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

### Organizers

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
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
Engineering, University of Bristol
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
Per Lundberg
Biology, Department of Biology, Lund University
Jon Machta
Physics, University of Massachusetts
School of Geography, Planning, and Environmental Management, University of Queensland
Andrew Morozov
Mathematics, University of Leicester
Steve Munch
Ecology and Evolutionary Biology, University of California, Santa Cruz
Natalia Petrovskaya
Mathematics, University of Birmingham
Jean-Christophe Poggiale
Institut Pytheas (OSU), Aix-Marseille University
Axel Rossberg
School of Biological and Chemical Sciences, Queen Mary, University of London
Monday, October 26, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:15 AM

Greetings and info from MBI - Marty Golubitsky

09:15 AM
09:30 AM

Welcome and Overview - The Organizers

09:30 AM
10:30 AM
Alan Hastings - Predictability, uncertainty and the persistence of ecological populations

I will develop simple approaches for the incorporation of large noise in ecological models, and indicate how this leads to open questions, both mathematical and biological. I will provide examples from both response to resource pulses and the dynamics of spatiotemporal synchrony in masting (production of seeds).

10:30 AM
11:00 AM

Break

11:00 AM
12:00 PM
Andrew Morozov - 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.

12:00 PM
12:50 PM

Group discussion: choosing relevant topics

12:50 PM
02:20 PM

Lunch Break

02:20 PM
03:20 PM
Gregor Fussmann - Structural sensitivity in food web models

Food webs are interaction networks that link predator and prey populations. The so-called functional response is the linking function that determines the uptake of prey by the predator. While it is clear that this function should be nonlinear and saturating with increasing prey densities, there is no single “right” function that describes the predator-prey interaction. A number of functions with vastly different mathematical properties (e.g., polynomial, exponential, trigonometric) are used in food web models. It has been shown previously that, already for two-species models, predictions about predator-prey dynamics and stability strongly depend on the choice of functional response. In this talk, I show the consequences of multiplying the sources of uncertainty by varying functional responses for the large number of predator-prey interactions that occur in complex food webs.

03:20 PM
03:50 PM

Break

03:50 PM
04:50 PM

Group discussion

04:50 PM
06:30 PM

Reception and Poster Session

06:30 PM

Shuttle pick-up from MBI

Tuesday, October 27, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
10:00 AM
Jean-Christophe Poggiale - 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.

10:00 AM
11:00 AM
Thilo Gross - Generalized Models of Food Web Dynamics

Much of the fascination for ecology stems from the complexity of ecological communities. The often subtle ways in which populations interact gives rise to a complex dynamical interplay. Understanding this interplay- both in concrete examples and by uncovering general principles- is a central goal of Ecology. By depicting the system as a network- we simplify the system while retaining the complexity of the structure of interactions. We can then ask how this structure relates to dynamical properties- such as stability to various perturbations- and heterogeneity in time and space. The obstacles to overcome in this search- are the complexity of the communities themselves- which can assemble in a myriad of different structures- and the lack of detailed data- such as the precise kinetic rate laws.

In this talk I present the approach of generalized modeling. A generalized model describes the dynamics of a complex 'networked' system- without restricting the dynamics in the network nodes to specific functional forms. Despite their generality- generalized models can be explored highly efficiently and can be used to find precise answers to specific ecological questions. They thus enable us to analyze a large ensemble of network structures based on limited information. I will illustrate this power of generalized models by showing recent results on the impact of climate change on mammal communities in Egypt- identification of key species in an aquatic systems and general results on important factors contributing to the stability of food webs.

11:00 AM
11:20 AM

Break

11:20 AM
12:10 PM

Discussions in groups

12:10 PM
01:40 PM

Lunch Break

01:40 PM
02:40 PM
Bill Fagan - 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.

02:40 PM
03:40 PM
Natalia Petrovskaya - 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.

03:40 PM
04:20 PM

Break

04:20 PM
05:25 PM

Discussion in groups + 20 min of General Discussion in the main lecture room

05:30 PM

Shuttle pick-up from MBI

Wednesday, October 28, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
10:00 AM
Ottar Bjornstad - 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'.

