Detecting dynamic system regime boundaries with Fisher information: the case of ecosystems
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Abstract
The direct measurement of the resilience (resistance to disturbances) of an ecosystem’s current regime (or “alternative stable state”) remains a key concern for managing human impacts on these ecosystems and their risk of collapse. Approaches which utilize statistics or information theory have demonstrated utility in identifying regime boundaries. Here, we use Fisher information to establish the limits of the resilience of a dynamic regime of a predator–prey system. This is important because previous studies using Fisher information focused on detecting whether a regime change has occurred, whereas here we are interested in determining how much an ecological system can vary its properties without a regime change occurring. We illustrate the theory with simple two species systems. We apply it first to a predator–prey model and then to a 60year wolf–moose population dataset from Isle Royale National Park in Michigan, USA. We assess the resilience boundaries and the operating range of a system’s parameters without a regime change from entirely new criteria for Fisher information, oriented toward regime stability. The approach allows us to use system measurements to determine the shape and depth of the “cup” as defined by the broader resilience concept.
Graphic abstract
Keywords
Fisher information Ecosystems Dynamic regime Resilience Isle Royale Wolf Moose Regime thresholdIntroduction
Ecosystems are dynamic and constantly interact with a range of external and internal drivers including species extinction, climate change, human activity, and other causes (ValienteBanuet and Verdú 2013; Suding et al. 2015; Seddon et al. 2016). The resilience of an ecosystem, as defined by the system’s ability to remain within a particular regime in the presence of disturbances, determines how and to what magnitude ecosystems will change in response to these drivers (Holling 1973; Grimm and Wissel 1997; Carpenter et al. 2001; Dai et al. 2015; Gao et al. 2016). To measure the vulnerability of systems to regimechanging disturbances, it is essential to understand the mechanisms of ecological resilience to natural and anthropogenic disturbances (Veraart et al. 2012; MacDougall et al. 2013; Suding and Hobbs 2014; Suding et al. 2015; Levine et al. 2016). This knowledge then contributes to effective environmental policy, identifying pressure points in the system which can be influenced through policies which reduce stressors (such as pollutants, invasive species or diseases, or land use change) or boost stabilizing factors (such as increasing native species populations).
Regime change, or the movement of a system from one regime (or alternative stable state) to another, can be triggered by exogenous disturbances (such as fire or the introduction of disease), or internal causes (e.g., loss of species, increased mortality, etc.; Spanbauer et al. 2014). The likelihood of regime change is determined by the system’s resilience to that disturbance and, in other words, its ability to maintain itself in that regime through internal feedbacks and interactions (Scheffer and Carpenter 2003; Folke et al. 2004). Note that in this paper, we will only be focused on one regime as our measure of resilience, and not multiple regimes or the recovery of a system to a previous regime after disturbance [where recovery time is an alternative measure of resilience; see Grimm and Wissel (1997)]. The identification of the location of regime boundaries, also known as thresholds or tipping points, is of critical importance as early warning systems for the management and sustainability of coupled human–environment systems (Guttall and Jayaprakash 2009; Scheffer et al. 2009; Scheffer 2010; Horan et al. 2011; Spanbauer et al. 2014; Suding and Hobbs 2014).
Holling (1973) adopted a quantitative view of the behavior of ecological systems (Carpenter et al. 2001). Perspectives on ecosystem resilience have been expanded and refined since Holling (1973) to explicitly consider nonlinear dynamics, boundaries, uncertainty and unpredictability, and how such dynamics interact across different time and spatial scales (Carpenter et al. 2001; Folke 2006; Brand and Jax 2007; Scheffer 2010; Veraart et al. 2012; Scheffer et al. 2015). Generally, resilience may be estimated by computing the eigenvalues of the system at its equilibrium (Lade and Niiranen 2017), but this approach does not provide any information about the behavior of a system right before the patterns decay.
