Effects of stochasticity on the length and behaviour of ecological transients

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

Structural stability of invasion graphs for Lotka-Volterra systems

In this paper, we study in detail the structure of the global attractor for the Lotka-Volterra system with a Volterra-Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in Hofbauer and Schreiber (J Math Biol 85:54, 2022) and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

Parameter Sensitivity of Transient Community Dynamics

AbstractTransient dynamics have always intrigued ecologists, but current rapid environmental change (inducing transients even in previously undisturbed systems) has highlighted their importance more than ever. Here, I introduce a method for analyzing the sensitivity of transient ecological dynamics to parameter perturbations. The question the method answers is: how would the community dynamics have unfolded for some time horizon had the parameters been slightly different? I apply the method to three empirically parameterized models: competition between native forbs and exotic grasses in California, a host-parasitoid system, and an experimental chemostat predator-prey model. These applications showcase the ecological insights one can gain from models using transient sensitivity analysis. First, one can find parameters and their combinations whose perturbations disproportionately affect a system. Second, one can identify particular windows of time during which the predicted deviation from the unperturbed trajectories is especially large and utilize this information for management purposes. Third, there is an inverse relationship between transient and long-term sensitivities whenever the interacting populations are ecologically similar; paradoxically, the smaller the immediate response of the system, the more extreme its long-term response will be.

Understanding change in benthic marine systems

BACKGROUND

The unprecedented influence of human activities on natural ecosystems in the 21st century has resulted in increasingly frequent large-scale changes in ecological communities. This has heightened interest in understanding such changes and effective means to manage them. Accurate interpretation of state changes is challenging because of difficulties translating theory to empirical study, and most theory emphasizes systems near equilibrium, which may not be relevant in rapidly changing environments.

SCOPE

We review concepts of long-transient stages and phase shifts between stable community states, both smooth, continuous and discontinuous shifts, and the relationships among them. Three principal challenges emerge when applying these concepts. The first is how to interpret observed change in communities - distinguishing multiple stable states from long transients, or reversible shifts in the phase portrait of single attractor systems. The second is how to quantify the magnitudes of three sources of variability that cause switches between community states: (1) 'noise' in species' abundances, (2) 'wiggle' in system parameters and (3) trends in parameters that affect the topography of the basin of attraction. The third challenge is how variability of the system shapes evidence used to interpret community changes. We outline a novel approach using critical length scales to potentially address these challenges. These concepts are highlighted by a review of recent examples involving macroalgae as key players in marine benthic ecosystems.

CONCLUSIONS

Real-world examples show three or more stable configurations of ecological communities may exist for a given set of parameters, and transient stages may persist for long periods necessitating their respective consideration. The characteristic length scale (CLS) is a useful metric that uniquely identifies a community 'basin of attraction', enabling phase shifts to be distinguished from long transients. Variabilities of CLSs and time series data may likewise provide proactive management measures to mitigate phase shifts and loss of ecosystem services. Continued challenges remain in distinguishing continuous from discontinuous phase shifts because their respective dynamics lack unique signatures.

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.

Landscape quantifies the intermediate state and transition dynamics in ecological networks

Understanding the ecological mechanisms associated with the collapse and restoration is especially critical in promoting harmonious coexistence between humans and nature. So far, it remains challenging to elucidate the mechanisms of stochastic dynamical transitions for ecological systems. Using an example of plant-pollinator network, we quantified the energy landscape of ecological system. The landscape displays multiple attractors characterizing the high, low and intermediate abundance stable states. Interestingly, we detected the intermediate states under pollinator decline, and demonstrated the indispensable role of the intermediate state in state transitions. From the landscape, we define the barrier height (BH) as a global quantity to evaluate the transition feasibility. We propose that the BH can serve as a new early-warning signal (EWS) for upcoming catastrophic breakdown, which provides an earlier and more accurate warning signal than traditional metrics based on time series. Our results promote developing better management strategies to achieve environmental sustainability.