10:00 AM
11:00 AM
Robert Holt - Evolutionary dimensions of ecological uncertainty and predictability

All species comprising ecological communities contain a standing crop of genetic variation that can likely affect key ecological traits that determine their responses to the abiotic environment, and how they interact with each other, and novel mutations can arise that change ecological interactions as well. Incorporating evolutionary perspectives into ecology can suggest several potential sources of uncertainty in ecological predictions. A species may adaptively response to for instance a deteriorating environment in a completely directional way, yet because of how genetic variation is sampled in finite populations, the response might be stochastic so that some focal populations rapidly adapt, whereas other exhibit long?term stasis and may even go extinct. Moreover, introducing genetic variation and evolution introduces additional modalities of density?dependent and frequency?dependent feedbacks into ecological theory, which can stabilize otherwise unstable dynamics, or lead to novel and unexpected ecological instabilities. These points will be illustrated with a review of theoretical studies from the past several years (including unpublished work) of niche conservatism, evolutionary rescue, and the coevolution of interacting species.

11:00 AM
11:20 AM

Break

11:20 AM
12:00 PM

Discussions in groups

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:30 PM
Axel Rossberg - 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.

02:30 PM
03:30 PM
Donald De Angelis - 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.

03:30 PM
04:30 PM
Per Lundberg - The irreducible uncertainty of the demography-environment interaction

The interpretation of ecological data has been greatly improved by bridging the gap between ecological and statistical models. The major challenge is to separate competing hypotheses concerning demography, or other ecological relationships, and environmental variability (noise). This may be an impossible task. A reconstruction of underlying ecological processes can only be done if we are certain of either the demographic or the noise model, which is something that can only be achieved by an improved theory of stochastic ecological processes. Ignoring the fact that this is a real problem may mislead ecologists and result in erroneous conclusions about the relative importance of endogenous and exogenous factors in natural ecosystems. The problem will be illustrated by a few model analyses and some thoughts on the epistemology of ecology.

04:30 PM
05:00 PM

Break

05:00 PM
05:30 PM

General Discussion

05:30 PM

Shuttle pick-up from MBI

Thursday, October 29, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
10:00 AM
Christian Kuehn - 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.

10:00 AM
11:00 AM
Jon Machta - What can statistical physics say about the synchronization of ecological populations?

Long-range synchronization of ecological populations is usually attributed to global perturbations--the Moran effect. However, short range dispersal may initiate and sustain global synchrony over distances much larger than the dispersal length scale, even in the presence of strong local noise and inhomogeneities. Statistical physics provides tools for understanding the onset and maintenance of long-range order in thermodynamic systems and these tools can be applied to spatially explicit population models relevant to ecology. In this talk, I will introduce some of the relevant concepts from statistical physics and show how they apply to locally-coupled, noisy ecological population models.

11:00 AM
11:30 AM

Break

11:30 AM
12:30 PM
Ben Bolker - Fitting and forecasting epidemic models: what should we worry about?

The last decade's explosion of statistical and computational tools for fitting nonlinear stochastic dynamical systems raises the question of what our priorities should be when trying to fit models to emerging epidemics. Given the primary goal of accurate forecasting and assessment of control measures, and the constraints of data availability, complexity of implementation, and computational cost, what are the real costs and benefits of simple (e.g. trajectory matching with a deterministic model to the cumulative incidence curve) vs. complex (e.g. particle filtering) modeling and estimation methods?

What is the value of incorporating various levels of stochastic uncertainty into the model, such as spatial and temporal variation in contact rate? Of treating auxiliary parameters such as the length and distribution of the latent and infectious periods as perfectly known, completely unknown, or governed by a Bayesian prior?

