A probabilistic framework for windows of opportunity: the role of temporal variability in critical transitions
The establishment of young organisms in harsh environments often requires a window of opportunity (WoO). That is, a short time window in which environmental conditions drop long enough below the hostile average level, giving the organism time to develop tolerance and transition into stable existence. It has been suggested that this kind of establishment dynamics is a noise-induced transition between two alternate states. Understanding how temporal variability (i.e. noise) in environmental conditions affects establishment of organisms is therefore key, yet not well understood or included explicitly in the WoO framework. In this paper, we develop a coherent theoretical framework for understanding when the WoO open or close based on simple dichotomous environmental variation. We reveal that understanding of the intrinsic timescales of both the developing organism and the environment is fundamental to predict if organisms can or cannot establish. These insights have allowed us to develop statistical laws for predicting establishment probabilities based on the period and variance of the fluctuations in naturally variable environments. Based on this framework, we now get a clear understanding of how changes in the timing and magnitude of climate variability or management can mediate establishment chances.
In many harsh environments, the establishment of young organisms is the most difficult phase of the life cycle [1–3]. In arid environments, for example, the successful establishment of trees is episodic and requires a year of exceptional rainfall [4,5] or a period of low consumption pressure due to unusually low grazer abundance [4,6]. In coastal wetlands, such as tidal marshes and mangrove forests, the episodic establishment of vegetation has been linked to rare periods of mild hydrodynamic exposure [7,8]. Moreover, the episodic recruitment of shellfish reefs has been suggested to be the result of unusually low predation pressure by crustaceans [9–11]. Such time intervals of benign conditions in which organisms suddenly establish have been coined windows of opportunity (WoOs) [4,12].
It has been suggested that the observed episodic establishment dynamics, typical for many harsh environments, can be explained as noise-induced critical transitions between two alternative stable states . Under this framework, environmental variability is vital in creating periods of unusually benign conditions. Such windows allow young organisms that are unsuited to the average environmental condition to reach a stable existence by rapidly growing and developing sufficient tolerance to that environment [7,13]. Such establishment events are thus the result of the fact that, while long-term environmental conditions may on average remain stationary (i.e. no increasing or decreasing trends in the long-term average or variability of the stressor), environments constantly vary on shorter timescales. Without such environmental variability no transitions can occur [7,8,14]. To date, the role of the period (i.e. determining the intermittence of benign and harsh conditions) and variance (i.e. the magnitude of the fluctuation) of the temporal variability in such noise-induced transitions that ultimately lead to the emergence of ecosystems remains a topic that is ill explored. Hence, this hampers the understanding and prediction of organism dynamics in variable environments [5,15].
In this paper, we further develop the concept of WoO into a coherent theoretical framework to systematically explore such episodic dynamics and derive establishment probabilities. We reveal that for young organisms to exploit windows of benign conditions, sufficiently large and temporally correlated environmental variability is the most critical requirement. Yet, for the organisms to be able to turn such benign windows into opportunities for establishment they need to exhibit stage-dependent tolerance (as the result of, e.g. physiological maturation or density-dependent feedbacks) and the ability to grow and develop their tolerance swiftly. We support our argumentation by (1) developing a theoretical framework based on organisms developing in a regular dichotomous environment. (2) Explaining how correlated variability (i.e. noise that has a non-negligible correlation time) shapes the statistical probability to allow critical transitions towards establishment of these organisms, and eventually the ecosystem. We then show (3) that the conceptual framework also holds for continuous environmental signals. Finally, (4) we discuss the consequences of these insights for the interpretation of stability and recoverability to real-world ecosystems and how establishment success can be mediated to facilitate management and restoration.
2. Theoretical framework for windows of opportunity
To establish, organisms need to rapidly gain tolerance against hostile conditions of the environment it is embedded within. Due to growth and development, organisms physiologically mature or can invoke density-dependent feedbacks that improve their environmental tolerance. Consider for instance a plant that develops a root system over time that allows it to reach deeper groundwater layers, increasing its tolerance to drought episodes [5,16] or prevent them from uprooting in intertidal sediments due to tidal current and wave action [8,12,17]. Moreover, becoming larger is often a good way to escape from predation or herbivory: trees that grow tall enough may escape the herbivory of their foliage . Bivalves are also known to become less vulnerable to predation by crustaceans once a size threshold is surpassed . Becoming larger can even increase an organism's ability to tolerate toxic conditions [19,20]. In the above examples, the size or biomass of an organism is thus directly linked to its tolerance, especially in the earliest stages of life, underlining the key concept of stage-dependent tolerance.
