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New graduate students in biology programs may lack the quantitative skills necessary for their research and professional careers. The acquisition of these skills may be impeded by teaching and mentoring experiences that decrease rather than increase students’ beliefs in their ability to learn and apply quantitative approaches. In this opinion piece, we argue that revising instructional experiences to ensure that both student confidence and quantitative skills are enhanced may improve both educational outcomes and professional success. A few studies suggest that explicitly addressing productive failure in an instructional setting and ensuring effective mentoring may be the most effective routes to simultaneously increasing both quantitative self-efficacy and quantitative skills. However, there is little work that specifically addresses graduate student needs, and more research is required to reach evidence-backed conclusions.

Understanding population responses to discrete 'pulsed' environmental disturbances is essential to conservation and adaptive management. Populations of concern can be driven to low levels by disturbance, and understanding interspecific differences in recovery trajectories is necessary for evaluating management options. We analysed single-species models to investigate the demographic and management factors determining the two components of population 'resilience': the magnitude of initial impact on population abundance, and duration of the recovery time. We simulated age-structured populations with density-dependent recruitment, subjected to a pulse disturbance consisting of a period of increased mortality of either the juvenile age class or all age classes, and calculated both impact and return time. For illustration, we used demographic parameters from a suite of 16 fish species. We formulated the model as a renewal equation, allowing us to describe disturbance impacts mathematically as a convolution. We also included nonlinear dynamics, representing populations that recover to a steady state; this is more realistic (in most cases) than prior analyses of resilience in linear models without density-dependence. When the disturbance affected only one or a few young age-classes, longevity was the major life-history determinant of impact and recovery time. Shorter-lived species endured greater impacts when disturbed because each age class is a greater proportion of the population. However, shorter-lived species also had faster recovery times, for the same reason. When disturbance affected adult age-classes, the impact was more immediate and no longer affected by species' longevity, though the effect of longevity on recovery time remained. These results improve our understanding of interspecific differences in resilience and increase our ability to make predictions for adaptive management. Additionally, formulating the problem as a renewal equation and using mathematical convolutions allows us to quantify how disturbances with different time courses (not just an immediate, constant level of disturbance but gradually increasing or decreasing levels of disturbance) would have different effects on population resilience: delayed responses for species in which biomass is concentrated in older age classes, and for disturbances that become progressively more severe.Entender las respuestas de la población a perturbaciones ambientales, específicamente a pulsadas individuales, es esencial para la conservación y la gestión adaptativa. Las poblaciones de interés pueden reducirse a niveles bajas debido a la perturbación, y es necesario entender las diferencias interespecíficas en las trayectorias de recuperación para evaluar las opciones de gestión. Analizamos modelos para especies individuales para investigar los factores demográficos y de gestión que determinan los dos componentes de la ‘resiliencia’ de la población: la magnitud del impacto inicial sobre la abundancia de la población y la duración del tiempo de recuperación. Simulamos poblaciones estructuradas por edad con reclutamiento que depende de la densidad, las sometimos a una perturbación pulsada que consiste en un período de mayor mortalidad del grupo etário juvenil o de todos los grupos etários, y calculamos tanto el impacto como el tiempo de retorno. A modo de ilustración, utilizamos parámetros demográficos de un conjunto de 16 especies de peces. Formulamos el modelo como una ecuación de renovación, lo que nos permite describir matemáticamente los impactos de las perturbaciones como una convolución. También incluimos dinámicas no lineales que representan poblaciones que se recuperan hacia un estado estable; esto es más realista (en la mayoría de los casos) que los análisis previos de resiliencia en modelos lineales sin la dependencia de la densidad. Cuando la perturbación ha afectado a uno o a algunos pocos grupos etários jóvenes, la longevidad fue el principal determinante de la historia de vida del impacto y el tiempo de recuperación. Las especies de vida más corta sufrieron mayores impactos cuando fueron perturbadas porque cada grupo etáreo representa una mayor proporción de la población. Sin embargo, las especies con vidas más cortas también tuvieron tiempos de recuperación más rápidos, por la misma razón. Cuando la perturbación afectó a los grupos etários adultos, el impacto fue más inmediato y ya no se vio afectado por la longevidad de las especies, aunque se mantuvo el efecto de la longevidad sobre el tiempo de recuperación. Estos resultados mejoran nuestra comprensión de las diferencias interespecíficas de la resiliencia y aumentan nuestra capacidad para hacer predicciones con fin a la gestión adaptativa. Además, formular el problema como una ecuación de renovación y usar convoluciones matemáticas nos permite cuantificar cómo las perturbaciones con distintos lapsos de tiempo (no solo un nivel de perturbación constante e inmediato, sino niveles de perturbación que aumentan o disminuyen gradualmente) tendrían diferentes efectos sobre la resiliencia de la población: respuestas tardías para especies en las que la biomasa se concentra en grupos etários de mayor edad y para perturbaciones que se vuelven progresivamente más severas.