12:30 PM
02:00 PM

Lunch Break

02:00 PM
03:00 PM
Eve McDonald-Madden - Understanding the perverse implications of conservation actions: modelling ecosystems with limited information and guiding system-wide conservation monitoring and management

Decisions about the allocation of conservation resources are often made with a focus on individual species. The management of any one species is, however, likely to impact other species in an ecosystem. For example, the re-establishment of wolves in Yellowstone National Park had dramatic and unexpected indirect impacts on vegetation and water flows via the wolves’ predation on elk. Considering individual species in isolation when making conservation management decisions may be detrimental not only to non-focal species in the system, but also ultimately to the very species we are aiming to protect. A decade or more of food web theory highlights the potential catastrophic cascading impacts of ecosystem modification and collateral impacts have been well documented for the introduction of invasive species. Predicting these ecosystem-level outcomes is notoriously difficult because they depend on accurate and quantitative understanding of the ecosystem dynamics. However for the majority of ecosystems information on these interactions are at best limited and in most cases unknown. In this talk I will present a novel modelling approach using generalized Lotka-Volterra equations to model the uncertain, coupled dynamics of a large system of species and to predict plausible ecosystem models using ‘backcasting’ and limited quantitative or qualitative system observations. I will then explore our ability to understand the potential adverse outcomes from planned management interventions and to inform effective monitoring to detect adverse species responses and hence guide strategic mitigation actions. I will illustrate this work with two Australian case studies, the impacts of cat eradication of Christmas Island, and the risk of perverse outcomes from reintroductions into Booderee National Park.

03:00 PM
04:00 PM
Steve Munch - Nonparametric approaches to ecological dynamics and ecosystem management

Nonlinear forecasting methods have been successful in fields ranging from physics and neurobiology to finance. Although these methods should apply to understanding ecological dynamics, they need to be extended to handle short, noisy time series from nonstationary systems. Adopting a Bayesian nonparametric approach based on Gaussian process regression allows us to easily define hierarchical, nonstationary models to integrate information across these series and make robust forecasts. Near-optimal policies may be derived from these short term forecasts using approximate dynamic programming. In simulations, the policies generated under this framework are much more robust to structural uncertainty than methods based on parametric model selection and in some cases produce significantly greater long-term rewards.

04:00 PM
04:20 PM

Break

04:20 PM
05:30 PM

Discussions in groups + General Discussion. Preparing short reports

05:30 PM

Shuttle pick-up from MBI

06:20 PM
06:50 PM

Cash Bar at Crowne Plaza

06:50 PM
08:30 PM

Banquet at Crowne Plaza

Friday, October 30, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
10:00 AM
Karen Abbott - Giving "noise" the respect it deserves: ways to understand and visualize effects of stochasticity in ecological dynamics

Population dynamics result from a combination of deterministic mechanisms (e.g. competition, predation) that drive density-dependent dynamics and stochastic forces that disrupt the neat patterns that would otherwise result. Stochastic noise is often effectively viewed as a nuisance, seen as creating uncertainty and unpredictability without contributing in interesting ways to the list of mechanisms driving dynamics. However, it is becoming increasingly clear that in some situations, stochasticity itself plays an important qualitative role in shaping overall dynamical patterns, such that the dynamics cannot be fully understood by studying the deterministic mechanisms alone. Classical approaches to studying theoretical models are not well-equipped to make insights about these situations. Alternative analytical approaches exist but are not yet widely used in ecology. In this talk, I will present some useful ways to interpret and visualize effects of stochasticity in noisy ecological models, as well as some proof-of-concept examples to show the value of these approaches.

10:00 AM
11:00 AM
Ludek Berec - 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.

11:00 AM
11:20 AM

Break

11:20 AM
12:20 PM
Odo Diekmann - 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.