Stability of ecosystems under oscillatory driving with frequency modulation

Consumer-resource cycles are widespread in ecosystems, and seasonal forcing is known to influence them profoundly. Typically, seasonal forcing perturbs an ecosystem with time-varying frequency; however, previous studies have explored the dynamics of such systems under oscillatory forcing with constant frequency. Studies of the effect of time-varying frequency on ecosystem stability are lacking. Here we investigate isolated and network models of a cyclic consumer-resource ecosystem with oscillatory driving subjected to frequency modulation. We show that frequency modulation can induce stability in the system in the form of stable synchronized solutions, depending on intrinsic model parameters and extrinsic modulation strength. The stability of synchronous solutions is determined by calculating the maximal Lyapunov exponent, which determines that the fraction of stable synchronous solution increases with an increase in the modulation strength. We also uncover intermittent synchronization when synchronous dynamics are intermingled with episodes of asynchronous dynamics. Using the phase-reduction method for the network model, we reduce the system into a phase equation that clearly distinguishes synchronous, intermittently synchronous, and asynchronous solutions. While investigating the role of network topology, we find that variation in rewiring probability has a negligible effect on the stability of synchronous solutions. This study deepens our understanding of ecosystems under seasonal perturbations.

Stochastic resonance in climate reddening increases the risk of cyclic ecosystem extinction via phase-tipping

Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator-prey ecosystems to extinction via so-called phase-tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator-prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator-prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.

A framework for long-lasting, slowly varying transient dynamics

Much of the focus of applied dynamical systems is on asymptotic dynamics such as equilibria and periodic solutions. However, in many systems there are transient phenomena, such as temporary population collapses and the honeymoon period after the start of mass vaccination, that can last for a very long time and play an important role in ecological and epidemiological applications. In previous work we defined transient centers which are points in state space that give rise to arbitrarily long and arbitrarily slow transient dynamics. Here we present the mathematical properties of transient centers and provide further insight into these special points. We show that under certain conditions, the entire forward and backward trajectory of a transient center, as well as all its limit points must also be transient centers. We also derive conditions that can be used to verify which points are transient centers and whether those are reachable transient centers. Finally we present examples to demonstrate the utility of the theory, including applications to predatory-prey systems and disease transmission models, and show that the long transience noted in these models are generated by transient centers.

Life-history speed, population disappearances and noise-induced ratchet effects

Nature is replete with variation in the body sizes, reproductive output and generation times of species that produce life-history responses known to vary from small and fast to large and slow. Although researchers recognize that life-history speed likely dictates fundamental processes in consumer-resource interactions like productivity and stability, theoretical work remains incomplete in this critical area. Here, we examine the role of life-history speed on consumer-resource interactions by using a well-used mathematical approach that manipulates the speed of the consumer's growth rate in a consumer-resource interaction. Importantly, this approach holds the isocline geometry intact, allowing us to assess the impacts of altered life-history speed on stability (coefficient of variation, CV) without changing the underlying qualitative dynamics. Although slowing life history can be initially stabilizing, we find that in stochastic settings slowing ultimately drives highly destabilizing population disappearances, especially under reddened noise. Our results suggest that human-driven reddening of noise may decrease species stability because the autocorrelation of red noise enlarges the period and magnitude of perturbations, overwhelming a species' natural compensatory responses via a ratchet-like effect. This ratchet-like effect then pushes species' population dynamics far away from equilibria, which can lead to precipitous local extinction.

Detecting hidden transient events in noisy nonlinear time-series

The information impulse function (IIF), running Variance, and local Hölder Exponent are three conceptually different time-series evaluation techniques. These techniques examine time-series for local changes in information content, statistical variation, and point-wise smoothness, respectively. Using simulated data emulating a randomly excited nonlinear dynamical system, this study interrogates the utility of each method to correctly differentiate a transient event from the background while simultaneously locating it in time. Computational experiments are designed and conducted to evaluate the efficacy of each technique by varying pulse size, time location, and noise level in time-series. Our findings reveal that, in most cases, the first instance of a transient event is more easily observed with the information-based approach of IIF than with the Variance and local Hölder Exponent methods. While our study highlights the unique strengths of each technique, the results suggest that very robust and reliable event detection for nonlinear systems producing noisy time-series data can be obtained by incorporating the IIF into the analysis.