The storage effect is a general explanation for coexistence in a variable environment. Unfortunately, the storage effect is poorly understood, in part because the generality of the storage effect precludes an interpretation that is simultaneously simple, intuitive and correct. Here, we explicate the storage effect by dividing one of its key conditions-covariance between environment and competition-into two pieces, namely that there must be a strong 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 the apparent competition, and (3) why the storage effect arises readily in models with either stage structure or environmental autocorrelation. Additionally, our expanded definition suggests two novel mechanisms by which the temporal storage effect can arise-transgenerational plasticity and causal chains of environmental variables-thus suggesting that the storage effect is a more common phenomenon than previously thought.

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 the 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 shows that many more species can coexist than originally thought. More importantly, it allows empiricists to use realistic models and more data to better infer the mechanisms of coexistence in real communities.

Sudden changes in populations are ubiquitous in ecological systems, especially under perturbations. The agents of global change may increase the frequency and severity of anthropogenic perturbations, but complex populations’ responses hamper our understanding of their dynamics and resilience. Furthermore, the long-term environmental and demographic data required to study those sudden changes are rare. Fitting dynamical models with an artificial intelligence algorithm to population fluctuations over 40 y in a social bird reveals that feedback in dispersal after a cumulative perturbation drives a population collapse. The collapse is well described by a nonlinear function mimicking social copying, whereby dispersal made by a few individuals induces others to leave the patch in a behavioral cascade for decision-making to disperse. Once a threshold for deterioration of the quality of the patch is crossed, there is a tipping point for a social response of runaway dispersal corresponding to social copying feedback. Finally, dispersal decreases at low population densities, which is likely due to the unwillingness of the more philopatric individuals to disperse. In providing the evidence of copying for the emergence of feedback in dispersal in a social organism, our results suggest a broader impact of self-organized collective dispersal in complex population dynamics. This has implications for the theoretical study of population and metapopulation nonlinear dynamics, including population extinction, and managing of endangered and harvested populations of social animals subjected to behavioral feedback loops.

Compartmental models play an important role to describe the dynamics of systems that involve mass movements between different types of pools. We develop a theory to analyse the average ages of mass in different pools in a linear compartmental system with time-dependent (i.e. non-autonomous) transfer rates, which involves transit times that characterize the average time a particle has spent in a particular pool. We apply our theoretical results to investigate a nine-dimensional compartmental system with time-dependent fluxes between pools modelling the carbon cycle which is a modification of the Carnegie–Ames–Stanford approach model. Knowledge of transit time and mean age allows calculation of carbon storage in a pool as a function of time. The general result that has important implications for understanding and managing carbon storage is that the change in storage in different pools does not change monotonically through time: as rates change monotonically a pool which initially shows a decrease may then show an increase in storage or vice versa. Thus caution is needed in extrapolating even the direction of future changes in storage in carbon storage in different pools with global change.