Neubert and Caswell (1997) investigated several measures of a transient response, such as the maximal possible growth rate that directly follows the perturbation, the biggest proportional deviation that can be generated by any perturbations, and the time at which the amplification occurs. Scheffer et al. (2015) presented methods based on the critical slowingdown phenomena, which implies that recovery from small perturbations becomes slower as a system approaches a regime threshold. They also characterized the resilience of alternative regimes in probabilistic terms, measuring critical slowing down by using generic indicators related to the fundamental properties of a dynamic system (Scheffer et al. 2015). Levine et al. (2016) reported contradictory predictions in the sensitivity and ecological resilience of Amazon forests to changes in climate, sometimes resulting in biomass stability, other times in catastrophic biomass loss; transitions between regimes was continuous (no thresholds observed). Other drivers, including fire disturbances, grazing, logging, or other anthropogenic activities, are also capable of amplifying these climate changedriven transitions between forests and savanna globally (Mayer and Henareh Khalyani 2011). The identification of these ecosystem transitions depends upon the availability of longterm data, which is expensive and resource intensive.
Information theory has been applied to assess the sustainability of dynamic systems (Eason and Cabezas 2012), mainly to detect transitions from one dynamic regime to another (Mayer et al. 2006; Karunanithi et al. 2008; Spanbauer et al. 2014; Eason et al. 2016; Sundstrom et al. 2017; Vance et al. 2017). The “ball and cup” mental model has been central to this work (Gunderson 2000). As a common analogy for dynamic regimes, a system (the ball) moves within a cup—a specific regime. The ability of the ball to remain in that same cup (or basin of attraction) is the resilience of the system (Grimm and Wissel 1997). To functionally relate resilience to regimes and regime change, we must determine (1) how large the cup is (regime resilience) and (2) whether the system is in the cup or outside of it (regime shift). In this paper, we apply Fisher information to identify the boundaries of the regime (the size and depth of the cup) relative to the position of the ecological system (the ball) from actual values of system variables. This is important because it moves the state of the science beyond discussing symbolic cups meant to represent basins of attraction to working with the actual basin of attraction for the system. Unlike in prior studies (e.g., Sundstrom et al. 2017), where boundaries were identified postregime shift, we identify regime boundaries before the system has a regime change. Knowing the size and shape of the basin of attraction makes it possible to take remedial action, to keep the system away from the regime boundaries before a shift has occurred. (Or, conversely in a restoration attempt, how far a system will need to be pushed in order to flip it into a more desirable regime). We illustrate the concept with a simple modeled system and with a twospecies predator–prey system (the wolves and moose of Isle Royale National Park, Michigan, USA). We further show that Fisher information can determine the range of predator–prey abundance over which the ecosystem remains in one regime and hence exhibits resilience.
Fisher information theory
Fisher information is also closely related to the concept of order in dynamic systems. A very ordered dynamic system is one where repeated observations of the system yield about the same result. In the case of a system with one observable variable \(s\), this means that repeated measurements of \(s\) give about the same value. In that case, \(p\left( s \right)\) is very narrow and sharp around the mean value of \(s\), and the slope \({\text{d}}p\left( s \right) / {\text{d}}s\) is a high number. Since the Fisher information is proportional to \({\text{d}}p\left( s \right) / {\text{d}}s\) squared, the Fisher information has a correspondingly high value as well. In the extreme example of a system where the measurable variables are constant, the system is said to be perfectly orderly, \({\text{d}}p\left( s \right) / {\text{d}}s \to +\, \infty\), and the Fisher information is positive infinity. In the case of a very disorderly dynamic system with again one observable variable \(s\), each measurement of s yields a more or less different value. Therefore, \(p\left( s \right)\) is broad and relatively flat, and the slope \({\text{d}}p\left( s \right) / {\text{d}}s\) of \(p\left( s \right)\) is near zero. Correspondingly, the Fisher information for a very disorderly dynamic system is near zero. In the extreme, of a system completely lacking order, each measurement of \(s\) yields a different value. Then,\(p\left( s \right)\) is flat, \({\text{d}}p\left( s \right) / {\text{d}}s\) is zero, and the Fisher information for this completely disorderly system is exactly zero. In summary, the Fisher information of an ordered system is high and that of a disordered system is low. One should also note that work of AlSaffar and Kim (2017) explored the mathematical behavior of Fisher information under different perturbations and oscillatory regimes with possible implications for small populations of one species.