To develop an intuitive understanding of how individual young organisms can establish in such a hostile environment and to encapsulate the concept of stage-dependent tolerance in the theoretical framework, we explore a model (figure 1) in which a young organism is trying to establish within an environment E that fluctuates as a regular dichotomous signal between a low and high value, Elow and Ehigh, respectively (statistical properties of the regular dichotomous signal are given in electronic supplementary material, note S1). The environment is stationary at a longer timescale, characterized by periodic transitions between Elow lasting for a fixed time window τlow, and Ehigh lasting for τhigh. The state of the young organism is defined by its tolerance T to this environmental stressor. From inception (that is e.g. birth, germination, attachment or transplantation), the organism starts developing from an initial tolerance T0 with a growth rate λ (green lines, figure 1). If the organism's tolerance T to the environment E has not developed high enough before the environment returns to the harsh condition Ehigh, it dies instantaneously (as T < E at τlow, lowest green arrow with red cross in figure 1 is a failing individual as it developed tolerance at a too slow rate). If the tolerance T is developed sufficiently (T > Ehigh), the individual establishes. Hence, the individual needs to overcome a threshold δ = Ehigh− T0 to establish by developing tolerance with a growth rate λ ≥ λcrit. Considering this framework, we can identify four types of regimes from the perspective of the young organism: (i) a benign environment in which T0≥ Ehigh > Eavg; (ii) a benign environment in which Ehigh > T0≥ Eavg; (iii) a harsh environment in which Eavg > T0 > Elow; and (iv) a harsh environment in which Eavg > Elow > T0. In the benign environment (i) establishment can always occur, while in the harsh environment (iv) it never does. In situation (ii), environmental variability will destroy possibilities for establishment rather than creating them. Thus, our interest and focus of this paper is mainly on the third set (iii) of conditions (figure 1) in which establishment is only possible if tolerance is increased enough after inception (e.g. due to growth) before the environment switches to harsh conditions Ehigh.
From this simple conceptual framework based on the regular dichotomous signal (figure 1), we can easily derive (see electronic supplementary material, note S2) that the critical growth rate λcrit (i.e. minimum rate) at which an organism needs to develop tolerance to be able to establish successfully is
Fulfilling criterium λ ≥ λcrit implies that a time window of benign conditions τlow becomes a WoO, in which establishment will be successful. Environmental variability thus needs to have a long enough time windows τlow for establishment to be possible.
From this framework, we can furthermore understand that the existence of sufficient environmental variability is key to create opportunities to get into stable existence. Without sufficient variance (i.e. σ2(E) = 0, thus Eavg = Elow = Ehigh, figure 2a) WoOs (or non-opportunity) do not exist (i.e. the system collapses to regimes i and iv). Instead, the only pathway to establishment is for an organism to have an initial environmental tolerance T0 > Eavg. Sufficiently large variance (i.e. σ2(E) < 0 in which Elow < T0) must be introduced to the system so that benign intervals will appear in which an organism of tolerance Ehigh > T0 ≥ Elow would have the potential to establish (figure 2b). Once the variance σ2(E) is large enough, noise-induced transitions towards establishment can occur for organisms with a growth rate exceeding λcrit. The null-isocline (thick black line in figure 2b) reflects what value of λcrit corresponds to what value of T0 (figure 2b). Here we can see there is a range of growth rates λ that allow establishment if T0 < Eavg (regime iii). Yet, at the same time environmental variability also poses a threat to organisms by increasing the distance between Elow and Ehigh, resulting in the need for a critical growth rate λcrit even if T0 ≥ Eavg (regime ii). Therefore, an organism that can establish in the benign environment without temporal variability (figure 2a) must have a sufficiently high growth rate λ in order to overcome the impending switch to a hostile condition Ehigh in the case of the environment with variability (figure 2b).