While much of theoretical ecology has focused on the study of long-term asymptotic dynamics of populations, a growing realization is that the study of dynamics on more ecologically relevant timescales is of equal or greater importance. Often, dynamics that occur on shorter, ecologically relevant, timescales are referred to as transient dynamics, and include examples such as disease outbreaks and regime shifts. While any perturbation or environmental change that moves a system away from equilibrium will trigger a transient, this article focuses specifically on those “long transients” whose impact persists over a meaningful timescale (very roughly, dozens of generations or longer). Transient dynamics often lead to significant qualitative changes in an ecological system’s state that can occur with little to no warning at all. In the context of management, transients can cloud the manager’s judgment of the stability of a system. It would be easy to assume the system is in a stable state when it may be near a regime shift (long transients), which may result in inaccurate predictions of long-term dynamics based solely on short-term data. Consequently, the study of transient dynamics involves several avenues, including mathematical theory, management implications, and empirical understanding. New literature in these areas has begun to emerge. The notion of transient dynamics was discussed as early as the mid-1900s, where the idea that steady states are the most ecologically relevant quantities was challenged and the exploration of more complex ecological dynamics was sought. In more recent literature, transient dynamics, the underlying theory, empirical evidence, management implications, and ecological implications have all been more explicitly discussed. One major challenge when dealing with transient dynamics is the infancy of the mathematical theory that underpins much of the necessary analysis. Additionally, the implications transient dynamics have on management and ecological applications are still being developed. However, the significance of transients is now appreciated and research in such areas is attracting much attention. Several important avenues for future research have been made apparent in ecology, and new insights continue to be developed.

Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favourable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Our results broaden the theory of species coexistence in heterogeneous space and provide empirical confirmation of the mathematical predictions.

One of the main factors that determines habitat suitability for sessile and territorial organisms is the presence or absence of another competing individual in that habitat. This type of competition arises in populations occupying patches in a metacommunity. Previous studies have looked at this process using a continuous-time modeling framework, where colonizations and extinctions occur simultaneously. However, different colonization processes may be performed by different species, which may affect the metacommunity dynamics. We address this issue by developing a discrete-time framework that describes these kinds of metacommunity interactions, and we consider different colonization dynamics. To understand potential dynamics, we consider specific functional forms that characterize the colonization and extinction processes of metapopulations competing for space as their limiting factor. We then provide a mathematical analysis of the models generated by this framework, and we compare these results to what is seen in nature and in previous models.

Adaptive management of marine protected areas (MPAs) to determine whether they are meeting their intended goals requires predicting how soon those goals will be realized. Such predictions have been made for increases in fish abundance and biomass inside MPAs. However, projecting increases in fishery yield (“fishery spillover”) is more complex because it involves both how the fishery is managed and uncertainty in larval connectivity. We developed a two-patch, age-structured population model, based on a renewal equation approach, to project the initial timing of increase in fishery yield from larvae exported from a no-take MPA. Our results link our understanding of the predicted timing of increases in biomass (and thus reproduction) in MPAs with the time-lags associated with new recruits entering the fishery. We show that the time-lag between biomass peaking within the MPA and the increased fishery yield outside the MPA reaching its maximum depends, in a predictable way, on the age-dependent patterns of growth, natural mortality, and fishing mortality. We apply this analysis to 16 fishery species from the US Pacific coast; this difference ranged from 7 to 18 years. This model provides broadly applicable guidance for this important emerging aspect of fisheries management.

Many benthic animals begin life with a planktonic larval stage during which coastal currents may move individuals far from shore. This trait is believed to allow individuals to develop away from nearshore predators and sibling competition, based on the assumption that mortality rates are weaker offshore. However, larvae developing offshore often fail to locate suitable coastal habitats. This results in a trade-off between nearshore mortality and offshore wastage with consequences for larval delivery to adult habitats that have not been fully appreciated. We use a reaction-diffusion model to show that when the nearshore larval mortality rate is high, larval supply can vary more than 10-fold with the offshore mortality rate. If this offshore rate is weak, then larval supply is maximized by an intermediate diffusion rate or larval duration. While a low-diffusivity coastal boundary layer typically improves the larval supply by decreasing wastage, it can also reduce the larval supply by preventing individuals from exploiting low offshore mortality rates. Finally, the cross-shore structure of the mortality rate may influence the alongshore transport of larvae by determining how far offshore they reside prior to settling, and, consequently, the alongshore currents they experience. Our observations contrast with the prior argument that larval supply decreases with diffusivity and larval duration due to wastage, and challenge the widespread decision to omit cross-shore heterogeneity from studies of alongshore movement. Scenarios in which spatial variability in the mortality rate has a large effect on recruitment are important both for understanding the biological consequences of the larval stage and from a modeling perspective.