For systems that have more than one observable variable, the aforementioned arguments apply, except that \(s\) now represents an ndimensional state of the system which depends on all of the observable variables of the system. Hence, a state of the system \(s\) for a dynamic system with \(n\) measurable variables \(x_{1} ,x_{2} , \ldots x_{n}\) is defined by a particular value of each of the \(n\) variables. Even two states that differ by the value of only one variable are different states of the system. Note that this can lead to a very large number of states of the system, each one being unique.
Resilience from Fisher Information
The resilience of an ecological system has been defined by Holling (1973) as the ability of the system to continue functioning within the same dynamic regime despite externally inflicted perturbations. Within the same regime, the system can be very resilient to some kinds of disturbances over a long period of time, and not at all resilient to others. The resilience of an ecological system in a regime can change over time, such as with the loss of species or gradually changing external conditions, at the same time that stability can appear constant. (The system does not change regimes.) Regime shift occurs when one or more thresholds have been reached (e.g., a catastrophic disturbance, or the loss of too many species). In previous research, Fisher information has been used retroactively, to identify regime thresholds after regime shifts have occurred (Mayer et al. 2006; Sundstrom et al. 2017; Vance et al. 2017). Identifying regime thresholds without first observing a regime shift is a different problem.
Consider that it is possible to compute Fisher information for an ecosystem as a function of any of its characteristic parameters (species mortality, growth rate, etc.). A perturbation or perturbations can be represented as changes in the characteristic parameters—note that the characteristic parameters of an ecosystem can change for other reasons as well. However, the Fisher information would be relatively low within the range of parameter values consistent with the existence of a functioning ecosystem since the system is dynamic, and the Fisher information would have a relatively high value for the range of parameter values leading to a nonfunctional or static and dead system. A Fisher information calculation, however, is an observational process. It provides information about the system dynamic regimes and the changes in those regimes. It can provide hints at what changes in the system parameters may be driving the changes, but its primary purpose is not to determine cause and effect. That requires either an explicit mathematical model of the system such as the prey–predator model, or an implicit model such as the observations for the moose–wolf population data for Isle Royale, both of which are discussed later.
The image for the behavior of Fisher information as a function of three or more model parameters would lie in a four or higherdimensional space. This is unfortunately outside the range of human perception. But the mathematical approach is still valid. The algorithm that one would pursue in investigating such a system would be similar to the one used here for one and two parameter systems. Hence, we could start by varying parameter \(\alpha_{1}\) over the range of interest while holding all parameters \(\alpha_{i \ne 1}\) constant at some predetermined value. One would then proceed to varying \(\alpha_{2}\) while holding all parameters \(\alpha_{i \ne 2}\) constant. At the end, we would have a set of Fisher information values that depend on the aforementioned \(n\) parameters, i.e., \(I\left( {\alpha_{1} , \alpha_{2} , \ldots \alpha_{n} } \right)\). The process for identifying the parameter range over which the system is resilient would involve looking for regions where the Fisher information is flat in this \(n\) parameter space. These are ranges of parameter values where the Fisher information does not significantly vary as given in Eqs. 7, 8, and 9.
The result of these conjectures emanating from Fisher information considerations is that of providing the mathematical machinery that is necessary to estimate how much the system parameters can vary, without inducing a change in the dynamic regime of the system. One would then argue that the wider the range of parameter variation that can be tolerated without a regime change, the more resilient the system.
Predator–prey model system
 \(y_{1}\)

