Schedule for: 19w5108 - New Mathematical Methods for Complex Systems in Ecology

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

When we construct mathematical models to represent a given real-world system, there is always a degree of uncertainty with regards to the model specification - whether with respect to the choice of parameters or to the choice of formulation of model functions. This can become a real problem in some cases, where choosing two different functions with close shapes in a model can result in substantially different model predictions. This phenomenon is known as structural sensitivity, and is a significant obstacle to improving the predictive power of models - particularly in fields where it is not possible to derive the functions suitable for representing system processes from theory or physical laws, such as the biological sciences. In this talk, I shall revisit the notion of structural sensitivity and propose a general approach to reveal structural sensitivity which is a far more powerful technique than the conventional approach consisting of fixing a particular functional form and varying its parameters. I will demonstrate that conventional methods based on variation of parameters alone will often miss structural sensitivity. I shall discuss the consequences that structural sensitivity and the resulting model uncertainty may have for the modelling of biological systems. In particular, it will be shown the concept of a 'concrete' bifurcation structure may no longer be relevant in the case of structural sensitivity, thus we can only describe bifurcations of completely deterministic systems with a certain probability. Finally, I will show that structural sensitivity can be a possible explanation of the observed irregularity of oscillations of population densities in nature. At the end, we will discuss the current challenges related to structural sensitivity in models and data.

Many systems from the natural world have to adapt to continuously changing external conditions. Some systems have dangerous levels of external conditions, defined by catastrophic bifurcations, above which they undergo a critical transition (B-tipping) to a different state; e.g. forest-desert transitions. Other systems can be very sensitive to how fast the external conditions change and have dangerous rates - they undergo an unexpected critical transition (R-tipping) if the external conditions change slowly but faster than some critical rate; e.g. critical rates of climatic changes. R-tipping is a genuine non-autonomous instability which captures ``failure to adapt to changing environments" [1,2]. However, it cannot be described by classical bifurcations and requires an alternative mathematical framework. In the first part of the talk, we demonstrate the nonlinear phenomenon of R-tipping in a simple ecosystem model where environmental changes are represented by time-varying parameters [Scheffer et al. Ecosystems 11 2008]. We define R-tipping as a critical transition from the herbivore-dominating equilibrium to the plant-only equilibrium, triggered by a smooth parameter shift [1]. We then show how to complement classical bifurcation diagrams with information on nonautonomous R-tipping that cannot be captured by the classical bifurcation analysis. We produce tipping diagrams in the plane of the magnitude and `rate’ of a parameter shift to uncover nontrivial R-tipping phenomena. In the second part of the talk, we develop a general framework for R-tipping based on thresholds, edge states and a suitable compactification of the nonautonomous system. This allows us to define R-tipping in terms of connecting heteroclinic orbits in the compactified system, which greatly simplifies the analysis. We explain the key concept of threshold instability and give rigorous testable criteria for R-tipping in arbitrary dimensions. References: [1] PE O'Keeffe and S Wieczorek,'Tipping phenomena and points of no return in ecosystems: beyond classical bifurcations', arXiv preprint arXiv:1902.01796 [2] A Vanselow, S Wieczorek, U Feudel, 'When very slow is too fast: Collapse of a predator-prey system' Journal of Theoretical Biology (2019)

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Classical ecological theory relies heavily on the principles of deterministic dynamical systems, and methods from mathematics and physics that are more appropriate for stochastic systems are unfamiliar to many ecologists. As a result, when stochasticity plays an important role in shaping ecological dynamics — as it often does — our ability to fully address certain questions can be limited. In this talk, I will give an overview of some new (or at least newly extended for ecological applications) mathematical methods that bring important new classes of questions into reach.

The dynamics of species’ densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. I will present recent theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables, and will illustrate their utility with applications to models of eco-evolutionary dynamics

Understanding stability of ecosystems and communities has always been major challenge for ecologists. Definitions and measures of stability abound and at times are confusing. Nowadays it is in general accepted that stability is multidimensional and it needs to be measured in different ways. Some of the metrics are used to highlight resistance of ecological systems to a specific type of perturbations (like an invasion of an alien). Others have been developed to highlight the approach to tipping points (that is catastrophic transitions between different dynamical states). As long-term data become increasingly available and experimental approaches are improving, the challenge is how to apply our theoretical metrics on these ecological dynamics to understand stability. In the talk, I will present a possible way for identifying best suitable metrics for measuring stability in ecological communities. More in depth, I will also focus on how changes in dynamical properties of ecological dynamics can be used as early warnings to abrupt ecological changes using examples from ecology and the climate.