Unexpected population crashes are an important feature of natural systems, yet many observed crashes have not been explained. Two difficulties in explaining population crashes are their relative rarity and the multi-causal nature of ecological systems. We approach this issue with experimental microcosms, with large numbers of replicates of red flour beetle populations (Tribolium castaneum). We determined that population crashes are caused by an interaction between stochasticity and successive episodes of density dependence: demographic stochasticity in oviposition rates occasionally produces a high density of eggs; so high that there are insufficient flour resources for subsequent larvae. This mechanism can explain unexpected population crashes in more general settings: stochasticity 'pushes' population into a regime where density dependence is severely overcompensatory. The interaction between nonlinearity and stochasticity also produces chaotic population dynamics and a double-humped one-generation population map, suggesting further possibilities for unexpected behaviour in a range of systems. We discuss the generality of our proposed mechanism, which could potentially account for previously inexplicable population crashes.

The underlying biological processes that govern many ecological systems can create very long periods of transient dynamics. It is often difficult or impossible to distinguish this transient behaviour from similar dynamics that would persist indefinitely. In some cases, a shift from the transient to the long-term, stable dynamics may occur in the absence of any exogenous forces. Recognizing the possibility that the state of an ecosystem may be less stable than it appears is crucial to the long-term success of management strategies in systems with long transient periods. Here we demonstrate the importance of considering the potential of transient system behaviour for management actions across a range of ecosystem organizational scales and natural system types. Developing mechanistic models that capture essential system dynamics will be crucial for promoting system resilience and avoiding system collapses.

We present a single-species metapopulation model structured by population size that is discrete in time. The novel formulation of our model allows for explicit incorporation of both the local, in space, dynamics and new details of the dispersal process. To study the impact of between-patch dynamics in the model, we construct various functions to describe density-dependent dispersal, recolonization, and between-patch stochasticity. Due to the complexity of the model, numerical simulations are used to obtain the distribution of the metapopulation. Moreover, we can use our model to predict the proportion of patches occupied, expected patch density, and variance in patch density. Our results provide insight into the influence that each of these processes play in the distribution of the metapopulation. In particular, our findings emphasize the benefit of explicitly modelling these processes. We consider two density-dependent dispersal strategies based upon individuals wanting to form average-sized patches and illustrate the differences in the metapopulation distributions. For recolonization of empty suitable patches, we look at two simplistic redistribution strategies modeled as either a continuous uniform or Laplace distribution. The between-patch stochasticity is modelled using redistribution kernels where we consider both thin-tailed and fat-tailed kernels. We find that the form of density-dependent dispersal has a major influence on the distribution of the metapopulation while recolonization is a key process for metapopulation persistence. In addition, fatter tails for the between-patch stochasticity can decrease the proportion of occupied patches and increase the expected patch density and variance in patch density.

Species expanding into new habitats as a result of climate change or human introductions will frequently encounter resident competitors. Theoretical models suggest that such interspecific competition can alter the speed of expansion and the shape of expanding range boundaries. However, competitive interactions are rarely considered when forecasting the success or speed of expansion, in part because there has been no direct experimental evidence that competition affects either expansion speed or boundary shape. Here we demonstrate that interspecific competition alters both expansion speed and range boundary shape. Using a two-species experimental system of the flour beetles Tribolium castaneum and Tribolium confusum, we show that interspecific competition dramatically slows expansion across a landscape over multiple generations. Using a parameterized stochastic model of expansion, we find that this slowdown can persist over the long term. We also find that the shape of the moving range boundary changes continuously over many generations of expansion, first steepening and then becoming shallower, due to the competitive effect of the resident and density-dependent dispersal of the invader. This dynamic boundary shape suggests that current forecasting approaches assuming a constant shape could be misleading. More broadly, our results demonstrate that interactions between competing species can play a large role during range expansions and thus should be included in models and studies that monitor, forecast, or manage expansions in natural systems.

