How noise and coupling influence leading indicators of population extinction in a spatially extended ecological system

Anticipating critical transitions in spatially extended systems is a key topic of interest to ecologists. Gradually declining metapopulations are an important example of a spatially extended biological system that may exhibit a critical transition. Theory for spatially extended systems approaching extinction that accounts for environmental stochasticity and coupling is currently lacking. Here, we develop spatially implicit two-patch models with additive and multiplicative forms of environmental stochasticity that are slowly forced through population collapse, through changing environmental conditions. We derive patch-specific expressions for candidate indicators of extinction and test their performance via a simulation study. Coupling and spatial heterogeneities decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations, and the noise regime and the degree of coupling together determine trends in summary statistics. This theory may be readily applied to other spatially extended ecological systems, such as coupled infectious disease systems on the verge of elimination.     

Coupled land use and ecological models reveal emergence and feedbacks in socio‐ecological systems

Understanding the dynamics of socio‐ecological systems is crucial to the development of environmentally sustainable practices. Models of social or ecological sub‐systems have greatly enhanced such understanding, but at the risk of obscuring important feedbacks and emergent effects. Integrated modelling approaches have the potential to address this shortcoming by explicitly representing linked socio‐ecological dynamics. We developed a socio‐ecological system model by coupling an existing agent‐based model of land‐use dynamics and an individual‐based model of demography and dispersal. A hypothetical case‐study was established to simulate the interaction of crops and their pollinators in a changing agricultural landscape, initialised from a spatially random distribution of natural assets. The bi‐directional coupled model predicted larger changes in crop yield and pollinator populations than a unidirectional uncoupled version. The spatial properties of the system also differed, the coupled version revealing the emergence of spatial land‐use clusters that neither supported nor required pollinators. These findings suggest that important dynamics may be missed by uncoupled modelling approaches, but that these can be captured through the combination of currently‐available, compatible model frameworks. Such model integrations are required to further fundamental understanding of socio‐ecological dynamics and thus improve management of socio‐ecological systems.     

Transient dynamics and persistence of ecological systems

Using spatially coupled predator–prey systems as an example of a cyclic ecological system where coexistence depends on oscillations, transient dynamics of models where there are no stable persistent solutions are shown to be a reasonable explanation of persistence over ecological time scales. The parameter values leading to transients within the context of a particular model may be far from parameter values that lead to stable solutions, so transients will need to be explicitly considered in model analysis. Since natural systems with many coupled oscillating species are common, and natural communities are often reset by disturbances or seasonality, transients should play a central role in understanding natural systems.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ₁-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.     

Modeling experimental time series with ordinary differential equations

Recently some methods have been presented to extract ordinary differential equations (ODE) directly from an experimental time series. Here, we introduce a new method to find an ODE which models both the short time and the long time dynamics. The experimental data are represented in a state space and the corresponding flow vectors are approximated by polynomials of the state vector components. We apply these methods both to simulated data and experimental data from human limb movements, which like many other biological systems can exhibit limit cycle dynamics. In systems with only one oscillator there is excellent agreement between the limit cycling displayed by the experimental system and the reconstructed model, even if the data are very noisy. Furthermore, we study systems of two coupled limit cycle oscillators. There, a reconstruction was only successful for data with a sufficiently long transient trajectory and relatively low noise level.     

Dispersal and transient dynamics in metapopulations

Abstract Recent studies of spatially explicit metapopulation models have shown the existence of complex transient behaviour (supertransients and mesotransients) characterized by spontaneous changes in the system's dynamics after thousands or hundreds of generations, respectively. Their detection in simple ecological models has been taken as evidence that transient dynamics may be common in nature. In this study, we explore the generality of these phenomena in a simple one‐dimensional spatially explicit metapopulation model. We investigate how frequently supertransient behaviour emerges in relation to the shape and type of the dispersal kernel used (normal and Laplace), system size, boundary conditions and how sensitive they are to initial conditions. Our results show that supertransients are rare, are heavily affected by initial conditions and occur for a small set of dispersal parameter values, which vary according to kernel type, system size, and boundary conditions. Similarly, mesotransients emerge over a very narrow range of dispersal parameter values and are rare under all circumstances. Thus, transient dynamics are not likely to be either common or widespread in simple models of ecological systems.     

