Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities.

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.     

Alternative stable states in inherently unstable systems

Alternative stable states are nontransitory states within which communities can exist. However, even highly dynamic communities can be viewed within the framework of stable‐state theory if an appropriate “ecologically relevant” time scale is identified. The ecologically relevant time scale for dynamic systems needs to conform to the amount of time needed for a system's community to complete an entire cycle through its normal range of variation. For some systems, the ecologically relevant period can be relatively short (eg, tidal systems), for others it can be decadal (eg, prairie wetlands). We explore the concept of alternative stable states in unstable systems using the highly dynamic wetland ecosystems of North America's Prairie Pothole Region. The communities in these wetland ecosystems transition through multiple states in response to decadal‐long climate oscillations that cyclically influence ponded‐water depth, permanence, and chemistry. The perspective gained by considering dynamic systems in the context of stable‐state theory allows for an increased understanding of how these systems respond to changing drivers that can push them past tipping points into alternative states. Incorporation of concepts inherent to stable‐state theory has been suggested as a key scientific element upon which to base sustainable environmental management.     

Tracking unstable states: ecosystem dynamics in a changing world

Ecological systems can show complex and sometimes abrupt responses to environmental change, with important implications for their resilience. Theories of alternate stable states have been used to predict regime shifts of ecosystems as equilibrium responses to sufficiently slow environmental change. The actual rate of environmental change is a key factor affecting the response, yet we are still lacking a non-equilibrium theory that explicitly considers the influence of this rate of environmental change. We present a metacommunity model of predator–prey interactions displaying multiple stable states, and we impose an explicit rate of environmental change in habitat quality (carrying capacity) and connectivity (dispersal rate). We study how regime shifts depend on the rate of environmental change and compare the outcome with a stability analysis in the corresponding constant environment. Our results reveal that in a changing environment, the community can track states that are unstable in the constant environment. This tracking can lead to regime shifts, including local extinctions, that are not predicted by alternative stable state theory. In our metacommunity, tracking unstable states also controls the maintenance of spatial heterogeneity and spatial synchrony. Tracking unstable states can also lead to regime shifts that may be reversible or irreversible. Our study extends current regime shift theories to integrate rate-dependent responses to environmental change. It reveals the key role of unstable states for predicting transient dynamics and long-term resilience of ecological systems to climate change.     

Dispersal and noise: Various modes of synchrony in ecological oscillators

We use the theory of noise-induced phase synchronization to analyze the effects of dispersal on the synchronization of a pair of predator–prey systems within a fluctuating environment (Moran effect). Assuming that each isolated local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive a Fokker–Planck equation describing the evolution of the probability density for pairwise phase differences between the oscillators. In the case of common environmental noise, the oscillators ultimately synchronize. However the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of partially correlated noise with dispersal can lead to a multistable steady-state probability density.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

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.     

Levels of multiallelic overdominance fitness, heterozygote excess and heterozygote deficiency

Concepts and results on selection balance in multiallelic systems are described. These include a multidimensional concept of heterozygote excess and heterozygote deficiency, a hierarchy of means of assessment of heterozygote advantage, comparisons and contrasts of allelic versus gametic polymorphic states, and conditions defining stable equilibria of complementary gametic sets. The concepts are illustrated in the context of viability selection and behavioral models of kin selection and for two major categories of multilocus selection regimes.     

Evolution of local facilitation in arid ecosystems

In harsh environments, sessile organisms can make their habitat more hospitable by buffering environmental stress or increasing resource availability. Although the ecological significance of such local facilitation is widely established, the evolutionary aspects have been seldom investigated. Yet addressing the evolutionary aspects of local facilitation is important because theoretical studies show that systems with such positive interactions can exhibit alternative stable states and that such systems may suddenly become extinct when they evolve (evolutionary suicide). Arid ecosystems currently experience strong changes in climate and human pressures, but little is known about the effects of these changes on the selective pressures exerted on the vegetation. Here, we focus on the evolution of local facilitation in arid ecosystems, using a lattice-structured model explicitly considering local interactions among plants. We found that the evolution of local facilitation depends on the seed dispersal strategy. In systems characterized by short-distance seed dispersal, adaptation to a more stressful environment leads to high local facilitation, allowing the population to escape extinction. In contrast, systems characterized by long-distance seed dispersal become extinct under increased stress even when allowed to adapt. In this case, adaptation in response to climate change and human pressures could give the final push to the desertification of arid ecosystems.     

