CTW: Uncertainty, Sensitivity and Predictability in Ecology: Mathematical Challenges and Ecological Applications

(October 26,2015 - October 30,2015 )


Jennifer Dunne
n/a, Santa Fe Institute
Alan Hastings
Department of Environmental Science and Policy, University of California, Davis
Andrew Morozov
Mathematics, University of Leicester

Uncertainty underlines almost every problem in mathematical ecology, and understanding its implications leads to substantial new mathematical challenges. Issues of uncertainty arise particularly in the structure of models, as reflected by the choice of state variables and model functions, uncertainty in parameters, initial conditions, etc. Uncertainty can greatly affect the determination of the current ecosystem state (e.g., stochastic versus deterministic description) and hence prediction of its dynamics. In ecological models uncertainty can be a real nuisance due to the phenomenon known as model sensitivity: models can be sensitive to the mathematical formulations of the constituent functions. This structural sensitivity can substantially reduce predictability of models. Whereas parameter-based sensitivity methods are now relatively well-developed, the mathematical framework to investigate structural sensitivity, when the entire function is unknown, is in its early stage and this represents a major challenge both in mathematics and ecology. In particular, there is a strong need for reliable mathematical tools to investigate structural sensitivity of biological models directly from data.

In addition, ecosystems are known to sometimes exhibit a sudden (catastrophic) regime shift, which is referred to as the tipping points, and this can be linked to a bifurcation in the model as a response to parameter changes (e.g., due to global climate changes). Development of robust techniques to identify reliable early warning signals of approaching catastrophic transition is a major challenge since the current methods are not always reliable and could result in false alarms, which can be very costly.

One of the goals of the ecosystem management is to estimate the risk of undesirable events. Coping with uncertainty (e.g., by providing the minimal required amount of information about the system) is therefore crucial to enable ecosystem managers to make the right decision in order to guarantee that the risk of undesirable event will not exceed the critical level. Lack of information about underlining processes calls into question the assumption that classical optimal control theory will always be successful. More research is needed to develop the mathematical framework for ecosystem management, in particular looking for an optimal balance between models complexity and their predictive power under a given level of uncertainty.

The main goal of the workshop is to bring together applied mathematicians, theoretical ecologists, empiricists and statisticians in order to address the above raised issues related to ecosystem understanding, modelling, and management to cope with uncertainty