Some basic ideas behind optimal control of ODEs and PDEs will be introduced. Control of a flow rate in a PDE model for a population in a river will be illustrated. Harvesting in a system of ODEs representing an anchovy ecosystem in the Black Sea will be discussed. Issues and new features of control approaches will be presented at the end and could lead to discussions later at this workshop.

Ecosystem-based approaches to management are desirable for many reasons. However quantitative approaches to ecosystem management are hampered by incomplete knowledge of the system state and uncertainty in the underlying dynamics. In principal, we can circumvent these difficulties using nonparametric approaches to model the uncertain dynamics and using time-delay embedding to implicit account for missing state variables. However, these methods are incredibly data-hungry and tend to be sensitive to observation noise. Here I propose to mitigate these practical obstacles a) by adopting a state-space perspective that allows us to partition observation and process uncertainty and b) by combining data from multiple locations using hierarchical and spatial modeling approaches. We find that noise reduction substantially improves attractor reconstruction and reduces bias in the estimation of Lyapunov exponents from noisy time series. Spatial-delay embedding significantly increases the time horizon over which useful predictions can be made compared to more traditional local embedding.

One of the features that distinguishes biological systems is the wide range of scales in time and space on which processes and interactions occur. This is a form of complexity, but it is one that can sometimes be turned into an advantage. I will describe models for a couple of systems where my collaborators and I have found that to be the case. The first (from long ago) is a system with ladybugs preying on aphids. The ladybugs (which are highly mobile but reproduce slowly) experience the environment as a system of patches, while the aphids (which are much less mobile but reproduce quickly) experience each patch as spatial continuum. The second (more recent) is a system aimed at describing the evolution of dispersal. Dispersal starts with the movement of individuals, which can be observed by tracks or tracking and described in terms of random walks. That then produces spatial patterns, which then influence ecological interactions within and among populations. Those in turn exert selective pressure on traits that determine the spatial patterns, and finally the selective pressure together with the occasional the appearance of mutants results in the evolution of dispersal traits. All of these processes can, in some cases, operate on different scales in time and space. It turns out that this when this occurs it can be exploited to produce relatively simple models in some situations. The older research I will discuss was conducted in collaboration with Steve Cantrell; the newer was with Steve Cantrell, Mark Lewis, and Yuan Lou

Two predator-prey model formulations will be studied. In the classical Rosenzweig-MacArthur (RM) model in absence of the predator the prey grows logistically. Consequently, no nutrient, resource for the prey, is modeled explicitly and the predator-prey model is described by a twodimensional system. In a mass balance (MB) chemostat model the nutrient is explicitly modeled leading to a three-dimensional system. Because this model is based on mass conservation laws, by perfect aggregation the dimension of the system can be reduced by one leading to a twodimensional system just as the RM-model. Assuming that the growth and loss rates of the predator are much smaller than the growth rate of the prey gives so called fast-slow systems. In the RM-model formulation the slow-fast assumption leads to an often-unrealistic assumption that the conversion from prey biomass to predator biomass efficiency needs to be small. On the other hand, in the MB model the unrealistic assumption is avoided. We will show using geometrically singular perturbation theory for the RM-model and bifurcation theory for the MB-model that the predicted long-term dynamics differ significantly when in both models the time scales at the two population levels differ a lot, [1, 2]. For instance, in the RM-model a so-called canard explosion occurs [2] and not in the MB-model [1]. A brief description of the blow-up method is given to proof that the RM model exhibits a maximal canard, [2]. We will also discuss a numerical method to calculate the parameter value where the canard explosion occurs using an asymptotic expansion in the small parameter, [1]. References [1] B. W. Kooi and J-C. Poggiale, Modelling, singular perturbation and bifurcation analyses of bitrophic food chains, Mathematical Bioscience, 301:93-110 2018. [2] J-C. Poggiale, C. Aldebert, B. Girardot and B.W. Kooi, Analysis of a predator-prey model with specific time scales : a geometrical approach proving the occurrence of canard solutions Journal of Mathematical Biology, in Press 2019.

In this talk I am going to first outline the recent progress made in the mathematical theory of early-warning signs for critical transitions. Then I am going to proceed to show new ideas, how to develop early-warning signs in the context of adaptive networks. Adaptive, or co-evolutionary, networks are complex systems with dynamics on and of the network. I will show in the context of epidemic models, how we can use network topology to better understand the transition from endemic to epidemic and vice versa.

