Climate change fluctuations can increase population abundance and range size.

Climate change threatens many species by a poleward/upward movement of their thermal niche. While we know that faster movement has stronger impacts, little is known on how fluctuations of niche movement affect population outcomes. Environmental fluctuations often affect populations negatively, but theory and experiments have revealed some positive effects. We study how fluctuations around the average speed of the niche impact a species' persistence, abundance and realized niche width under climate change. We find that the outcome depends on how fluctuations manifest and what the relative time scale of population growth and climate fluctuations are. When populations are close to extinction with the average speed, fluctuations around this average accelerate population decline. However, populations not yet close to extinction can increase in abundance and/or realized niche width from such fluctuations. Long-lived species increase more when their niche size remains constant, short-lived species increase more when their niche size varies.

Pushed to the edge: Spatial sorting can slow down invasions.

Our ability to understand population spread dynamics is complicated by rapid evolution, which renders simple ecological models insufficient. If dispersal ability evolves, more highly dispersive individuals may arrive at the population edge than less dispersive individuals (spatial sorting), accelerating spread. If individuals at the low-density population edge benefit (escape competition), high dispersers have a selective advantage (spatial selection). These two processes are often described as forming a positive feedback loop; they reinforce each other, leading to faster spread. Although spatial sorting is close to universal, this form of spatial selection is not: low densities can be detrimental for organisms with Allee effects. Here, we present two conceptual models to explore the feedback loops that form between spatial sorting and spatial selection. We show that the presence of an Allee effect can reverse the positive feedback loop between spatial sorting and spatial selection, creating a negative feedback loop that slows population spread.

Integrodifference models for evolutionary processes in biological invasions.

Individual variability in dispersal and reproduction abilities can lead to evolutionary processes that may have significant effects on the speed and shape of biological invasions. Spatial sorting, an evolutionary process through which individuals with the highest dispersal ability tend to agglomerate at the leading edge of an invasion front, and spatial selection, spatially heterogeneous forces of selection, are among the fundamental evolutionary forces that can change range expansions. Most mathematical models for these processes are based on reaction-diffusion equations, i.e., time is continuous and dispersal is Gaussian. We develop novel theory for how evolution shapes biological invasions with integrodifference equations, i.e., time is discrete and dispersal can follow a variety of kernels. Our model tracks how the distribution of growth rates and dispersal ability in the population changes from one generation to the next in continuous space. We include mutation between types and a potential trade-off between dispersal ability and growth rate. We perform the analysis of such models in continuous and discrete trait spaces, i.e., we determine the existence of travelling wave solutions, asymptotic spreading speeds and their linear determinacy, as well as the population distributions at the leading edge. We also establish the relation between asymptotic spreading speeds and mutation probabilities. We observe conditions for when spatial sorting emerges and when it does not and also explore conditions where anomalous spreading speeds occur, as well as possible effects of deleterious mutations in the population.

A seasonal hybrid model for the evolution of flowering onset.

In strongly seasonal environments, many plants tend to divide the favorable season into an earlier part, where they allocate resources to vegetative growth, and a later part, where they allocate resources to reproduction. The onset of flowering typically indicates the shift from one to the other. We derive and analyze a model for the evolution of flowering onset on the phenotypic level. Our model tracks a continuous phenotype distribution through the various seasons from year to year. We analyze a special case of a monomorphic population with the tools of adaptive dynamics. We analyze the general case by a moment approximation. We find that (the mean of) flowering onset converges to some intermediate time within the favorable season. In the monomorphic case, we prove that this is an ESS. The moment approach reveals that there are different time scales involved on which the plant density, the mean flowering onset, and its variance converge.

Towards Building a Sustainable Future: Positioning Ecological Modelling for Impact in Ecosystems Management.