12:20 PM
12:50 PM

Summary Discussion

12:50 PM
12:55 PM

Closing

Name Email Affiliation
Abbott, Karen kcabbott@case.edu Biology, Case Western Reserve University
Adamson, Matthew mwa4@le.ac.uk. Environmental Systems Research, Universit""at Osnabr""uck
Barabás, György dysordys@uchicago.edu Ecology & Evolution, The University of Chicago
Bearup, Daniel daniel.bearup@uni-oldenburg.de Institute for Chemistry and Biology of the Marine Environment, University of Oldenburg
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 Mathematics, University of Miami
Cuddington, Kim kcuddington@uwaterloo.ca Biology, University of Waterloo
Cushing, Jim cushing@math.arizona.edu Mathematics, University of Arizona
Dahlin, Kyle kdahlin@purdue.edu Mathematics, Purdue University
Dakos, Vasilis vasilis.dakos@ebd.csic.es Institute of Integrative Biology, Eidgen""ossische TH H""onggerberg
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
Enyi, Cyril enyicyrildennis@gmail.com Mathematics and Statistics, King Fahd University of Petroleum and Minerals
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 Engineering Mathematics, Dalhousie University
Gross, Thilo thilo@biond.org Engineering, University of Bristol
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
Hyder, Ayaz College of Public Health, The Ohio State University
Iyiola, Olaniyi samuel@kfupm.edu.sa Department of Mathematics and Statistics, University of Wisconsin
Johnson, Leah lrjohnson0@gmail.com Integrative Biology, University of South Florida
Karatayev, Vadim vak32@cornell.edu Ecology, University of California, Davis
Kooi, Bote Department of Theoretical Biology, Vrije Universiteit
Kramer, Peter kramep@rpi.edu Mathematical Sciences, Rensselaer Polytechnic Institute
Kuehn, Christian ck274@cornell.edu Mathematics, Vienna University of Technology
Lamba, Sanjay sanjaylamba1@gmail.com Mathematics, Central University of Rajasthan
Laurie, Henri henri.laurie@gmail.com Mathematics and Applied Mathematics, University of Cape Town
Lister, Bradford listeb@rpi.edu Biological Sciences, RPI
Lundberg, Per per.lundberg@biol.lu.se Biology, Department of Biology, Lund University
Machta, Jon machta@physics.umass.edu Physics, University of Massachusetts
Massie, Thomas thomas.massie@ieu.uzh.ch IEU, Institute for Evolutionary Biology & Environmental Studies, Universit""at Z""urich
McDonald-Madden, Eve e.mcdonaldmadden@uq.edu.au School of Geography, Planning, and Environmental Management, University of Queensland
Meszena, Geza geza.meszena@elte.hu Department of Biological Physics, EÃ¶tvÃ¶s University
Moritz, Mark moritz.42@osu.edu Anthropology, The Ohio State University
Morozov, Andrew am379@leicester.ac.uk Mathematics, University of Leicester
Mubayi, Anuj a-mubayi@neiu.edu Mathematics, Northeastern Illinois University
Munch, Steve steve.munch@noaa.gov Ecology and Evolutionary Biology, University of California, Santa Cruz
Nerini, David david.nerini@univ-amu.fr Observatoire des Sciences de l'Univers, Universit'e d'Aix-Marseille II (Universit'e de la M'editerran'ee)
Noble, Andrew andrew.e.noble@gmail.com Environmental Science and Policy, University of California, Davis
Pennekamp, Frank frank.pennekamp@ieu.uzh.ch Institute of Evolutionary Biology and Environmental Studies, Universit""at Z""urich
Petrovskaya, Natalia n.b.petrovskaya@bham.ac.uk Mathematics, University of Birmingham
Petrovskii, Sergei sp237@le.ac.uk Mathematics,
Poggiale, Jean-Christophe jean-christophe.poggiale@univ-amu.fr Institut Pytheas (OSU), Aix-Marseille University
Rossberg, Axel Axel@Rossberg.net School of Biological and Chemical Sciences, Queen Mary, University of London
Sudakov, Ivan isudakov1@udayton.edu Physics, University of Dayton
Tilles, Paulo paulotilles@hotmail.com Mathematics Department, State University of Londrina (UEL)
White, Easton eawhite@ucdavis.edu Center for Population Biology, University of California, Davis
Xie, Dexuan dxie@uwm.edu Department of Mathematical Sciences, University of Wisconsin
Giving "noise" the respect it deserves: ways to understand and visualize effects of stochasticity in ecological dynamics