Summarizing, from the established conceptual framework, we developed a sound intuitive understanding of when WoO can occur. It is revealed that the understanding of the intrinsic timescales of both the organism developing tolerance to the environment and the environmental variability itself is key to understand if young organisms can or cannot establish in a hostile environment. It further highlights the key role that environmental variability plays in the occurrence of transitions between unestablished and established states. Specifically, it reveals that (1) without environmental variability, either no establishment can occur, or establishment is always possible; (2) benign time intervals must be long enough for organisms, whose growth rates are intrinsically limited, to be able to take advantage of them. Based on these insights, we can now further develop a probabilistic framework.
3. Statistical regularities in random environments
Now we have set up the theoretical framework for WoO, we can apply this to a more variable environmental signal in order to derive statistical rules and explore the role of environmental variability more systematically. To focus on understanding how environmental variability determines the probability of WoO to occur in variable environments, we first consider the simple dichotomous Markov noise (DMN) signal (figure 3a, statistical properties of DMN signal are given in electronic supplementary material, note S1). Like the regular dichotomous signal, the environment can only be in a high or low condition (Elow and Ehigh). However, the length of the time windows (τ, or ‘waiting time') is no longer constant, but varies stochastically. The transition from one to the other condition depends on the transitions rates, klow and khigh. The mean benign window size is inversely related to the transition rate, thus 〈τlow〉 = 1/klow. As a result, DMN can capture the finite correlation time (i.e. the time at which the temporal autocorrelation ξ > 1/e) of a noise signal (i.e. coloured noise) very well and in a simple fashion. The finite correlation time is proportional to the approximate period 〈τ〉 of the environmental signal (i.e. the average period until/before the signal returns to the original position after an excursion). Furthermore, as we will keep 〈τhigh〉 constant throughout the analysis and focus on varying 〈τlow〉 only, we therefore can use the approximate period 〈τ〉 to refer to the temporal aspect of the environmental variability throughout the remainder of this work. It furthermore makes it easier to study the interplay of the intrinsic timescale of the DMN and the timescale of the establishing organisms. It has been shown before that such interactions can lead to non-trivial effects such as multistability and hysteresis . Thus, using the DMN signal, we can first focus on analytically solvable situations before we go into more complicated continuous environmental signals.
The probability of a time window with a specific length occurring in the stationary DMN signal corresponds to an exponential distribution as (figure 3b)
Based on equation (3.2), we systematically explore the consequences on the probability of WoO occurring. First, we focus on the environmental threshold δ, which can be increased due to a higher harsh environmental condition Ehigh (i.e. larger variance σ2(E) ∝ Ehigh − Elow) or lower initial tolerance T0. When keeping the growth rate constant (e.g. λ = 0.1, red arrows in figure 3b and red dashed line figure 3c), a larger threshold δ requires increasingly larger critical window sizes τlow, and these are less likely to occur (red arrows in figure 3b). Increasing the variance σ2(E), or lowering T0, thus consequently requires the organism to grow at a higher rate to acquire the same probability for establishment (figure 3c).
When focusing on the gradual increase of the growth rate (e.g. as would occur along a gradient from high to low physiological stress), the model reveals that at growth rates that are too low, the WoO (PWoO) becomes highly unlikely, but not nil (figure 3c). Yet, above a critical growth rate, the WoO for establishment suddenly becomes much more likely and rises with further increasing growth rate towards the asymptote at PWoO = 1. The critical growth rate at which the probability for a WoO significantly improves depending strongly on the height of the threshold. Hence, the critical growth rate required for significant establishment chances increases proportionally with increasing environmental variance σ2(E).
The role of changing temporal autocorrelation (i.e. the period) can be revealed by changing the transition rate klow (not shown). Increasing the transition rate klow cuts the average window length (i.e. whitening of environmental signal E) and vice versa decreasing the transition rate klow increases the average window length (i.e. reddening of E) and therewith the probability of establishment. Thus, reddening of the fluctuating signal increases the probability that a WoO will occur. Whitening of the environmental signal (i.e. approaching complete random noise, thus a fluctuating environment with neglectable temporal autocorrelation length) on the other hand will result in a nonlinear decrease of probability for a WoO to occur (i.e. sharp drop in PWoO). Generally, this agrees with recent work showing climate reddening increases the chances of critical transitions to occur .