Accepted Speakers

Karen Abbott
Biology, Case Western Reserve University
Ludek Berec
Department of Biosystematics and Ecology, Biology Centre CAS, Institute of Entomology
Ottar Bjornstad
Entomology, Pennsylvania State University
Ben Bolker
Math & statistics and Biology, McMaster University
Donald De Angelis
Department of Biology, University of Miami
Odo Diekmann
Mathematics, Utrecht University
Jennifer Dunne
n/a, Santa Fe Institute
Bill Fagan
Biology, University of Maryland
Gregor Fussmann
Department of Biology, McGill University
Thilo Gross
Alan Hastings
Department of Environmental Science and Policy, University of California, Davis
Robert Holt
Zoology, University of Florida
Christian Kuehn
Mathematics, Vienna University of Technology
Mark Lewis
Canada Research Chair in Mathematical Biology, University of Alberta
Per Lundberg
Biology, Department of Biology, Lund University
Eve McDonald-Madden
Andrew Morozov
Mathematics, University of Leicester
Steve Munch
Ecology and Evolutionary Biology, University of California, Santa Cruz
Natalia Petrovskaya
Jean-Christophe Poggiale
Institut Pytheas (OSU), Aix-Marseille University
Axel Rossberg
Environment and Ecosystems Division, Centre for Environment, Fisheries & Aquaculture Science
Monday, October 26, 2015
Time Session
Tuesday, October 27, 2015
Time Session
Wednesday, October 28, 2015
Time Session
Thursday, October 29, 2015
Time Session
Friday, October 30, 2015
Time Session
Name Email Affiliation
Abbott, Karen kcabbott@case.edu Biology, Case Western Reserve University
Adamson, Matthew mwa4@le.ac.uk.
Barabás, György dysordys@uchicago.edu
Bearup, Daniel daniel.bearup@uni-oldenburg.de
Berec, Ludek berec@entu.cas.cz Department of Biosystematics and Ecology, Biology Centre CAS, Institute of Entomology
Bjornstad, Ottar onb1@psu.edu Entomology, Pennsylvania State University
Blasius, Bernd blasius@icbm.de Institute of Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg
Bolker, Ben bolker@mcmaster.ca Math & statistics and Biology, McMaster University
Cosner, Chris gcc@math.miami.edu Department of Mathematics, University of Miami
Cuddington, Kim kcuddington@uwaterloo.ca Biology, University of Waterloo
Cushing, Jim cushing@math.arizona.edu Mathematics, University of Arizona
Dakos, Vasilis vasilis.dakos@ebd.csic.es Integrative Ecology, Eustachian Biologica de Donana
De Angelis, Donald ddeangelis@bio.miami.edu Department of Biology, University of Miami
Diekmann, Odo O.Diekmann@uu.nl Mathematics, Utrecht University
Dunne, Jennifer jdunne@santafe.edu n/a, Santa Fe Institute
Englund, Göran goran.englund@emg.umu.se
Fagan, Bill bfagan@umd.edu Biology, University of Maryland
Fryxell, John jfryxell@uoguelph.ca
Fussmann, Gregor gregor.fussmann@mcgill.ca Department of Biology, McGill University
Gentleman, Wendy Wendy.Gentleman@Dal.Ca
Gross, Thilo thilo@biond.org
Hastings, Alan amhastings@ucdavis.edu Department of Environmental Science and Policy, University of California, Davis
Holt, Robert rdholt@zoo.ufl.edu Zoology, University of Florida
Johnson, Leah lrjohnson0@gmail.com Integrative Biology, University of South Florida
Kooi, Bob bob.kooi@vu.nl Department of Theoretical Biology,
Kuehn, Christian ck274@cornell.edu Mathematics, Vienna University of Technology
Laurie, Henri henri.laurie@gmail.com
Lewis, Mark mark.lewis@ualberta.ca Canada Research Chair in Mathematical Biology, University of Alberta
Liu, Rongsong Rongsong.Liu@uwyo.edu Mathematics, University of Wyoming
Lundberg, Per per.lundberg@biol.lu.se Biology, Department of Biology, Lund University
McDonald-Madden, Eve e.mcdonaldmadden@uq.edu.au
Meszena, Geza geza.meszena@elte.hu Department of Biological Physics, Eötvös University
Morozov, Andrew am379@leicester.ac.uk Mathematics, University of Leicester
Munch, Steve steve.munch@noaa.gov Ecology and Evolutionary Biology, University of California, Santa Cruz
Nerini, David david.nerini@univ-amu.fr
Petrovskaya, Natalia n.b.petrovskaya@bham.ac.uk
Petrovskii, Sergei sp237@le.ac.uk Mathematics, University of Leicester
Poggiale, Jean-Christophe jean-christophe.poggiale@univ-amu.fr Institut Pytheas (OSU), Aix-Marseille University
Rossberg, Axel Axel@Rossberg.net Environment and Ecosystems Division, Centre for Environment, Fisheries & Aquaculture Science
Terry, Alan aterry.maths@outlook.com
Tilles, Paulo paulotilles@hotmail.com
Uncertainty behind estimation of Allee effects and population density may significantly impact risk of population extinction

Estimation of extinction thresholds arising from Allee effects is notoriously difficult. Traditionally, a point estimate is substituted for the Allee effect strength in adequately formulated population models. However, since any point estimate entails an underlying uncertainty, accounting for this uncertainty inevitably affects risk of population extinction. I will show that the probability of population extinction decreases sigmoidally with increasing population density, even in the absence of any stochasticity, and predict how adding stochastic noise modifies the effect. Modelling suggests that the impact of uncertainty in the Allee effect strength estimate increases as the Allee effect strength itself increases and decreases as the species recovery potential increases. This is by no means good, since we aim to preferentially and efficiently manage slowly recovering populations prone to strong Allee effects. I will argue, somewhat paradoxically, that the impact of the uncertainty can be mitigated by diversifying Allee effect experiments such that we put more emphasis on larger groups. Uncertainty also surrounds estimates of population density, especially in low-density populations. The joint effect of both these kinds of uncertainty on the probability of population extinction will also be discussed.