Dispersal is a key process in shaping species spatial distributions. Species interactions and variation in dispersal probabilities may jointly influence species spatial dynamics. However, many studies examine dispersal as a neutral process, independent of community context or intraspecific variation in dispersal behaviour. Here, we use controlled, replicated communities of two Tribolium species (T. castaneum and T. confusum) to examine how intraspecific variation in dispersal behaviour and community context influence dispersal dynamics in simple experimental landscapes composed of homogeneous habitat patches. We found considerable individual-level variation in dispersal probability that was unrelated to body size variation. Further, the context of dispersal mattered, as T. castaneum dispersal was reduced in two-species communities, while T. confusum dispersal was unaffected by community composition. Incorporating individual-level variation into a two-species stochastic spatial Ricker model, we provide evidence that individual-level variability in dispersal behaviour results in more variable spatial spread than assuming individuals have the same dispersal probability. Further, interspecific competition resulted in more variable spatial spread. The variability in spatial spread observed in our tightly controlled and replicated experimental system and in our stochastic model simulations points to potential fundamental limitations in forecasting species shifting ranges without considering potential interspecific interactions and demographic variability in dispersal behaviour.

This paper discusses the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms along with the knowledge of scaling laws of the transient's lifetime should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This text aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and critically examining a broad variety of mathematical models used in ecological applications as well as empirical facts, we reveal several main mechanisms leading to the emergence of long transients and hence lays the basis for a unifying theory.

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., 8 pages, 2 figures

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. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource, and a transcritical bifurcation, under which a ghost limit cycle appears. 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.

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., 40 pages, 1 figure, 2 tables

Climate-driven habitat shifts pose challenges for dispersal-limited, late-maturing taxa such as trees. Older trees are often the most reproductive individuals in the population, but as habitats shift, these individuals can be left behind in the trailing range edge, generating “zombie forests” that may persist long after the suitable habitat has shifted. Are these zombie forests vestiges of ecosystems past or do they play an ecological role? To understand how zombie forests affect population persistence, we developed a spatially explicit, stage-structured model of tree populations occupying a shifting habitat. Our model shows that seed dispersal from zombie forests to the range core can considerably increase the maximum rate of climate change that a population can withstand. Moreover, the entire core population can ultimately descend from recruitment-limited zombie forests, highlighting their demographic value. Our results suggest that preserving trailing-edge zombie forests can greatly increase population persistence in the face of climate change.

Integrodifference equations are a discrete time spatially explicit model that describes dispersal of ecological populations through space. This framework is useful to study spread dynamics of organisms and how ecological interactions can affect their spread. When studying interactions such as consumption, dispersal rates might vary with life cycle stage, such as cases with dispersive juveniles and sessile adults. In the non-dispersive stage, resources may engage in group defense to protect themselves from consumption. These local nondispersive interactions may limit the number of dispersing recruits that are produced and therefore affect how fast populations can spread. We present a spatial consumer-resource system using an integrodifference framework with limited movement of their adult stages and group defense mechanisms in the resource population. We model group defense using a Type IV Holling functional response, which limits survival of adult resource population and enhances juvenile consumers production. We find that high mortality levels for sessile adults can destabilize resource at carrying capacity. Furthermore, we find that at high resource densities, group defense leads to a slower local growth of resource in newly invaded regions due to intraspecific competition outweighing the effect of consumption on resource growth.