Human activities or natural events may perturb locally stable equilibrium communities. One can ask how long the community will take to return to its equilibrium and how "far" from the equilibrium it may get in the process. To answer those questions, we can measure the "resilience" and "reactivity" of a system. These concepts thus quantify one particular form of transient dynamics in ecological models. I will briefly review these measures and give some examples and known but still surprising insights. Then I will suggest extensions to periodically forced systems and periodic orbits in autonomous systems and examine some of their properties.

We will discuss the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This work aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and examining a variety of mathematical models used in ecological applications as well as some empirical facts, we reveal main mechanisms leading to the emergence of long transients both is spatial and nonspatial systems.

In many ecological problems spatial data are collected to satisfy the requirement that the population spatial distributions can be reconstructed with high accuracy. The situation, however, may be different when reconstruction of spatial distributions is required in the context of monitoring and control (M&C) protocol. In my talk I will argue that the M&C protocol can be thought of as a data filter as its application transforms the original dataset and it often results in a spatial distribution with essentially different properties. That transformation may, in turn, alleviate negative impact of uncertainty on the accuracy of results when spatial distributions are reconstructed from data with measurement errors. While original data are affected by the measurement errors, the filtered data may or may not be affected depending on the filter definition. In some cases, there is no need to ask for more accurate data collection as measurement errors will be `eliminated' by application of the M&C protocol. Meanwhile, it will also be shown in the talk that if inherent uncertainty presents in the model, the M&C data filter may become useless, no matter how accurate the data are. This is a joint work with John Ellis and Wenxin Zhang.

In order for models to make relevant predictions about real ecological systems, it is helpful to have data to guide the selection of, for example, parameter values, interaction functions, and dispersal kernels. For many ecological systems, however, data is sparse. Nonetheless, these data can still lead to the development of models that give rise to theoretical predictions with real relevance. Several examples will be presented and discussed in this talk.

Several ecosystems exhibit long transient behavior and experience sudden transition to another state under seemingly constant environment. In the first part of my talk, I will present a singularly perturbed three-species predator-prey model, where such long transients are observed in a neighborhood of a “singular Hopf” bifurcation. The transient dynamics consist of aperiodic mixed mode oscillations (MMOs), featuring concatenation of long epochs of small amplitude oscillations and large amplitude oscillations as the system asymptotically approaches a small amplitude periodic attractor [1]. I will discuss a method that will help to predict whether the system settles down to the periodic attractor or undergoes another long time scale MMO transient. In the second part of my talk, I will discuss the effect of stochasticity on the population dynamics. The interplay between noise strength and distance from the Hopf bifurcation will be explored to study noise induced mixed mode oscillations seen in the model [2]. The distribution of random number of small oscillations between two large oscillations is numerically simulated, which can be related to the return time between the outbreaks. References: 1. S. Sadhu, Mixed-mode oscillations and chaotic dynamics near singular Hopf bifurcation in a two time-scale ecosystem, arXiv preprint arXiv:1901.02974. 2. S. Sadhu and C. Kuehn, Stochastic mixed-mode oscillations in a three-species predator-prey model, Chaos 28, 3, 033606 (2018).

Ecosystems are nonlinear spatially extended systems characterized by multiplicity of stable and unstable states, spatially uniform or patterned, stationary or oscillatory. Two major concerns are associated with multi-stable ecosystems in variable environments. The first is related to the predictions of a warmer and drier climate, and to possible transitions to malfunctioning ecosystem states. These transitions can be abrupt, but also gradual, induced by local disturbances and the subsequent propagation of degradation fronts. The second concern is related to the dominant role played by humans in shaping and transforming the ecology of the Earth, and to the detrimental effects that such transformations often have. Using mathematical models of dryland ecosystems, I will discuss recent results that shed new light on these concerns. Following a brief introduction to vegetation pattern formation in drylands as a population-level mechanism to cope with water stress, I will discuss the dynamics of desertification fronts and describe a front instability that results in the growth of vegetation fingers backward into degraded areas. Possible ways of triggering this instability, and, thereby, inducing self-recovery processes, will be suggested. I will then address the concern of human intervention in ecosystem dynamics, focusing on a simple example – feeding livestock by grazing in drought-prone grasslands. By mapping the unstable states of the system and their existence ranges, a surprising conclusion will be described: grazing can improve the resilience to droughts, rather than impose an additional stress, if managed non-uniformly in space. I will conclude the presentation by generalizing these results to broader ecological contexts and by outlining mathematical challenges in transforming model predictions into practical methodologies of sustainable land management

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.