The above image is by USFWS Midwest (creative commons license).
Guest Post by Adam Clark at the University of Minnesota
Related PLOS ONE paper:
How might you go about finding a moose?
It turns out that this is a matter of some consequence. Back in 1996, when I was growing up in Massachusetts, a moose somewhat famously wandered through Cleveland Circle in Boston, and ultimately disappeared into the city’s western suburbs. While the moose was unusual enough to bump coverage of the armed standoff between the FBI and the “Montana Freedmen” from local newspapers, its presence wasn’t really all that surprising. Throughout New England, moose are abundant enough to be something of an annoyance to hikers and gardeners, and remain a serious traffic hazard. It therefore came as a shock to me to learn that in Minnesota, moose populations are rapidly declining and retreating north. While the decline is thought to be the result of some combination of climate change, habitat loss, and pathogen pressure, the jury is still out on the specifics, though it appears that similar pressures are beginning to endanger moose populations throughout North America.
This is a particularly good example of the state of predictive modeling in ecology today. It is tempting to imagine that ecologists have some widely agreed upon, quantitative method for figuring out where species live.
In reality, though, the best way to find a species is still to go somewhere that is similar to where others have seen that species in the past.
If you want to improve your chances of finding a moose in North America, it’s a good idea to look in forested habitats near lakes or bogs, which experience snow cover for part of the year. Models that take advantage of these kinds of associations, but don’t seek to test why those associations exist, are often called phenomenological models. Alternatively, you might want to test potential reasons for these associations. For example, moose may be confined to northern habitats because they begin to experience heat stress when temperatures exceed 27°C for long periods. Models that ascribe these kinds of reasons to relationships are often referred to as mechanistic models.
In reality, the lines that determine which models are mechanistic and which models are phenomenological are often blurred. Amy Huford points to a passage from theoretical ecologist Ben Bolker, who notes that in many cases “the same function could be classified as either phenomenological or mechanistic depending on why it was chosen.” This might seem strange –
How could a modeler’s intentions change what a model does?
The simple answer is that we can interpret most relationships to be either associative (moose are found in cold places because they were found there in the past) or causative (moose are found in cold places because they can’t survive in hot climates).
I’d argue that rather than focusing on how to interpretat components of a model, a much easier starting point is to focus on what the models are intended to test. Specifically: Where models are intended solely to reproduce observed patterns, the modeler is choosing a phenomenological approach to understanding his or her data, and the model tests whether there is a consistent correspondence among variables. Where a model is meant to mimic the underlying processes that bring about those patterns, the method is more mechanistic, and the model tests whether the posited mechanism would be capable of generating the observed outcomes.
Some disciplinary context
Understanding why modelers care so much about this distinction between methods (particularly in ecology) requires a bit of context. To many researchers, the term “phenomenological” connotes something somewhat negative – sort of a “second best” approach only to be used when we don’t understand mechanisms well enough to apply them.
Part of the concern is reliability: because phenomenological models follow observed patterns without any underlying logic, they can sometimes give absurdly wrong predictions. A phenomenological species distribution model, for example, might predict high moose abundance in sites that have similar climate and habitat characteristics to current moose habitat, but that are too far from existing moose populations to ever be colonized by them. A mechanistic model would hopefully be able to correctly exclude these sites based on reasonable dispersal distances for moose. However, there are many cases where phenomenological approaches such as correlative species distribution models or empirical dynamic modeling significantly out-perform their mechanistic counterparts in terms of predictive power. Because of this, concerns over reliability alone may not be a good reason to eschew phenomenological methods.
A second, more broadly relevant concern, is interpretability. At a recent symposium at the annual ESA meeting, the discussion panel pretty much unanimously agreed that they would not be satisfied with a model that gave perfect predictions and extrapolations, but offered no insight into how the process it described actually worked. I think that general sentiment is shared by most ecologists: ecologists are much more interested in what their model is meant to test than how it goes about doing the testing. Because a mechanistic model is intended to mimic the underlying processes that lead to an observed pattern, fitting a mechanistic model is really just a way to test hypotheses about how the world works (rather than to predict how it will behave in the future). Thus, even fitting a simple linear regression might qualify as a mechanistic model, provided that the regression is meant to test a convincing story.
To illustrate this, consider the following example from a classic paper by Bill Schaffer’s on ecological abstraction:
Schaffer’s resource abstraction
Here, Schaffer presents a simple model of population dynamics for a consumer with population size N, limited by a single resource with abundance R. For example, N might describe moose abundance, and R could describe the abundance of a food source such as aspen trees:
In this case, dN/dt 1/N represents the “per-capita” change in moose population per unit time (i.e. the total change in population size, divided by the number of moose), and dR/dt 1/R describes the per-capita change in aspens. The parameter a represents the capture rate of aspens by moose, w is the energy content of the aspens, T is the minimum energy requirement of the moose, c is the rate at which energy is converted into new moose, and r and K are the intrinsic growth rate and carrying capacity of the aspens. Thus, in the absence of moose, aspen populations would increase in abundance until they reach their carrying capacity K, while when moose are present, moose populations would grow, and reduce aspen abundance until there were just enough apens left to support the resulting moose population.