In restoration ecology, the Field of Dreams Hypothesis posits that restoration efforts that create a suitable environment could lead to eventual recovery of the remaining aspects of the ecosystem through natural processes. Natural processes following partial restoration has lead to ecosystem recovery in both terrestrial and aquatic systems. However, understanding the efficacy of a “field of dreams” approach requires comparison of different approaches to partial restoration in terms of spatial, temporal, and ecological scale to what would happen with more comprehensive restoration efforts. We explore the relative effect of partial restoration and ongoing recovery on restoration efficacy with a dynamical model based on temperate rocky reefs in Northern California. We analyze our model for both the ability and rate of bull kelp forest recovery under different restoration strategies. We compare the efficacy of a partial restoration approach with a more comprehensive restoration effort by exploring how kelp recovery likelihood and rate change with varying intensities of urchin removal and kelp outplanting over different time periods and spatial scales. We find that, for the case of bull kelp forests, setting more favorable initial conditions for kelp recovery through implementing both urchin harvesting and kelp outplanting at the start of the restoration project has a bigger impact on the kelp recovery rate than applying restoration efforts through a longer period of time. Therefore partial restoration efforts, in terms of spatial and temporal scale, can be significantly more effective when applied across multiple ecological scales in terms of both the capacity and rate of achieving the target outcomes.

Nonlinear physics and agroecosystems can be of great relevance in the synchronisations of chaotic oscillators. The endogenous dynamics of the seed production of perennial plant species which include alternate bearing and masting, portray typical synchronisation patterns in nature and can be modelled using a tent map known as a resource budget model (RBM). This study investigates the collective rhythm in 9,562 pistachio trees caused by their endogenous network dynamics and exogenous forces (common noise). Common noise and a local coupling of RBMs are the two primary factors emerging from the bearing phase synchronisation in this orchard. The in-phase/out-of-phase analysis technique quantifying the strength of the phase synchronisation in trees (population /individual) allows us to study the observed spatial synchrony in detail. We demonstrate how three essential factors, i.e. (a) common noise, (b) local direct coupling, and (c) the gradient of the cropping coefficient, explain the spatial synchrony of the orchard. Here, we also show that the methodology employing nonlinear physics to study agroecological systems can be useful for resolving practical problems in agriculture including yield variability and spatial synchrony which often compromise efficient resource management.

As many ecosystems worldwide are in peril, efforts to manage them sustainably require scientific advice. While numerous researchers around the world use a great variety of models to understand ecological dynamics and their responses to disturbances, only a small fraction of these models are ever used to inform ecosystem management. There seems to be a perception that ecological models are not useful for management, even though mathematical models are indispensable in many other fields. We were curious about this mismatch, its roots, and potential ways to overcome it. We searched the literature on recommendations and best practices for how to make ecological models useful to the management of ecosystems and we searched for ‘success stories’ from the past. We selected and examined several cases where models were instrumental in ecosystem management. We documented their success and asked whether and to what extent they followed recommended best practices. We found that there is not a unique way to conduct a research project that is useful in management decisions. While research is more likely to have impact when conducted with many stakeholders involved and specific to a situation for which data are available, there are great examples of small groups or individuals conducting highly influential research even in the absence of detailed data. We put the question of modelling for ecosystem management into a socio-economic and national context and give our perspectives on how the discipline could move forward.

Forecasting tipping points in spatially extended systems is a key area of interest to ecologists. A slowly declining spatially distributed population is an important example of an ecological system that could exhibit a cascade of tipping points. Here, we develop a spatial two-patch model with environmental stochasticity that is slowly forced through population collapse, in the presence of changing environmental conditions. We begin with a basic spatial model, then introduce a fast–slow version of the model using geometric singular perturbation theory, followed by the inclusion of stochasticity. Using the spectral density of the fluctuating subpopulation in each patch, we derive analytic expressions for candidate indicators of population extinction and evaluate their performance through a simulation study. We find that coupling and spatial heterogeneity decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations. Moreover, the degree of coupling dictates the trends in summary statistics. We conclude that this theory may be applied to other contexts, including the control of invasive species.