Synchronization in ecological systems by weak dispersal coupling with time delay

One of the most salient spatiotemporal patterns in population ecology is the synchronization of fluctuating local populations across vast spatial extent. Synchronization of abundance has been widely observed across a range of spatial scales in relation to the rate of dispersal among discrete populations. However, the dependence of synchrony on patterns of among-patch movement across heterogeneous landscapes has been largely ignored. Here, we consider the duration of movement between two predator–prey communities connected by weak dispersal and its effect on population synchrony. More specifically, we introduce time-delayed dispersal to incorporate the finite transmission time between discrete populations across a continuous landscape. Reducing the system to a phase model using weakly connected network theory, it is found that the time delay is an important factor determining the nature and stability of phase-locked states. Our analysis predicts enhanced convergence to stable synchronous fluctuations in general and a decreased ability of systems to produce in-phase synchronization dynamics in the presence of delayed dispersal. These results introduce delayed dispersal as a tool for understanding the importance of dispersal time across a landscape matrix in affecting metacommunity dynamics. They further highlight the importance of landscape and dispersal patterns for predicting the onset of synchrony between weakly coupled populations.     

How parasite-mediated costs drive the evolution of disease state-dependent dispersal

The process of dispersal is central to population biology and evolutionary ecology. Because of negative impacts on host fitness, parasite infection generates potential costs of dispersal. However, theoretical predictions that address this issue are lacking. Here, we develop a mathematical model to demonstrate how the dispersal rate of hosts evolves under the influence of parasites in ecological scenarios incorporating pre-, during-, and post-dispersal infection/recovery events. We show that (1) the dispersal tendency is strongly biased towards either infected individuals or susceptible individuals, (2) the bias is inherently determined by the parasite-mediated relative cost of dispersal, and (3) the dispersal costs are determined by the autocorrelation of disease states (susceptible and infected) between pre- and post-dispersal. Our results suggest that parasite virulence in concert with the timing of infection drive the evolution of disease state-biased dispersal. To understand the evolutionary processes in spatial host–parasite systems, the parasite-induced costs of dispersal need to be taken into account.     

Stability threshold of a tissue system

Studies of changes in the cells adhesive force during spontaneous and induced cancerogenesis, as well as under the effect of cancerogenesis promotors permits to reveal the cells adhesion forces critical values which separate tissue states stable and unstable to blastomogenesis. These data are analysed proceeding from the concepts on generalized cooperative transitions in the membranes and their role in the control over biological systems. It is taken into account that the membranes and cytoskeleton elements are the most important systems of the tissue mechanical integration, and phase transitions proceeding in them must be reflected in the tissue mechanical properties.     

Detection of alternative stable states in marine communities

During the last 20 years there have been several attempts to test the theory of alternative stable states using marine systems. The results have been mixed, and there have been conflicting interpretations of the outcomes. In an attempt to resolve some of the problems, the theory and evidence for alternative stable states are reviewed. There are several different views of what is meant by alternative stable states, and so the key elements of the theory of alternative stable states are set in the context of marine systems. Appropriate experimental designs for detecting alternative states are discussed, and recent experimental studies in coral reefs, rocky intertidal systems, soft-sediment beds, and subtidal systems are briefly reviewed. Suggestions for improving experimental designs are proposed and unresolved issues are highlighted.     

Panbiogeography and the public

Abstract

Progress is reported on the systematics and biogeography of some small land snails in Australasia. Observed patterns of dispersal and massings of taxa appear to support basic concepts in panbiogeographic methodology. The role of the amateur in science is related to the use of computers, curatorial systems, and the current political environment for science.     

