MBI Videos

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

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    Odo Diekmann
    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.
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    Karen Abbott
    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.
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    Sergei Petrovskii
    A conventional view of the spatial spread of invasive species dating back to the works by Fisher (1937) and Kolmogorov et al. (1937) is that it occurs via the propagation of a travelling population front. In a realistic 2D system, such a front normally separates the invaded area behind the front from the uninvaded areas in front of the front. This view has eventually been challenged by discovering an alternative scenario called “patchy invasion� where the spread takes place via the spatial dynamics of separate patches of high population density with a very low density between them, and a continuous population front does not exist at any time. Patchy invasion was studied theoretically in much detail using diffusion-reaction models. However, diffusion-reaction models have many limitations; in particular, they almost completely ignore long-distance dispersal. In this talk, I will present some new results showing that patchy invasion can occur as well when long-distance dispersal is taken into account. Mathematically, the system is described by integral-difference equations with fat-tailed dispersal kernels. I will also show that apparently minor details of kernel parametrization may have a relatively strong effect on the rate of species spread, which evokes the general issues of understanding the uncertainty and the limits of predictability in ecology.
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    Axel Rossberg
    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 anthropogenic 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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    Alan Hastings
    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).
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    Gregor Fussmann
    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.
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    Jean-Christophe Poggiale
    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.
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    Ottar Bjornstad
    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'.
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    Donald De Angelis
    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.
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    Per Lundberg
    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.
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    Eve McDonald-Madden
    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.

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