Nonlinearity and chaos in ecological dynamics revisited

Historically, both experimental and theoretical ecologists have sought to emulate the development of early theory in the physical sciences: the ideal that a few simple equations may accurately predict the complex movement of celestial bodies or interactions among molecules in mixing gasses. In the environmental sciences, such simple clockworks have rarely been found, and rather than predictable stable or recurring patterns, erratic patterns abound. The discovery in the late 1970s through mid-1980s of certain ecological models—such as the Ricker or discrete logistic maps—suggesting erratic fluctuations through dynamic chaos caused what cautionaries may characterize as ecology’s period of “rational exuberance” with respect to hoping that a small set of mathematical equations may explain the erratic dynamics of real-world ecological communities. Upon much discussion, the field as a whole grew skeptical of this idea during the late 90s. During the subsequent 3 decades, mathematical theories of the sensitivity and predictability of ecological and epidemiological systems have been much refined. I will discuss a handful of case studies that I believe were pivotal in changing our more recent understanding of 'Uncertainty, Sensitivity and Predictability in Ecology'.

Approaches and Uncertainties in Predicting Coastal Ecosystem Changes Due to Rising Sea Level

Sea level rise (SLR) is causing changes in coastal vegetation in some locations, negatively affecting freshwater terrestrial ecosystems through salinity intrusion of groundwater and through increased instances of salinity overwash from hurricane-induced storm surges. These effects of SLR cause shifts in the ecotone from freshwater (glycophytic) and salinity tolerant (halophytic) vegetation. Numerous uncertainties make predictions of these shifts difficult. The uncertainties include the obvious difficulty in predicting hurricanes and their effects, but they also include uncertainty in the internal feedbacks between each vegetation type and its local associated soil conditions. These feedbacks may promote resilience to change from disturbances such as storm surges, but disturbances of sufficient size may overcome resilience and lead to vegetation regime shifts. We review a series of models with increasing resolution intended to make predictions concerning effects of both gradual SLR and storm surges on coastal vegetation in southern Florida. In combination with modeling, use of stable isotopes is described as an early indicator of future changes from glycophytic (freshwater hardwood trees) to halophytic (mangrove) trees.

Top down, bottom up or pragmatism?

Despite the pretentious title, the talk will just consist of a few loose remarks followed by a brief description of Linear Chain Trickery (i.e., a characterization of kernels for delay equations that allow reduction to ordinary differential equations) mainly in the context of epidemic models.

Animal movement in dynamic landscapes: Quantifying patterns and modeling processes

Terrestrial landscapes change on spatial and temporal scales that are relevant for the movement of large-bodied vertebrates. Using several empirical datasets about vertebrate migrations, I will outline the critical role that food resources play in determining variation in migration distance among populations and in driving changes in migration distance over time within populations. Continuous-time, continuous-space stochastic processes, which can be characterized in terms of a population's critical spatial and temporal scales of autocorrelation, provide one useful mathematical framework for the analysis of animal movement trajectories. This framework lends itself naturally to a variety of extensions that are useful in ecological applications relying on movement tracks, including the delineation of animal home ranges (and shifts in range) and probabilistic path reconstruction.

A Survey of the Mathematical Theory of Early-Warning Signs

In this talk, I am going to illustrate current techniques based upon multiple time scale dynamical systems and stochastic analysis that can be used to detect early-warning signs for drastic transitions. In particular, the role of scaling laws will be emphasized and several examples from mathematical biology will be given including theoretical modelling components as well as data analysis.