As many ecosystems worldwide are in peril, efforts to manage them sustainably require scientific advice. While numerous researchers around the world use a great variety of models to understand ecological dynamics and their responses to disturbances, only a small fraction of these models are ever used to inform ecosystem management. There seems to be a perception that ecological models are not useful for management, even though mathematical models are indispensable in many other fields. We were curious about this mismatch, its roots, and potential ways to overcome it. We searched the literature on recommendations and best practices for how to make ecological models useful to the management of ecosystems and we searched for 'success stories' from the past. We selected and examined several cases where models were instrumental in ecosystem management. We documented their success and asked whether and to what extent they followed recommended best practices. We found that there is not a unique way to conduct a research project that is useful in management decisions. While research is more likely to have impact when conducted with many stakeholders involved and specific to a situation for which data are available, there are great examples of small groups or individuals conducting highly influential research even in the absence of detailed data. We put the question of modelling for ecosystem management into a socio-economic and national context and give our perspectives on how the discipline could move forward.

Competitive coexistence of seasonal breeders.

Many species are annual breeders who, between reproductive events, consume resources and may die. Their resource often reproduces continuously or has short, overlapping generations. An accurate model for such life cycles needs to represent both, the discrete- and the continuous-time processes in the community. The dynamics of a single discrete breeder and its resource can differ significantly from that of a fully continuous consumer-resource community (e.g., Lotka-Volterra) and that of a fully discrete one (e.g., Nicholson-Bailey). We study the dynamics of multiple discrete breeders on a single resource and identify a number of coexistence mechanisms and complex dynamics. The resource grows logistically, resource consumption is linear and consumer reproduction can be linear or nonlinear. We derive explicit conditions for the positive equilibrium state to exist and for mutual invasion to occur at that equilibrium. Stable equilibrium coexistence of more than one consumer is possible only when reproduction is nonlinear. Higher resource growth rate generally allows more consumers to stably coexist. Our explicit formulas allow us to generate communities of many coexisting consumers. Total biomass in the system seems to increase with the number of coexisting consumers. Complex patterns of coexistence arise, including bistability of equilibrium and non-equilibrium coexistence. The mixed continuous-discrete modeling approach can easily be adapted to study how certain aspects of global change affect discrete breeder communities.

Reactivity of communities at equilibrium and periodic orbits.

Reactivity measures the transient response of a system following a perturbation from a stable state. For steady states, the theory of reactivity is well developed and frequently applied. However, we find that reactivity depends critically on the scaling used in the equations. We therefore caution that calculations of reactivity from nondimensionalized models may be misleading. The attempt to extend reactivity theory to stable periodic orbits is very recent. We study reactivity of periodically forced and intrinsically generated periodic orbits. For periodically forced systems, we contribute a number of observations and examples that had previously received less attention. In particular, we systematically explore how reactivity depends on the timing of the perturbation. We then suggest ways to extend the theory to intrinsically generated periodic orbits. We investigate several possible global measures of reactivity of a periodic orbit and show that there likely is no single quantity to consistently measure the transient response of a system near a periodic orbit.

Pushing the Boundaries: Models for the Spatial Spread of Ecosystem Engineers.

Ecosystems engineers are species that can substantially alter their abiotic environment and thereby enhance their population growth. The net growth rate of obligate engineers is even negative unless they modify the environment. We derive and analyze a model for the spread and invasion of such species. Prior to engineering, the landscape consists of unsuitable habitat; after engineering, the habitat is suitable. The boundary between the two types of habitat is moved by the species through their engineering activity. Our model is a novel type of a reaction-diffusion free boundary problem. We prove the existence of traveling waves and give upper and lower bounds for their speeds. We illustrate how the speed depends on individual movement and engineering behavior near the boundary.

Evolutionarily stable movement strategies in reaction-diffusion models with edge behavior.