Evolution of dispersal in spatially and temporally variable environments: The importance of life cycles

Spatiotemporal variability of the environment is bound to affect the evolution of dispersal, and yet model predictions strongly differ on this particular effect. Recent studies on the evolution of local adaptation have shown that the life cycle chosen to model the selective effects of spatiotemporal variability of the environment is a critical factor determining evolutionary outcomes. Here, we investigate the effect of the order of events in the life cycle on the evolution of unconditional dispersal in a spatially heterogeneous, temporally varying landscape. Our results show that the occurrence of intermediate singular strategies and disruptive selection are conditioned by the temporal autocorrelation of the environment and by the life cycle. Life cycles with dispersal of adults versus dispersal of juveniles, local versus global density regulation, give radically different evolutionary outcomes that include selection for total philopatry, evolutionary bistability, selection for intermediate stable states, and evolutionary branching points. Our results highlight the importance of accounting for life‐cycle specifics when predicting the effects of the environment on evolutionarily selected trait values, such as dispersal, as well as the need to check the robustness of model conclusions against modifications of the life cycle.     

Sociogenetic structure, kin associations and bonding in delphinids

Social systems are the outcomes of natural and sexual selection on individuals' efforts to maximize reproductive success. Ecological conditions, life history, demography traits and social aspects have been recognized as important factors shaping social systems. Delphinids show a wide range of social structures and large variation in life history traits and inhabit several aquatic environments. They are therefore an excellent group in which to investigate the interplay of ecological and intrinsic factors on the evolution of mammalian social systems in these environments. Here I synthetize results from genetic studies on dispersal patterns, genetic relatedness, kin associations and mating patterns and combine with ecological, life history and phylogenetic data to predict the formation of kin associations and bonding in these animals. I show that environment type impacts upon dispersal tendencies, with small delphinids generally exhibiting female-biased philopatry in inshore waters and bisexual dispersal in coastal and pelagic waters. When female philopatry occurs, they develop moderate social bonds with related females. Male bonding occurs in species with small male-biased sexual size dimorphism and male-biased operational sex ratio, and it is independent of dispersal tendencies. By contrast, large delphinids, which live in coastal and pelagic waters, show bisexual philopatry and live in matrilineal societies. I propose that sexual conflict favoured the formation of these stable societies and in turn facilitated the development of kin-biased behaviours. Studies on populations of the same species inhabiting disparate environments, and of less related species living in similar habitats, would contribute towards a comprehensive framework for the evolution of delphinid social systems.     

Dispersal, disease and life-history evolution

Discrete-time susceptible-infective-susceptible (S-I-S) disease transmission models can exhibit bistability (alternative stable equilibria) over a wide range of parameter values. We illustrate the richness generated by such 'simple' non-linear systems in the study of two patch epidemic models with disease-enhanced or disease-suppressed dispersal. Dispersal between patches can have a profound impact on local patch disease dynamics. In fact, dispersal between patches may give rise to bistability in parameter regimes without bistability in single patch models.     

Molecular self-organization and multiple equilibrium systems

We have compared different types of natural self-organizing systems: crystal-like formation of multimolecular systems were conferred with self-organization in active media and dissipative structure formation. The comparison revealed a common feature of all such systems. They all have bi-(multi-)stable states. We propose a hypothesis that self-organization is impossible in systems that could not have bi-(multi-)stable states.     

A Mathematical Biologist’s Guide to Absolute and Convective Instability

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator–prey systems.     

Multiple Stable States of Disease Occurrence: A Note on the Implications for the Anthropological Study of Human Disease

Ecologists model human disease using predator-prey models, with humans as the prey and the pathogen as the predator. Use of these models permits the identification of stable equilibria and stable states of disease prevalence for human host-parasite systems. May (1977) has described the possibility of the presence of multiple stable states of disease prevalence for human disease systems. The model of multiple stable states implies that historical circumstances are important in determining the occurrence of disease. This model offers a theoretical framework in which to examine the occurrence of epidemics in human populations and to develop a more complete understanding of disease processes.