Overall, the probabilistic framework can be summarized in a generic probability diagram (figure 4). This diagram is in essence similar to the bifurcation diagram (figure 2b, and therefore plotted for reference as the black line in figure 4). Yet, the strict dichotomous outcomes are now replaced with the probabilities of a WoO occurring. For example, at any given initial tolerance T0 with corresponding critical growth rate λcrit specific for an environment with approximate period 〈τ〉, where it just crossed the threshold (thus PWoO = 1) in the regular dichotomous case (black line from figure 2b in figure 4), the probability PWoO of WoOs to occur is always 1/e (approx. 0.368). Thus, here WoOs for establishment occur in the DMN on average approximately once every third ‘cycle' the environmental signal is going through. This probability diagram for PWoO is not specific for a unique parameter set and can be drawn for any DMN signal and can be used to estimate the probability of establishment Pest (see electronic supplementary material, note S5). Yet, it is unclear how generic this diagram is beyond the realm of DMN signals, as insights derived from DMN signals might be too simplistic and overestimate establishment chances for more complicated continuous signals such as found in nature. Fortunately, we can now see if these statistical patterns are holding for other (non-dichotomous) environmental signals based on the developed statistical framework, which is the focus of next section.
4. Establishment in continuous environments
Thus far, we have derived understanding about how variance characteristics (i.e. period and variance) determine the occurrence of WoOs based on very simple dichotomous signals, as such signals are easy to understand and analytically tractable. However, most environmental signals are much more complicated. We therefore demonstrate next that the statistical rules derived remain largely intact when expanding the analysis to continuous environmental signals as can be encountered in real natural systems. To do so, we study how the occurrence probabilities of WoOs change as the result of changes in period and variance of temporally correlated Gaussian noise.
Correlated Gaussian noise signals, as the results of the Ornstein–Uhlenbeck process (see electronic supplementary material, note S6 for details and figure 5a,c for example signals) have been shown to reflect many characteristics of complex natural signals (e.g. [21,23]). In this section, we numerically solve how organisms can establish in environmental signals constructed based on Hasselmann  variation of the Ornstein–Uhlenbeck noise (OUN). Changes in the temporally correlated signal E have an approximate period 〈τ〉 becoming red noise in case τ is sufficiently large
The numerical results clearly show that period 〈τ〉 and variance σ2(E) of the environmental fluctuations have a similar effect on the ability of organisms to establish in a harsh environment as previous explorations with the stylized DMN signal (figure 5b,d). At too slow growth rates λ no establishment can occur as the windows are too short. Increasing the period 〈τ〉 (from blue to black to red lines in figure 5b) results in a higher fraction of windows that are attainable. At larger periods, more windows are attainable, and establishment can thus occur at lower growth rates. Increasing the variance σ2(E) on the other hand reduces the fraction attainable windows (from red to black to blue line in figure 5d; black lines in b and d are the same). Yet, some minimal level of variance is required because if the variance is too small it cannot create any potential windows at which E < T0. In this specific example, the low variance signal (σ2(E) = 0.25) results in episodic emergence as revealed by the especially high variability in the numerical results (red line in figure 5d). Establishment becomes highly unlikely at σ2(E) < 0.25 because no windows are occurring. Hence, a reduction in variance can result in a sudden drop in establishment probability (see also grey area in electronic supplementary material, figure S1). The results underline that studying how organisms can establish using DMN-signals gives good intuitive understanding of how the colour (i.e. temporal correlation) and magnitude of the environmental fluctuations determine the establishment probabilities in other more complex environments such as those constructed using OUN. Hence, we expect that these insights will also apply to more complex and composite natural environmental signals.