Many types of organisms disperse through heterogeneous environments as part of their life histories. For various models of dispersal, including reaction-advection-diffusion models in continuously varying environments, it has been shown by pairwise invasibility analysis that dispersal strategies which generate an ideal free distribution are evolutionarily steady strategies (ESS, also known as evolutionarily stable strategies) and are neighborhood invader strategies (NIS). That is, populations using such strategies can both invade and resist invasion by populations using strategies that do not produce an ideal free distribution. (The ideal free distribution arises from the assumption that organisms inhabiting heterogeneous environments should move to maximize their fitness, which allows a mathematical characterization in terms of fitness equalization.) Classical reaction diffusion models assume that landscapes vary continuously. Landscape ecologists consider landscapes as mosaics of patches where individuals can make movement decisions at sharp interfaces between patches of different quality. We use a recent formulation of reaction-diffusion systems in patchy landscapes to study dispersal strategies by using methods inspired by evolutionary game theory and adaptive dynamics. Specifically, we use a version of pairwise invasibility analysis to show that in patchy environments, the behavioral strategy for movement at boundaries between different patch types that generates an ideal free distribution is both globally evolutionarily steady (ESS) and is a global neighborhood invader strategy (NIS).

How Spatial Heterogeneity Affects Transient Behavior in Reaction-Diffusion Systems for Ecological Interactions?

Most studies of ecological interactions study asymptotic behavior, such as steady states and limit cycles. The transient behavior, i.e., qualitative aspects of solutions as and before they approach their asymptotic state, may differ significantly from asymptotic behavior. Understanding transient dynamics is crucial to predicting ecosystem responses to perturbations on short timescales. Several quantities have been proposed to measure transient dynamics in systems of ordinary differential equations. Here, we generalize these measures to reaction-diffusion systems in a rigorous way and prove various relations between the non-spatial and spatial effects, as well as an upper bound for transients. This extension of existing theory is crucial for studying how spatially heterogeneous perturbations and the movement of biological species involved affect transient behaviors. We illustrate several such effects with numerical simulations.

The Effect of Movement Behavior on Population Density in Patchy Landscapes.

Many biological populations reside in increasingly fragmented landscapes, where habitat quality may change abruptly in space. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. Several recent publications explore how such individual movement behavior affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We prove existence, uniqueness and global stability, and we classify their qualitative shape depending on movement behavior. We apply our results to study the question why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape.

Turing patterns in a predator-prey model with seasonality.

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction-diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the 'summer' season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington-DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).

Persistence and spread of stage-structured populations in heterogeneous landscapes.

Conditions for population persistence in heterogeneous landscapes and formulas for population spread rates are important tools for conservation ecology and invasion biology. To date, these tools have been developed for unstructured populations, yet many, if not all, species show two or more distinct phases in their life cycle. We formulate and analyze a stage-structured model for a population in a heterogeneous habitat. We divide the population into pre-reproductive and reproductive stages. We consider an environment consisting of two types of patches, one where population growth is positive, one where it is negative. Individuals move randomly within patches but can show preference towards one patch type at the interface between patches. We use linear stability analysis to determine persistence conditions, and we derive a dispersion relation to find spatial spread rates. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest.

Individual behavior at habitat edges may help populations persist in moving habitats.

Moving-habitat models aim to characterize conditions for population persistence under climate-change scenarios. Existing models do not incorporate individual-level movement behavior near habitat edges. These small-scale details have recently been shown to be crucially important for large-scale predictions of population spread and persistence in patchy landscapes. In this work, we extend previous moving-habitat models by including individual movement behavior. Our analysis shows that populations might be able to persist under faster climate change than previous models predicted. We also find that movement behavior at the trailing edge of the climatic niche is much more important for population persistence than at the leading edge.

Invasion pinning in a periodically fragmented habitat.

Biological invasions can cause great damage to existing ecosystems around the world. Most landscapes in which such invasions occur are heterogeneous. To evaluate possible management options, we need to understand the interplay between local growth conditions and individual movement behaviour. In this paper, we present a geometric approach to studying pinning or blocking of a bistable travelling wave, using ideas from the theory of symmetric dynamical systems. These ideas are exploited to make quantitative predictions about how spatial heterogeneities in dispersal and/or reproduction rates contribute to halting biological invasion fronts in reaction-diffusion models with an Allee effect. Our theoretical predictions are confirmed using numerical simulations, and their ecological implications are discussed.

Behavioral responses to resource heterogeneity can accelerate biological invasions.