Population dynamics result from a combination of deterministic mechanisms (e.g. competition, predation) that drive density-dependent dynamics and stochastic forces that disrupt the neat patterns that would otherwise result. Stochastic noise is often effectively viewed as a nuisance, seen as creating uncertainty and unpredictability without contributing in interesting ways to the list of mechanisms driving dynamics. However, it is becoming increasingly clear that in some situations, stochasticity itself plays an important qualitative role in shaping overall dynamical patterns, such that the dynamics cannot be fully understood by studying the deterministic mechanisms alone. Classical approaches to studying theoretical models are not well-equipped to make insights about these situations. Alternative analytical approaches exist but are not yet widely used in ecology. In this talk, I will present some useful ways to interpret and visualize effects of stochasticity in noisy ecological models, as well as some proof-of-concept examples to show the value of these approaches.

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'.

Fitting and forecasting epidemic models: what should we worry about?

The last decade's explosion of statistical and computational tools for fitting nonlinear stochastic dynamical systems raises the question of what our priorities should be when trying to fit models to emerging epidemics. Given the primary goal of accurate forecasting and assessment of control measures, and the constraints of data availability, complexity of implementation, and computational cost, what are the real costs and benefits of simple (e.g. trajectory matching with a deterministic model to the cumulative incidence curve) vs. complex (e.g. particle filtering) modeling and estimation methods?

What is the value of incorporating various levels of stochastic uncertainty into the model, such as spatial and temporal variation in contact rate? Of treating auxiliary parameters such as the length and distribution of the latent and infectious periods as perfectly known, completely unknown, or governed by a Bayesian prior?

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.

Structural sensitivity in food web models

Food webs are interaction networks that link predator and prey populations. The so-called functional response is the linking function that determines the uptake of prey by the predator. While it is clear that this function should be nonlinear and saturating with increasing prey densities, there is no single “right” function that describes the predator-prey interaction. A number of functions with vastly different mathematical properties (e.g., polynomial, exponential, trigonometric) are used in food web models. It has been shown previously that, already for two-species models, predictions about predator-prey dynamics and stability strongly depend on the choice of functional response. In this talk, I show the consequences of multiplying the sources of uncertainty by varying functional responses for the large number of predator-prey interactions that occur in complex food webs.

Generalized Models of Food Web Dynamics

Much of the fascination for ecology stems from the complexity of ecological communities. The often subtle ways in which populations interact gives rise to a complex dynamical interplay. Understanding this interplay- both in concrete examples and by uncovering general principles- is a central goal of Ecology. By depicting the system as a network- we simplify the system while retaining the complexity of the structure of interactions. We can then ask how this structure relates to dynamical properties- such as stability to various perturbations- and heterogeneity in time and space. The obstacles to overcome in this search- are the complexity of the communities themselves- which can assemble in a myriad of different structures- and the lack of detailed data- such as the precise kinetic rate laws.

In this talk I present the approach of generalized modeling. A generalized model describes the dynamics of a complex 'networked' system- without restricting the dynamics in the network nodes to specific functional forms. Despite their generality- generalized models can be explored highly efficiently and can be used to find precise answers to specific ecological questions. They thus enable us to analyze a large ensemble of network structures based on limited information. I will illustrate this power of generalized models by showing recent results on the impact of climate change on mammal communities in Egypt- identification of key species in an aquatic systems and general results on important factors contributing to the stability of food webs.

Predictability, uncertainty and the persistence of ecological populations

I will develop simple approaches for the incorporation of large noise in ecological models, and indicate how this leads to open questions, both mathematical and biological. I will provide examples from both response to resource pulses and the dynamics of spatiotemporal synchrony in masting (production of seeds).