Studies about WoOs have so far mainly focused on developing mechanistic understanding about sudden establishment events observed in various ecosystems under harsh conditions (e.g. [4,7,8]). These studies pointed out that sudden emergence of stable occupancy can be explained as critical transitions, where establishment thresholds are transgressed due to variability of the environmental conditions [5,7]. Using hindcasts, the origins of the establishment events could indeed be linked to infrequent short time intervals of benign conditions—the so-called WoOs—underlining the validity of these mechanistic insights [7,8]. In this study, we developed a theoretical and statistical framework furthering our conceptual understanding of this type of noise-induced transitions. Moreover, this framework allows us to forecast the probability of future establishment events based on both the period and variance of the environmental variability in relation to the intrinsic timescales over which the organisms develop tolerance to the environment. Finally, we can apply these insights to better understand ecosystems responses to climate change, or shape management and restoration practices to kickstart these ecosystems. As we have shown that the stylized DMN signals capture the qualitative behaviour of other more complex signals well, we further discuss these three aspects based on insights from DMN signals.
5.1. Windows of opportunity as a specific class of noise-induced critical transitions
Formally, the concept of WoOs dealt with noise-induced critical transitions (i.e. ‘N-tipping', in [25,26]). Although not explicitly mentioned as such in earlier work (e.g. [4,7]), this notion explains the typical dynamics observed where thresholds need to be transgressed to allow establishment. However, noise-induced transitions related to WoOs exhibit specific characteristics that set them apart from the common notion of noise-induced transitions:
(1) In this work, we considered noise solely in the independent variable (i.e. parameter noise). Noise is usually—but not exclusively—considered to affect the dependent variable (i.e. demographic noise) which means variability is in the biological state (see e.g. Ashwin et al.  and Bonciolini et al.  and references therein).
(2) The WoO concept only focuses on the transition towards establishment of the organism (or emergence of an ecosystem); however, most theoretical studies about noise-induced transitions deal with both emergence and extinction transitions at the same time (e.g. Ashwin et al. , Bonciolini et al.  and Dakos et al.,  and references therein) e.g. leading to back and forth ‘flickering’ between the two states (e.g. ). Therefore our work links better with the recently identified concept of phase tipping or partial tipping (P-tipping) in which a transition can only occur at a certain phase in the trajectory [28,29]. As ecosystems are usually more complex and multidimensional, the stressors limiting establishment (i.e. emergence) can be very different from the stressors responsible for their collapse (i.e. extinction). Therefore, it is reasonable to split the attention to develop appropriate mechanistic insights related to the different transition trajectories (i.e. emergence versus extinction transitions). There remains, however, scope to further develop the current framework to understand extinction transitions (see electronic supplementary material, note S3).
(3) The noise needs to be coloured because without any non-negligible temporal autocorrelation no WoO can exist. Then, the system remains locked in the unoccupied state. More specifically, for organisms to be at criticality (i.e. minimal requirements for establishment) the timescales of both the tolerance development by the organism and the environmental variability need to be proportional (following from equation (2.1)). That also means, that if noise whitens (i.e. the period shortens), the opportunities for establishment eventually collapse to a much smaller range of parameter space (regime i in figure 6). The fact that especially red noise and other temporally correlated signals (i.e. including periodic signals, such as the tidal, spring neap and seasonal cycles) are import drivers of critical transitions has only recently been recognized [22,30]. Hence, this special class of noise-induced transitions is likely to be common as such temporally correlated signals forcing ecosystem dynamics are widespread.
5.2. From individual establishment thresholds to landscape-scale patterns
The consequence of the ‘unlocking' of alternative states due to environmental variability is that the ecosystem's response to changing environmental conditions is no longer discontinuous, as is at the individual level, but becomes continuous at the population or landscape scale. This means that, although the occurrence of a benign window might be a low probability event, establishment inevitably happens once the window of opportunity opens (in regions ii and iii in figure 6). If at that time plenty of propagules are present a sudden large-scale colonization event occurs transitioning large parts of the suitable landscape from one to the other state. Such rare episodic tree recruitment events have for instance been reported for tropical trees with variable rainfall conditions . In case the presence of propagules is limited the successful establishment occurring during the WoOs can be the initiation of patch dynamics of bimodal patterns. These have been observed in a range of terrestrial and marine ecosystems [31,32]. Hence, the agreement between observations and our theoretical framework suggests that this class of noise-induced critical transitions is likely to be applicable to these ecosystems and might be for a larger range of ecosystems for which similar individual-level thresholds are reported or suggested.