A probabilistic framework for nutrient uptake length

The nutrient uptake length, the average displacement of a nutrient in a stream before being taken up by the biota, is an important quantity to characterize and compare streams and rivers, or to quantify certain aspects of their related ecosystems. This concept has been widely used for almost 30 years now, and uptake lengths have been estimated for several nutrients in many systems, but it also suffers from a number of limitations, one of them being the requirement of a spatially homogeneous stream or river. We combine recently advocated, transport-based models of stream processes with current concepts of dispersal theory into a novel framework for nutrient uptake length. The framework is based on the theory for dispersal kernels in terrestrial systems, where the entire distribution of dispersal distances is calculated and not only the average. Within this framework, we can re-derive all previous results and formulae for uptake length, and we can include spatially heterogeneous stream environments. In addition, we propose a number of new characteristic quantities that can complement nutrient uptake length when evaluating the health of a stream system or the impact of a source of nutrients. We illustrate our method with two examples of spatially non-homogeneous streams: point-source input of nutrients (e.g., wastewater treatment plant) and diffuse lateral input (e.g., agricultural run-off), and we show how to measure the relative contribution of the two sources to the uptake length and other characteristics.     

From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence–absence data to recover information on the population dynamics of predator–prey systems. We use a highly complex and spatially explicit simulation model of a predator–prey mite system to generate simple presence–absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the "footprints" of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator's functional response and the dispersal rates of both species) are reflected by different footprints
The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.     

Integrating Urbanization into Landscape-level Ecological Assessments

Economists and ecologists are often asked to collaborate on landscape-level analyses designed to jointly assess economic and ecological conditions resulting from environmental policy scenarios. This trend toward multidisciplinary projects, coupled with the growing use of geographic information systems, has led to the development of spatially explicit models that can be used to examine and project land-use change. Although spatial land-use models are still evolving, most published efforts have modeled the conversion of nonurban land to urban uses as a function of explanatory variables based on population density and the spatial proximity of land to roads, markets, and population centers. In this paper, we use a gravity model to describe the urbanization potential of forest and agricultural land as a combination of population and proximity. We develop an empirical model that describes the probability that forests and agricultural land in western Oregon and western Washington were transformed to residential, commercial, or industrial uses over a 30-year period as a function of urbanization potential, other socioeconomic factors, and geographic and physical land characteristics. Land-use data were provided by the USDA Forest Service's Forest Inventory and Analysis program. We use this empirical model to generate geographic information system maps depicting the probability of future land-use change that can be integrated with landscape-level ecological models developed for western Oregon's Coast Range. Received 14 April 2000; accepted 17 August 2000.     

Generalized theorems for nonlinear state space reconstruction

Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved ad hoc by including multiple dynamically coupled time series in the reconstructions, to provide a more mechanistic model. Here we provide three analytical proofs that add to the growing literature to generalize Takens' work and that demonstrate how multiple time series can be used in attractor reconstructions. These expanded results (Takens' theorem is a special case) apply to a wide variety of natural systems having parallel time series observations for variables believed to be related to the same dynamic manifold. The potential information leverage provided by multiple embeddings created from different combinations of variables (and their lags) can pave the way for new applied techniques to exploit the time-limited, but parallel observations of natural systems, such as coupled ecological systems, geophysical systems, and financial systems. This paper aims to justify and help open this potential growth area for SSR applications in the natural sciences.     

The Use of Electron Tomography for Structural Analysis of Disordered Protein Arrays

A method based on Fourier transforms is described for obtaining a 3-D reconstruction from a paracrystalline object with static disorder. The method is derived from the standard methods used in 3-D reconstruction of 2-D crystals except that all of the Fourier coefficients are used and not just the sampled data from the periodic lattice. Thus, not only is the spatially ordered part of the structure visualized in 3-D, but also the spatially disordered part. Application of the method to 3-D reconstructions of insect flight muscle is described as well as prospects for extension of the method to radiation-sensitive specimens.     

Parallel selective pressures drive convergent diversification of phenotypes in pythons and boas

Pythons and boas are globally distributed and distantly related radiations with remarkable phenotypic and ecological diversity. We tested whether pythons, boas and their relatives have evolved convergent phenotypes when they display similar ecology. We collected geometric morphometric data on head shape for 1073 specimens representing over 80% of species. We show that these two groups display strong and widespread convergence when they occupy equivalent ecological niches and that the history of phenotypic evolution strongly matches the history of ecological diversification, suggesting that both processes are strongly coupled. These results are consistent with replicated adaptive radiation in both groups. We argue that strong selective pressures related to habitat‐use have driven this convergence. Pythons and boas provide a new model system for the study of macro‐evolutionary patterns of morphological and ecological evolution and they do so at a deeper level of divergence and global scale than any well‐established adaptive radiation model systems.     