Evolutionary dimensions of ecological uncertainty and predictability

All species comprising ecological communities contain a standing crop of genetic variation that can likely affect key ecological traits that determine their responses to the abiotic environment, and how they interact with each other, and novel mutations can arise that change ecological interactions as well. Incorporating evolutionary perspectives into ecology can suggest several potential sources of uncertainty in ecological predictions. A species may adaptively response to for instance a deteriorating environment in a completely directional way, yet because of how genetic variation is sampled in finite populations, the response might be stochastic so that some focal populations rapidly adapt, whereas other exhibit long?term stasis and may even go extinct. Moreover, introducing genetic variation and evolution introduces additional modalities of density?dependent and frequency?dependent feedbacks into ecological theory, which can stabilize otherwise unstable dynamics, or lead to novel and unexpected ecological instabilities. These points will be illustrated with a review of theoretical studies from the past several years (including unpublished work) of niche conservatism, evolutionary rescue, and the coevolution of interacting species.

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.

The irreducible uncertainty of the demography-environment interaction

The interpretation of ecological data has been greatly improved by bridging the gap between ecological and statistical models. The major challenge is to separate competing hypotheses concerning demography, or other ecological relationships, and environmental variability (noise). This may be an impossible task. A reconstruction of underlying ecological processes can only be done if we are certain of either the demographic or the noise model, which is something that can only be achieved by an improved theory of stochastic ecological processes. Ignoring the fact that this is a real problem may mislead ecologists and result in erroneous conclusions about the relative importance of endogenous and exogenous factors in natural ecosystems. The problem will be illustrated by a few model analyses and some thoughts on the epistemology of ecology.

What can statistical physics say about the synchronization of ecological populations?

Long-range synchronization of ecological populations is usually attributed to global perturbations--the Moran effect. However, short range dispersal may initiate and sustain global synchrony over distances much larger than the dispersal length scale, even in the presence of strong local noise and inhomogeneities. Statistical physics provides tools for understanding the onset and maintenance of long-range order in thermodynamic systems and these tools can be applied to spatially explicit population models relevant to ecology. In this talk, I will introduce some of the relevant concepts from statistical physics and show how they apply to locally-coupled, noisy ecological population models.

Understanding the perverse implications of conservation actions: modelling ecosystems with limited information and guiding system-wide conservation monitoring and management

Decisions about the allocation of conservation resources are often made with a focus on individual species. The management of any one species is, however, likely to impact other species in an ecosystem. For example, the re-establishment of wolves in Yellowstone National Park had dramatic and unexpected indirect impacts on vegetation and water flows via the wolves’ predation on elk. Considering individual species in isolation when making conservation management decisions may be detrimental not only to non-focal species in the system, but also ultimately to the very species we are aiming to protect. A decade or more of food web theory highlights the potential catastrophic cascading impacts of ecosystem modification and collateral impacts have been well documented for the introduction of invasive species. Predicting these ecosystem-level outcomes is notoriously difficult because they depend on accurate and quantitative understanding of the ecosystem dynamics. However for the majority of ecosystems information on these interactions are at best limited and in most cases unknown. In this talk I will present a novel modelling approach using generalized Lotka-Volterra equations to model the uncertain, coupled dynamics of a large system of species and to predict plausible ecosystem models using ‘backcasting’ and limited quantitative or qualitative system observations. I will then explore our ability to understand the potential adverse outcomes from planned management interventions and to inform effective monitoring to detect adverse species responses and hence guide strategic mitigation actions. I will illustrate this work with two Australian case studies, the impacts of cat eradication of Christmas Island, and the risk of perverse outcomes from reintroductions into Booderee National Park.

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.

Nonparametric approaches to ecological dynamics and ecosystem management

Nonlinear forecasting methods have been successful in fields ranging from physics and neurobiology to finance. Although these methods should apply to understanding ecological dynamics, they need to be extended to handle short, noisy time series from nonstationary systems. Adopting a Bayesian nonparametric approach based on Gaussian process regression allows us to easily define hierarchical, nonstationary models to integrate information across these series and make robust forecasts. Near-optimal policies may be derived from these short term forecasts using approximate dynamic programming. In simulations, the policies generated under this framework are much more robust to structural uncertainty than methods based on parametric model selection and in some cases produce significantly greater long-term rewards.

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.

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