Robert (Bob) May, Baron May of Oxford, a long-time International Member of the National Academy of Sciences, passed away on April 28, 2020, in Oxford, United Kingdom, from “frailties of old age, Alzheimer's disease, and pneumonia,” at the age of 84. He is survived by his wife Judith, his daughter Nome, and legions of students and colleagues. Across six decades, Bob had a transformative impact on myriad fields: mathematical biology, ecology, epidemiology, public policy, and finance. His intellect, sense of loyalty to colleagues and students, and commitment to humanity made him a towering giant in science and science policy. Robert M. May, c. 2004. Image credit: Royal Society/Laurence Bulaitis. Bob May was born in Sydney, Australia, where his talents as a mathematician were evident from an early age. Bob was trained in theoretical physics at the University of Sydney by the distinguished physicist Robbie Schafroth, receiving his PhD at 24 years old for his work on bosons and superconductivity. Nonetheless, May traced his good luck in mentorships back to his high school chemistry teacher Lenny Basser and was always quick to give accolades to the many influences on his development, from teachers and colleagues alike. For those who knew him, it was clear that he made it a point to pay forward this debt through his investment in his own students and others. After a postdoctoral stint with Max Krook at Harvard University—during which he met his future wife, Judith—Bob accepted a faculty position at the University of Sydney, and within a few years had risen to one of the first “Personal Professorships” at the university. In his early work, he focused on problems in plasma physics and related fields and quickly rose to prominence. However, soon after, concerned for the fate of the planet and feeling he had a … [↵][1]1To whom correspondence may be addressed. Email: slevin{at}princeton.edu. [1]: #xref-corresp-1-1

When managing natural systems, the importance of recognizing the role of uncertainty has been formalized as the precautionary approach. However, it is difficult to determine the role of stochasticity in the success or failure of management because there is almost always no replication; typically, only a single observation exists for a particular site or management strategy. Yet, assessing the role of stochasticity is important for providing a strong foundation for the precautionary approach, and learning from past outcomes is critical for implementing adaptive management of species or ecosystems. In addition, adaptive management relies on being able to implement a variety of strategies in order to learn—an often difficult task in natural systems. Here, we show that there is large, stochastically driven variability in success for management treatments to control an invasive species, particularly for moderate, and more feasible, management strategies. This is exactly where the precautionary approach should be important. Even when combining management strategies, we show that moderate effort in management either fails or is highly variable in its success. This variability allows some management treatments to, on average, meet their target, even when failure is probable. Our study is an important quantitative replicated experimental test of the precautionary approach and can serve as a way to understand the variability in management outcomes in natural systems which have the potential to be more variable than our tightly controlled system.

Marine Protected Areas (MPAs) are being implemented worldwide, yet there are few cases where managers make specific predictions of the response of previously harvested populations to MPA implementation. Such predictions are needed to evaluate whether MPAs are working as expected, and if not, why. This evaluation is necessary to perform adaptive management, identifying whether and when adjustments to management might be necessary to achieve MPA goals. Using monitoring data and population models, we quantified expected responses of targeted species to MPA implementation and compared them to monitoring data. The model required two factors to explain observed responses in MPAs: (a) pre‐MPA harvest rates, which can vary at local spatial scales, and (b) recruitment variability before and after MPA establishment. Low recruitment years before MPA establishment in our study system drove deviations from expected equilibrium population size distributions and introduced an additional time lag to response detectability. Synthesis and applications. We combined monitoring data and population models to show how (a) harvest rates prior to Marine Protected Area (MPA) implementation, (b) variability in recruitment, and (c) initial population size structure determine whether a response to MPA establishment is detectable. Pre‐MPA harvest rates across MPAs plays a large role in MPA response detectability, demonstrating the importance of measuring this poorly known parameter. While an intuitive expectation is for response detectability to depend on recruitment variability and stochasticity in population trajectories after MPA establishment, we address the overlooked role of recruitment variability before MPA establishment, which alters the size structure at the time of MPA establishment. These factors provide MPA practitioners with reasons whether or not MPAs may lead to responses of targeted species. Our overall approach provides a framework for a critical step of adaptive management.