Models of central pattern generators for quadruped locomotion I. Primary gaits

In this paper we continue the analysis of a network of symmetrically coupled cells modeling central pattern generators for quadruped locomotion proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean a system of ordinary differential equations and by a coupled cell system we mean a network of identical cells with coupling terms. We have three main results in this paper. First, we show that the proposed network is the simplest one modeling the common quadruped gaits of walk, trot, and pace. In doing so we prove a general theorem classifying spatio-temporal symmetries of periodic solutions to equivariant systems of differential equations. We also specialize this theorem to coupled cell systems. Second, this paper focuses on primary gaits; that is, gaits that are modeled by output signals from the central pattern generator where each cell emits the same waveform along with exact phase shifts between cells. Our previous work showed that the network is capable of producing six primary gaits. Here, we show that under mild assumptions on the cells and the coupling of the network, primary gaits can be produced from Hopf bifurcation by varying only coupling strengths of the network. Third, we discuss the stability of primary gaits and exhibit these solutions by performing numerical simulations using the dimensionless Morris-Lecar equations for the cell dynamics.     

Modelling food web complexity: The consequences of individual-based, spatially explicit behavioural ecology on trophic interactions

We present a prototype simulator that enables one to explore the influence of individual behaviour on the dynamics and structural complexity of food webs. In the simulations, individuals act according to simple, biologically plausible rules in a spatially explicit setting. We present the results of a series of simulation experiments on artificial, tri-trophic level food chains used to calibrate the simulator against real-world systems and to demonstrate the simulators promise for ecological modelling. Our primary objective was to discover the biological features leading to stability of artificial food chains over ecological time and under different conditions of trophic efficiency. This involved a qualitative analysis of food chains comprised of a plant, a herbivore and a carnivore species. We explored the consequences of allowing individual heterotrophs to make active choices about resource selection (perception and intentional behaviour) under high and low degrees of trophic efficiency. We found that individuals had to adopt realistic behavioural ecological strategies, such as active resource selection, for systems to persist, especially under conditions in which trophic efficiencies were of the magnitude observed in real systems (e.g. 10%). Our results reaffirm previous convictions that a better understanding of food web interactions in real-world systems will require approaches that blend animal behavioural ecology with population and community ecology. However, the evidence comes from a new mathematical perspective.     

Attractivity of coherent manifolds in metapopulation models

The likelihood that coupled dynamical systems will completely synchronize, or become “coherent”, is often of great applied interest. Previous work has established conditions for local stability of coherent solutions and global attractivity of coherent manifolds in a variety of spatially explicit models. We consider models of communities coupled by dispersal and explore intermediate regimes in which it can be shown that states in phase space regions of positive measure are attracted to coherent solutions. Our methods yield rigorous and practically useful coherence criteria that facilitate useful analyses of ecological and epidemiological problems.     

Spatial correlation as leading indicator of catastrophic shifts

Generic early-warning signals such as increased autocorrelation and variance have been demonstrated in time-series of systems with alternative stable states approaching a critical transition. However, lag times for the detection of such leading indicators are typically long. Here, we show that increased spatial correlation may serve as a more powerful early-warning signal in systems consisting of many coupled units. We first show why from the universal phenomenon of critical slowing down, spatial correlation should be expected to increase in the vicinity of bifurcations. Subsequently, we explore the applicability of this idea in spatially explicit ecosystem models that can have alternative attractors. The analysis reveals that as a control parameter slowly pushes the system towards the threshold, spatial correlation between neighboring cells tends to increase well before the transition. We show that such increase in spatial correlation represents a better early-warning signal than indicators derived from time-series provided that there is sufficient spatial heterogeneity and connectivity in the system.     

Pruning the tree of life: k-core percolation as selection mechanism

We propose a model for evolution aiming to reproduce statistical features of fossil data, in particular the distributions of extinction events, the distribution of species per genus and the distribution of lifetimes, all of which are known to be of power-law type. The model incorporates both species-species interactions and ancestral relationships. The main novelty of this work is to show the feasibility of k-core percolation as a selection mechanism. We give theoretical predictions for the observable distributions, confirm their validity by computer simulation and compare with fossil data. A key feature of the proposed model is a co-evolving fitness landscape determined by the topology of the underlying species interactions, ecological niches emerge naturally. The predicted distributions are independent of the rate of speciation, i.e. whether one adopts an gradualist or punctuated view of evolution.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.