The abundance and spatial distribution of resources in a landscape and the behavioral response of individuals determines whether and how fast an invasive species spreads in an environment. Whether and how landscape manipulations can be used to slow invasive species is of great interest, in particular in forest ecosystems, where tree removal, thinning, and increasing tree diversity are discussed as management options. Classically, the focus is on availability and accessibility of resources; more recent considerations include individual-level behavioral movement responses to a spatially heterogeneous resource distribution. We derive a novel model for insect-host dynamics that includes three common behavioral aspects of foraging: higher movement rate in resource-poor areas, lower ovipositioning rate in resource-poor areas, and movement preference for resource-rich areas. We show that each of these basic mechanisms can increase the speed of invasion in a source-sink landscape above that in a homogeneous landscape with larger overall resource availability. We parameterize our model and illustrate our results with data for emerald ash borer, a recent highly destructive forest pest in North America. Our results highlight the importance of empirical work on movement behavior in different landscape types and near the interface between types.

Meandering Rivers: How Important is Lateral Variability for Species Persistence?

Models for population dynamics in rivers and streams have highlighted the importance of spatial and temporal variations for population persistence. We present a novel model that considers the longitudinal variation as introduced by the sinuosity of a meandering river where a main channel is laterally extended to point bars in bends. These regions offer different habitat conditions for aquatic populations and therefore may enhance population persistence. Our model is a nonstandard reaction-advection-diffusion model where the domain of definition consists of the real line (representing the main channel) with periodically added intervals (representing the point bars). We give an existence and uniqueness proof for solutions of the equations. We then study population persistence as the (in-) stability of the trivial solution and population spread as the minimal wave speed of traveling periodic waves. We conduct a sensitivity analysis to highlight the importance of each parameter on the model outcome. We find that sinuosity can enhance species persistence.

Seasonally Varying Predation Behavior and Climate Shifts Are Predicted to Affect Predator-Prey Cycles.

The functional response of some predator species changes from a pattern characteristic for a generalist to that for a specialist according to seasonally varying prey availability. Current theory does not address the dynamic consequences of this phenomenon. Since season length correlates strongly with altitude and latitude and is predicted to change under future climate scenarios, including this phenomenon in theoretical models seems essential for correct prediction of future ecosystem dynamics. We develop and analyze a two-season model for the great horned owl (Bubo virginialis) and snowshoe hare (Lepus americanus). These species form a predator-prey system in which the generalist to specialist shift in predation pattern has been documented empirically. We study the qualitative behavior of this predator-prey model community as summer season length changes. We find that relatively small changes in summer season length can have a profound impact on the system. In particular, when the predator has sufficient alternative resources available during the summer season, it can drive the prey to extinction, there can be coexisting stable states, and there can be stable large-amplitude limit cycles coexisting with a stable steady state. Our results illustrate that the impacts of global change on local ecosystems can be driven by internal system dynamics and can potentially have catastrophic consequences.

Analysis of spread and persistence for stream insects with winged adult stages.

Species such as stoneflies have complex life history details, with larval stages in the river flow and adult winged stages on or near the river bank. Winged adults often bias their dispersal in the upstream direction, and this bias provides a possible mechanism for population persistence in the face of unidirectional river flow. We use an impulsive reaction-diffusion equation with non-local impulse to describe the population dynamics of a stream-dwelling organism with a winged adult stage, such as stoneflies. We analyze this model from a variety of perspectives so as to understand the effect of upstream dispersal on population persistence. On the infinite domain we use the perspective of weak versus local persistence, and connect the concept of local persistence to positive up and downstream spreading speeds. These spreading speeds, in turn are connected to minimum travelling wave speeds for the linearized operator in upstream and downstream directions. We show that the conditions for weak and local persistence differ, and describe how weak persistence can give rise to a population whose numbers are growing but is being washed out because it cannot maintain a toe hold at any given location. On finite domains, we employ the concept of a critical domain size and dispersal success approximation to determine the ultimate fate of the populations. A simple, explicit formula for a special case allows us to quantify exactly the difference between weak and local persistence.