresponds to a combination of population thresholds. Analysis of this reduced model shows when the system is likely to recover, as well as when tipping cascades through the whole system., Comment: 18 pages, 5 figures

Long-range synchrony from short-range interactions is a familiar pattern in biological and physical systems, many of which share a common set of ``universal'' properties at the point of synchronization. Common biological systems of coupled oscillators have been shown to be members of the Ising universality class, meaning that the very simple Ising model replicates certain spatial statistics of these systems at stationarity. This observation is useful because it reveals which aspects of spatial pattern arise independently of the details governing local dynamics, resulting in both deeper understanding of and a simpler baseline model for biological synchrony. However, in many situations a system's dynamics are of greater interest than their static spatial properties. Here, we ask whether a dynamical Ising model can replicate universal and non-universal features of ecological systems, using noisy coupled metapopulation models with two-cycle dynamics as a case study. The standard Ising model makes unrealistic dynamical predictions, but the Ising model with memory corrects this by using an additional parameter to reflect the tendency for local dynamics to maintain their phase of oscillation. By fitting the two parameters of the Ising model with memory to simulated ecological dynamics, we assess the correspondence between the Ising and ecological models in several of their features (location of the critical boundary in parameter space between synchronous and asynchronous dynamics, probability of local phase changes, and ability to predict future dynamics). We find that the Ising model with memory is reasonably good at representing these properties of ecological metapopulations. The correspondence between these models creates the potential for the simple and well-known Ising class of models to become a valuable tool for understanding complex biological systems.

We present a discrete-time model of a spatially structured population and explore the effects of coupling when the local dynamics contain a strong Allee effect and overcompensation. While an isolated population can exhibit only bistability and essential extinction, a spatially structured population can exhibit numerous coexisting attractors. We identify mechanisms and parameter ranges that can protect the spatially structured population from essential extinction, whereas it is inevitable in the local system. In the case of weak coupling, a state where one subpopulation density lies above and the other one below the Allee threshold can prevent essential extinction. Strong coupling, on the other hand, enables both populations to persist above the Allee threshold when dynamics are (approximately) out-of-phase. In both cases, attractors have fractal basin boundaries. Outside of these parameter ranges, dispersal was not found to prevent essential extinction. We also demonstrate how spatial structure can lead to long transients of persistence before the population goes extinct., Comment: 25 pages, 9 figures, Submitted to Bulletin of Mathematical Biology

Mutualistic interactions are vital constituents of ecological and socio-economic systems. Empirical studies have found that the patterns of reciprocal relations among the participants often shows the salient features of being simultaneously nested and modular. Whether and how these two structural properties of mutualistic networks can emerge out of a common mechanism however remains unclear. We propose a unified dynamic model based on the adaptation of niche relations that gives rise to both structural features. We apply Hutchinson's concept of niche interaction to networked cooperative species. Their niche relation evolves under the assumption of fitness maximization. Modularity and nestedness emerge concurrently through the accumulated local advantages in the structural and demographic distribution. A rich ensemble of key dynamical behaviors are unveiled in the dynamical framework. We demonstrate that mutualism can exhibit either a stabilizing or destabilizing effect on the evolved network, which undergoes a drastic transition with the overall competition level. Most strikingly, the adaptive network may exhibit a profound nature of history-dependency in response to environmental changes, allowing it to be found in alternative stable structures. The adaptive nature of niche interactions, as captured in our framework, can underlie a broad class of ecological relations and also socio-economic networks that engage in bipartite cooperation., Comment: 18 pages, 4 figures

In addition to their unusually long life cycle, periodical cicadas, {\it Magicicada} spp., provide an exceptional example of spatially synchronized life stage phenology in nature. Within regions ("broods") spanning 50,000 to 500,000 km$^2$, adults emerge synchronously every 13 or 17 years. While satiation of avian predators is believed to be a key component of the ability of these populations to reach high densities, it is not clear why populations at a single location remain entirely synchronized. We develop nonlinear Leslie matrix-type models of periodical cicadas that include predation-driven Allee effects and competition in addition to reproduction and survival. Using both analytical and numerical techniques, we demonstrate the observed presence of a single brood critically depends on the relationship between fecundity, competition, and predation. We analyze the single-brood, two-brood and all-brood equilibria in the large life-span limit using a tractable hybrid approximation to the Leslie matrix model with continuous time competition in between discrete reproduction events. Within the hybrid model we prove that the single-brood equilibrium is the only stable equilibrium. This hybrid model allows us to quantitatively predict population sizes and the range of parameters for which the stable single-brood and unstable two-brood and all-brood equilibria exist. The hybrid model yields a good approximation to the numerical results for the Leslie matrix model for the biologically relevant case of a 17-year lifespan., Comment: 19 pages, 2 figures, v3 contains a general proof of the instability of multi-brood steady states