Alternative (un)stable states in a stochastic predator–prey model

First Passage Times of Long Transient Dynamics in Ecology

Long transient dynamics in ecological models are characterized by extended periods in one state or regime before an eventual, and often abrupt, transition. One mechanism leading to long transient dynamics is the presence of ghost attractors, states where system dynamics slow down and the system lingers before eventually transitioning to the true attractor. This transition results solely from system dynamics rather than external factors. This paper investigates the dynamics of a classical herbivore-grazer model with the potential for ghost attractors or alternative stable states. We propose an intuitive threshold for first passage time analysis applicable to both bistable and ghost attractor regimes. By formulating the first passage time problem as a backward Kolmogorov equation, we examine how the mean first passage time changes as parameters are varied from the ghost attractor regime to the bistable one, through a saddle-node bifurcation. Our results reveal that the mean and variance of first passage times vary smoothly across the bifurcation threshold, eliminating the deterministic distinction between ghost attractors and bistable regimes. This work suggests that first passage time analysis can be an informative way to classify the length of a long transient. A better understanding of the duration of long transients may contribute to greater ecological understanding and more effective environmental management.

Integrating the underlying structure of stochasticity into community ecology

Stochasticity is a core component of ecology, as it underlies key processes that structure and create variability in nature. Despite its fundamental importance in ecological systems, the concept is often treated as synonymous with unpredictability in community ecology, and studies tend to focus on single forms of stochasticity rather than taking a more holistic view. This has led to multiple narratives for how stochasticity mediates community dynamics. Here, we present a framework that describes how different forms of stochasticity (notably demographic and environmental stochasticity) combine to provide underlying and predictable structure in diverse communities. This framework builds on the deep ecological understanding of stochastic processes acting at individual and population levels and in modules of a few interacting species. We support our framework with a mathematical model that we use to synthesize key literature, demonstrating that stochasticity is more than simple uncertainty. Rather, stochasticity has profound and predictable effects on community dynamics that are critical for understanding how diversity is maintained. We propose next steps that ecologists might use to explore the role of stochasticity for structuring communities in theoretical and empirical systems, and thereby enhance our understanding of community dynamics.

Long transients in ecology: Theory and applications

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.

Ecological pleiotropy and indirect effects alter the potential for evolutionary rescue

Invading predators can negatively affect naïve prey populations due to a lack of evolved defenses. Many species therefore may be at risk of extinction due to overexploitation by exotic predators. Yet the strong selective effect of predation might drive evolution of imperiled prey toward more resistant forms, potentially allowing the prey to persist. We evaluated the potential for evolutionary rescue in an imperiled prey using Gillespie eco-evolutionary models (GEMs). We focused on a system parameterized for protists where changes in prey body size may influence intrinsic rate of population growth, space clearance rate (initial slope of the functional response), and the energetic benefit to predators. Our results show that the likelihood of rescue depends on (a) whether multiple parameters connected to the same evolving trait (i.e., ecological pleiotropy) combine to magnify selection, (b) whether the evolving trait causes negative indirect effects on the predator population by altering the energy gain per prey, (c) whether heritable trait variation is sufficient to foster rapid evolution, and (d) whether prey abundances are stable enough to avoid very rapid extinction. We also show that when evolution fosters rescue by increasing the prey equilibrium abundance, invasive predator populations also can be rescued, potentially leading to additional negative effects on other species. Thus, ecological pleiotropy, indirect effects, and system dynamics may be important factors influencing the potential for evolutionary rescue for both imperiled prey and invading predators. These results suggest that bolstering trait variation may be key to fostering evolutionary rescue, but also that the myriad direct and indirect effects of trait change could either make rescue outcomes unpredictable or, if they occur, cause rescue to have side effects such as bolstering the populations of invasive species.

Transient phenomena in ecology

The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.

Abrupt Change in Ecological Systems: Inference and Diagnosis

Abrupt ecological changes are, by definition, those that occur over short periods of time relative to typical rates of change for a given ecosystem. The potential for such changes is growing due to anthropogenic pressures, which challenges the resilience of societies and ecosystems. Abrupt ecological changes are difficult to diagnose because they can arise from a variety of circumstances, including rapid changes in external drivers (e.g., climate, or resource extraction), nonlinear responses to gradual changes in drivers, and interactions among multiple drivers and disturbances. We synthesize strategies for identifying causes of abrupt ecological change and highlight instances where abrupt changes are likely. Diagnosing abrupt changes and inferring causation are increasingly important as society seek to adapt to rapid, multifaceted environmental changes.