Because the model is intended to generate predictions by mimicking the interactions between moose and aspens, this seems like a good example of a “mechanistic” treatment of population growth.
Schaffer then shows that the general features of consumer population dynamics in this model can be approximated using a simple “logistic growth” model, following the form:
Here, rN=c(awK-T), which is the maximum growth rate of moose when aspens are at their carrying capacity, and, KN=ra(1 – T/awK), which is the equilibrium abundance of moose. This “abstracted” model therefore correctly predicts population dynamics when moose abundance is small, and when it is near equilibrium, but slightly mis-estimates the transient dynamics in between. Because the model is intended to fit the shape of the observed data, but is not intended to mimic any of interactions between moose and aspens that give rise to the pattern (aspens aren’t even directly modelled in this approximation), this logistic abstraction represents a somewhat more “phenomenological” treatment.
This is demonstrated in the figure below (with r = 1.2, K = 1, a = 2, w = 0.5, c = 0.1, and T = 0.2). This yields KN = 0.48, and rN = 0.08. For intermediate time periods, particularly between t = 50 and t = 100, the “phenomenological” approximation to the mechanistic model slightly underestimates true system dynamics:
However, this does not mean that a phenomenological model (such as this one) is incapable of reconstructing transient dynamics for the moose-aspen system. If instead of just considering initial growth rate and equilibrium, we track how moose per-capita growth rate changes as a function of resource availability, we find the following linear relationship:
By fitting a simple linear model, we find:
which perfectly matches:
since -cT = -0.02 , and caw = 0.10 (from the above parameterization). Our parameter fits for either case are identical. We could fit a similar relationship between dR/dt 1/R and R and N to obtain estimates for parameters in that equation as well. Thus, by simply fitting regressions to observed patterns, with no intention of understanding the underlying dynamics of the system, we could derive a perfectly predictive phenomenological abstraction of the system.
Interpretation & Intentions
The result is that we end up testing two different questions. In the phenomenological model, we hypothesize (somewhat arbitrarily) that moose reproduction rates will be about 10% faster when aspen populations are at their carrying capacity than when aspens are rare, and also hypothesize that aspens must be at least 20% of their carrying capacity for the moose population to grow at all.
In the mechanistic model, on the other hand, we have a more specific hypothesis: Moose go out into the world and eat aspens at a fixed rate of 2 units of mass for every aspen they encounter, each of those aspens contains 0.5 energy units per unit of mass, each unit of energy that moose consume allows them to add 0.1 new moose to the population, and moose populations require a minimum amount of energy (0.2 units) to reproduce. Thus, the model not only provides predictions, but also shows that a specific, biologically meaningful process (energy conversion from food) is capable of generating the kinds of dynamics that we observe. Thus, it tests the hypothesized functional form of a biological relationship.
Importantly, no matter what method we use to build models, there is no guarantee that our predictions will be right. Imagine that we had parameterized the moose-aspen model for a real system where we were able to manipulate resource availability to fall between 0 (no aspens) and 1.0 (aspens at their moose-free carrying capacity):
Both the “phenomenological” fit, and the “mechanistic” model above assume a simple linear relationship between aspen abundance (R) and moose per-capita growth rate (dN/dt 1/N) (blue line in the figure). While it might be tempting to assume that this relationship holds for higher resource abundance (for example, to predict what would happen if environmental changes caused the carrying capacity for aspens to double), this need not be the case. Think about a limiting resource such as sunlight in a forest: adding more available sunlight to the system does not suddenly make those trees able to grow 1000 meters tall. And, even if trees have enough sunlight to grow, there are also other physiological constraints on the system, such as water or soil nutrients, that would cause their growth rate to slow again.
This is where the storytelling aspect of mechanistic modeling can become useful. Because we derive the linear relationship between per-capita growth rates based on interactions between aspens and moose, we might be able to come up with a more convincing argument for why this relationship should (or should not) hold when aspen populations suddenly increase beyond expected or past concentrations. If we think that doubling aspen populations from R=1 to R=2 won’t change anything about how moose eat and grow, then maybe the extrapolation will hold. If we think that in environments where aspen carrying capacity has doubled, this may indicate less nutritional content per aspen, this may indicate a saturating relationship between R and moose population growth rates.
This seems like the main utility of including “mechanistic” parameters in a model. Since they are intended to represent real-world biological concepts, this means that we can think about them in terms of real-world properties. And, if we can cook up a story that predicts outcomes that look like what we observe in the real world, then we don’t just have a predictive model, but rather we also have quantitative evidence showing that our story could plausibly be the underlying cause of the patterns we see in the real world.
In other words, the mechanistic story helps test not only what will happen, but also why it happened, and also gives us the insight needed to make informed hypotheses about how complex systems may function in novel ecological situations.
Related PLOS ONE paper:
Adam Clark (www.adamclarktheecologist.com) is a PhD student in the University of Minnesota’s Department of Ecology Evolution, and Behavior, studying how plant communities assemble and change across space and time. Adam is interested in mathematical models that predict future dynamics of plant communities based on past trends. He hopes to someday find ways to harness the natural stability of ecosystems to better meet human needs.