Evaluating structural sensitivity of partially specified models in ecology

Mathematical models in ecology and evolution are highly simplified representations of a complex underlying reality. For this reason, there is always a high degree of uncertainty with regards to the model specification, not just in terms of parameters, but also in the form taken by the model equations themselves. This uncertainty becomes critical for models in which the use of two different functions fitting the same dataset can yield substantially different model predictions - a property known as structural sensitivity. In this case, even if the model is purely deterministic, the uncertainty in the model functions carries through into uncertainty in our model predictions, and new frameworks are required to tackle this fundamental problem. Here, we construct a framework that uses partially specified models in ecology: ODE models in which unknown functions are represented not by a specific functional form, but by an entire data range and constraints of biological realism. Partially specified models can be used to rigorously detect when models are structurally sensitive in their predictions concerning the character of an equilibrium point or a limit cycle by projecting the data range into a generalised bifurcation space formed of equilibrium values and derivatives of any unspecified functions (e.g. functional responses of predators, species growth rates, etc). The key question of how to carry out this projection is a serious mathematical challenge and an obstacle to the use of partially specified models. We address this challenge by developing several powerful techniques to perform such a projection, using geometrical methods and techniques from optimal control theory. Finally, we introduce the 'degree of sensitivity' of these models, which allows us to estimate uncertainty in partially specified biological models, and then show how this degree can be calculated using different techniques.

The effect of sparse and noisy data in ecological monitoring and pest control

Many ecological problems require monitoring and sampling of `alien' population, where the information obtained as a result of monitoring is then used for making decision about means of control. In ecological applications, data used for decision making are often sparse due to financial, labour, and other restrictions on the sampling routine. The same sparse data can also be noisy because of the inherent nature of the ecological problem.

One example of a monitoring procedure based on sparse and noisy data is given by a widespread and important problem of pest insect abundance evaluation from the insect density in an agricultural field. An inaccurate estimate of the pest abundance obtained because of uncertainty in data can result in the wrong decision about a control action (e.g. unnecessary application of pesticides). Thus in our talk we discuss how to quantify the effect of data sparseness and noise in the pest insect monitoring problem. It will be argued that noise is a negligible factor in comparison with the uncertainty of evaluation arising as a result of poor sampling.

Including individual properties in community and ecosystem models: is it useful and how should we proceed?

In a first part, some problems observed with ecosystem models are discussed, focusing on the choice of the formulations of the biological processes involved, with several examples from the literature. Providing explicit relations between individuals properties and population or community dynamics allows to build model formulations on a mechanistic basis. We discuss some examples where this approach can be useful for understanding the community dynamics. The functional response in predator - prey systems is an example of ecological process involving several levels of organization and time scales. Its mathematical formulation should depend on the applications of the model : which spatial scales are considered? Is the environment homogeneous or heterogeneous? These questions should shape the choice of the formulations used in models. Moreover, the data used to develop a model are often acquired in conditions which are different than those of the applications. For instance, some formulations are based on data obtained in laboratory experiments, while the models are used to describe natural environments. Scaling up methods, which provide explicit links between different organization levels or between several temporal/spatial scales, are then useful to build formulations adapted for models used in the natural environment. Several applications to marine systems modelling are then presented.

Why structural instability is inherent to ecological communities and how management can deal with it

Structural instability denotes situations where small changes in parameters (or external pressures) can fundamentally change the state of a system, in ecological communities typically through extirpations. I will argue based on models and data that structural instability increases with species richness and that natural communities tend to be packed to the point where invasion of any new species leads to extirpation of one other on average. As a result, ecological communities are inherently structurally unstable; detailed predictions of changes in ecosystem state in response to anthropogentic pressures are often impossible. Facing this challenge, managers have two options: to manage at the level of higher emergent properties, e.g. community size spectra, or to engineer desired ecosystem states and to stabilize them through adaptive management. I will discuss both options for the case of fisheries management.