Population mass of the prey [mass]
 \(y_{2}\)

Population mass of the predator [mass]
 \(g_{1}\)

Growth rate of prey [1/time]
 \(l_{12}\)

Loss rate to prey due to predator feeding [1/time]
 \(g_{21}\)

Feeding rate of predator [1/time]
 \(m_{2}\)

Mortality rate of predator [1/time]
 \(k\)

Density dependence of prey [mass]
 \(\beta\)

Reproduction rate of predator [mass/mass]
In summary, for purposes of this study of a model prey–predator system, we compute the Fisher information from Eq. 5 setting \(\Delta t = 1\) and using \(y_{1}\) and \(y_{2}\) computed as a function of time from Eqs. 10 and 12.
Results for a model prey–predator system
Results for a real prey–predator ecosystem
To study the application of the methodology, we include in our analysis the case of a real predator–prey system represented by the wolf–moose (Canis lupis, Alces alces) system from Isle Royale National Park in the USA. The data originate from a 60year research project (1957 to present) of the dynamics of wolf and moose populations (and their impacts on the vegetation) on Isle Royale, a remote 540 km^{2} island in Lake Superior (Vucetich and Peterson 2012; Mlot 2017). The population sizes of wolves and moose are surveyed each winter; the dataset includes the precise number of wolves and estimated number of moose. The system has been in the news in the past several years after the wolf population began an unsustainable decline in abundance; as of 2017, only one inbred pair of wolves lived on the island, and the moose population was increasing rapidly in the absence of sufficient predation (Mlot 2017).
A brief delay is perceptible in the Fisher information trend compared to the population trends, but as expected, Fisher information is high when population fluctuations are low and drops when the fluctuations intensify. The Fisher information calculated here indicates that there is, perhaps, a functional state with relatively high dynamic order that persisted in the 1970s, where wolf populations were around 40 individuals and moose around 1000. However, this region may not be entirely resilient, as since that time this system has spent the bulk of its time in a low Fisher information region of less than 20 wolves and well over 1000 moose. The sharp decline of wolves in 1981 (echoed in a decline in Fisher information) was due to the accidental introduction of canine parvovirus to the island (Wilmers et al. 2006). It is notable that Fisher information indicated (via a slight increase) a brief period between 2000 and 2007 when the wolf and moose populations appeared to be more stable (but were not, according to Fisher Information). In this period, the populations roughly echoed the numbers seen in the stable era of the 1970s.
However, this resilience degraded as the wolf population entered a sharp decline after 2009. Fisher information’s behavior for this realworld system is consistent with the behavior observed for the model system, although the impact of the noise in a realworld system on the clarity of Fisher information behavior is easy to see. This is to be expected with real data from real systems. However, broadly speaking, Fisher information indicates that some event (internal or external) occurred in the early 1980s, despite the appearance of some stability in population numbers in the early 2000s, which set this system on a less resilient pathway from which it has not yet recovered.
Discussion
The previous work related to system regimes and Fisher information focused mostly on regime changes when a system shifts from one regime into another. The goal of this research was to develop a method to calculate where a resilient system has its borders and to identify the ranges of the interacting parameters where the system persists in one regime independently of the perturbations. By the criterion formulated as Eq. 7, it is possible to decide whether a dynamic system is in a healthy, dynamically changing state, in a dysfunctional and therefore static state, or in transition from a healthy state into a dysfunctional one. The criterion, defined by Eqs. 8, 9, and 10, tells where a system is resilient when there is only one, two, or more varying system parameters, respectively.
Fisher information theory is well known and applied in several scientific fields, but it has not been utilized for measuring system resilience directly. The method described in this paper provides a technique to measure the resilience of a dynamic system by checking the criteria defined in Eqs. 7, 8, 9, and 10. As with previous iterations of Fisher information, it remains highly sensitive to the quality of the data (Mayer et al. 2006); accordingly, users must assure that the variables selected are relevant to characterizing changes in the system’s condition; otherwise, the Fisher information results are uninformative. In the wolf–moose example, other variables such as winter ice cover of Lake Superior and seasonal temperature and precipitation may be important to include in future iterations of calculating the Fisher information of the system. However, even with only the two species data, Fisher information may provide valuable information to the management of the resilience of the wolf–moose system on Isle Royale National Park. For example, in 2016–2017 the National Park Service debated which of several management options it should pursue to stabilize the wolf and moose populations, such as doing nothing (waiting to see if wolves return via an ice bridge over Lake Superior), or reintroducing several wolf packs from Canada over a period of 3 years (81 Federal Register 91192 2016; Mech et al. 2017). In 2018, the National Park Service decided to go ahead with a slow introduction of very small numbers of wolves each year, releasing the first four in October 2018 (Mlot 2018). With better refinement, Fisher information could help park managers and wildlife biologists determine whether this management option is having the desired effect (increasing the resilience of the wolf and moose populations). For example, Fisher information suggests that the island system with parvo present may not allow for a resilient wolf–moose regime, and a policy prescription of parvo vaccinations for all wolves may be warranted.
While the theory has been illustrated via the prey–predator model system and the wolf–moose population data, it can be applied in its present form to larger, more complicated systems. It should also be noted that the theory in its present form is applicable to any dynamic system as long as model differential equations or timeseries data are available for the system variables. The system can be biological, social, economic, or technological. This means that it is possible to generally assess the resilience of a system by assessing the impact of changes in system parameters on the value of Fisher information. It is easy to represent the line or the surface of Fisher information as a function of two varying parameters (as shown in Fig. 4 or Figs. 6, 7). By three or more varying parameters, the plot becomes four or higher dimensional, which is more difficult to visualize but the method is still valid. Further work will need to develop methods to interpret Fisher information accurately in these higher dimensions, particularly when recommending specific policy interventions.
Conclusions
The Fisher information of any system is a fundamental and computable property that is a measure of order. When applied to ecological systems, we find that living functioning systems have relatively low but steady Fisher information, while dysfunctional ecosystems can have either very high or very low Fisher information, depending upon the variability in the system parameters. Fisher information is very sensitive to the dynamic behavior of complex systems which makes it a good indicator of regime shift. Here, we use it to measure the range of system parameter values over which a system remains within the same regime; larger range indicates higher resilience. Resilience defined and measured in this manner can be accomplished irrespective of the specific perturbation affecting the ecosystem; we measure change without having information on the perturbation causing it. While it would be optimal to know which disturbance is responsible for observed resilience loss, this information is not always available. This form of resilience is, therefore, a measure of robustness or ruggedness in the face of often unpredictable perturbations. While much work remains to understand its strengths and limitations, the index shows promise as a way to characterize an important aspect of resilience in ecological systems and other dynamic systems generally.
Notes
Acknowledgements
Open access funding provided by Pázmány Péter Catholic University (PPKE). The authors would like to thank John Vucetich for the Isle Royale wolf/moose population data and three anonymous reviewers for the constructive comments.
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