5.3. Implications for management, restoration and climate change
The inherent stochastic nature of establishment dynamics can give the false impression that management or restoration cannot be fruitful. It has been argued that large-scale restoration efforts are needlessly expensive because organisms can recover by themselves without interventions. It only requires sufficient time and patience, as for example has been put forward for the establishment of bivalve beds  and seagrasses  in a soft-sediment environment where these ecosystems recovered naturally decades after landscape-scale loss. Given sufficient time, the entire feasible parameter space will eventually see establishment happen. However, our results also suggest that management actions can be taken to overcome establishment thresholds and speed up ecosystem recovery. In cases where stage- or density-dependent tolerance determines establishment success, the required minimal hospitable window (WoO) is essentially determined by the ratio of required tolerance gain to survive and the rate at which the organism develops this tolerance (equation (2.1)). This relationship leads to the insight that there are three basic options to affect establishment success (figure 7): (i) changing the length of the benign windowτlow; (ii) changing the height of the tolerance threshold (δ) and (iii) changing the rate of tolerance growth (λ).
By prolonging the time window of benign conditions τlow the establishment chances will be improved. However, changing the return time of the adverse condition might be the most difficult of the three options to achieve as the period is often set by regular environmental cycles, such as the season or tides. Attenuating the environmental condition is a more straightforward option to lower the tolerance threshold δ that needs to be overcome by the organisms. Especially attenuating high adverse conditions Ehigh can be a management option to create favorable conditions for establishment. Similarly, many species engineer their environment  e.g. by creating stress attenuating structures such as canopies [31,36]. It has recently been shown that structures mimicking these canopies can boost restoration success in coastal ecosystems . Alternatively, the initial tolerance T0 can be boosted e.g. by skipping the most vulnerable life-stage by transplanting or introducing older more tolerant individuals which is a commonly used approach . Improving growth rate also offers possibilities for management interventions by e.g. reducing local stressors or providing ample supply of otherwise limiting nutrients and other resources [38,39].
A somewhat counterintuitive, but important, conclusion of our study is that mere damping of environmental variance might be counterproductive, as it can reduce the occurrence of the benign conditions. For instance, natural water level fluctuations have been shown to be essential for creating WoOs . Restoring the natural variability might therefore be the best strategy in some systems to obtain natural patterns of recruitment. Similarly, climate change can have sudden unexpected effects if increased variability suddenly opens otherwise unattainable environments (e.g. from regime iv to iii in figure 6). Thus, whatever the strategy chosen, a basic understanding of the intrinsic timescales of the environmental fluctuations is at least as important as developments in average conditions if one wants to be able to better predict ecological responses to ecological management or climate change.
Predicting the possible outcome of climate change and restoration efforts critically depends on appropriate prognostics of both system characteristics (e.g. period and variance of environmental variability, unidirectional changes in drivers) and characteristics of the organisms, most notably their ontogenetic range of tolerance and their vital parameters. Intervening in the basic parameters of the system of interest may alter the outcome of restoration profoundly. Understanding for example how temperature affects growth rates, might be key to explaining why ecosystem responses to a changing climate can be unexpectedly nonlinear. The theoretical framework we developed here moves away from the strict dichotomous view on critical transitions and can therefore better inform societal debate on the appropriateness of interventions.
The code and generated synthetic data are made accessible via the NIOZ DAS open repository (https://doi.org/10.25850/nioz/7b.b.3c).
J.v.B.: conceptualization, formal analysis, methodology, software, visualization, writing—original draft, writing—review and editing; G.S.F.: methodology, writing—review and editing; Z.H.: conceptualization, methodology, writing—review and editing; T.J.B.: conceptualization, funding acquisition, project administration, supervision, writing—review and editing; P.M.J.H.: conceptualization, supervision, writing—review and editing.
All authors gave final approval for publication and agreed to be held accountable for the work performed therein.
Conflict of interest declaration
We declare we have no competing interests.
This work is part of the CoE-Rammegors, CoE-Perkpolder and CoE-Zuidgors project, which are financed by Rijkswaterstaat and the province of Zeeland.
We would like to thank Olivier Gourgue and Valerie de Witte for thoughtful feedback on earlier versions of the manuscript.
Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
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