Integrodifference equations, Allee effects, and invasions

 Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. We study an invasion model based on an integrodifference equation with an Allee effect. We derive a general result for the sign of the speed of invasion. We then examine a special, linear–constant, Allee function and introduce a numerical scheme that allows us to estimate the speed of traveling wave solutions. Received: 14 April 2000 / Revised version: 23 December 2000 / Published online: 8 February 2002     

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest-neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations.     

Population spread in patchy landscapes under a strong Allee effect

Many species of invasive insects establish and spread in regions around the world, causing enormous economical and environmental damage, in particular in forests. Some of these insects are subject to an Allee effect whereby the population must surpass a certain threshold in order to establish. Recent studies have examined the possibility of exploiting an Allee effect to improve existing control strategies. Forests and most other ecosystems show natural spatial variation, and human activities frequently increase the degree of spatial heterogeneity. It is therefore imperative to understand how the interplay between this spatial variation and individual movement behavior affects the overall speed of spread of an invasion. To this end, we study an integrodifference equation model in a patchy landscape and with Allee growth dynamics. Movement behavior of individuals varies according to landscape quality. Our study focuses on how the speed of the resulting traveling periodic wave depends on the interaction between landscape fragmentation, patch-dependent dispersal, and Allee population dynamics.     

Woodland birds in patchy landscapes: the evidence base for strategic networks

Habitat creation and management within wooded networks is a potentially effective strategy to reduce ecological isolation and the deleterious effects of fragmentation. However, questions remain over the relative advantages of different approaches, e.g. buffering patches vs. increasing connectivity. Potential effects of woodland fragmentation include reduction in regional woodland cover, reduced patch size, edge effects with loss of core habitat, and increased isolation with disruption of dispersal and metapopulation dynamics. We adopt an evidence-based approach to review how each of these affects woodland birds with an emphasis on studies from the UK and use this to identify management priorities for mitigation. There is evidence for both patch area and composition effects: larger woodlands support more woodland bird species, and woods located within sparsely wooded landscapes are less valuable to specialist woodland species. Bird assemblages show a nested pattern with respect to area, and thus species found in small woods also occur in large woods but not vice versa. However, small woods may be preferred by a few edge species, while small woods also have greater variability in bird species composition. Consideration of the metapopulation dynamics of specialist species with poor dispersal shows that creating or buffering large woodlands is more efficient than a greater total area of small fragments. Connectivity appears most useful for widespread generalist species with almost continuous populations. Woodland structure and quality are of overwhelming importance: as well as mature woodland, young growth, scrub and edges are also key components. There is an urgent need to examine the relationship between nest predation and landscape structure within UK woodlands.     

Preference in patchy landscapes: the influence of scale-specific intake rates and variance in reward

Understanding the responses of foragers to patchy distributionsof resources has formed a fundamental challenge in behavioralecology. Two currencies have been used to assess the patch preferencesof herbivores—intake rate maximization and risk sensitivity.We wished to understand if small mammalian foragers, collaredlemmings (Dichrostonyx groenlandicus), choose patches to maximizefood intake rate or to reduce risk of starvation in "variable"environments. Moreover, we examined the possibility that maximizingintake rate depends on the spatial scale of patchiness. We designedan experiment offering two alternative patches of food, varyingthe predictability of food rewards and the "potential intakerate" at different spatial scales. Collared lemmings did notconsistently select patches that maximized their intake rateat either scale studied. Instead, they chose patches offeringthe least variation in food reward over the course of the experiment.Collared lemmings used prior knowledge gained from previousforaging bouts to assess food variability. We interpret theseresults as evidence for risk-averse foraging strategies, whichare predicted for continuous foragers aiming to minimize riskof starvation.     

Edge effects in patchy seagrass landscapes: The role of predation in determining fish distribution

Predation is often described as an underlying mechanism to explain edge effects. We assessed the importance of predation in determining edge effects in seagrass using two approaches: a video survey to sample predators at small scales across seagrass edges, and a tethering experiment to determine if predation was an underlying mechanism causing edge effects. Underwater videos were placed at four positions: middle of seagrass patches; edge of seagrass; sand immediately adjacent to seagrass and sand distant from seagrass. Fish abundances and the time fish spent in view were measured. The main predatory fish (Australian salmon, Arripis spp.) spent more time over adjacent sand than other positions, while potential prey species (King George whiting, Sillaginodes punctata (Cuvier), recruits) were more common in the middle of seagrass patches. Other species, including the smooth toadfish, Tetractenos glaber (Freminville), and King George whiting adults, spent more time over sand adjacent to seagrass than distant sand, which may be related to feeding opportunities. King George whiting recruits and pipefish (Stigmatopora spp.) were tethered at each of the four positions. More whiting recruits were preyed upon at outer than inner seagrass patches, and survival time was greater in the middle of shallow seagrass patches than other positions. Relatively few pipefish were preyed upon, but of those that were, survival time was lower over sand adjacent to seagrass than at the seagrass edge or middle. Video footage revealed that salmon were the dominant predators of both tethered King George whiting recruits and pipefish. The distribution of predators and associated rate of predation can explain edge effects for some species (King George whiting) but other mechanisms, or combinations of mechanisms, are determining edge effects for other species (pipefish).     

Distribution and coexistence of shrews in patchy landscapes: A field test of multiple hypotheses

Despite the important role of shrews (Soricomorpha: Soricidae) in the functioning of ecosystems, as predators and prey, the effects of habitat loss and fragmentation on this guild of mammals are still unclear. We studied the distribution of 5 species (the greater white toothed shrew Crocidura leucodon; the lesser white toothed shrew Crocidura suaveolens; the pigmy shrew Sorex minutus; the Appennine shrew Sorex samniticus and the Etruscan shrew Suncus etruscus) in a fragmented landscape in central Italy.Shrews were trapped with pitfall traps made from plastic water bottles, the number of traps increased with patch size. A total of 170 individuals, of 5 species of shrews were captured. Shrews were widely distributed in our study area, however patch occupancy was determined mainly by vegetation and geometrical characteristics of the patches. Our data supports the hypotheses that patterns of habitat selection and the dynamics of seasonal abundance (habitat and temporal partitioning between similarly sized species) reduce competitive pressure, thus allowing coexistence of shrews in relatively species-rich assemblages, for such small amounts of habitat. The most important outcome of our results is the crucial role played by vegetation structure in determining distribution patterns. These results strongly suggest that measurements of the vegetation structure of habitat patches should always be included as explanatory variables when studying the distribution of shrews in fragmented landscapes.     

Integrodifference models for persistence in fragmented habitats

Integrodifference models of growth and dispersal are analyzed on finite domains to investigate the effects of emigration, local growth dynamics and habitat heterogeneity on population persistence. We derive the bifurcation structure for a range of population dynamics and present an approximation that allows straighforward calculation of the equilibrium populations in terms of local growth dynamics and dispersal success rates. We show how population persistence in a heterogeneous environment depends on the scale of the heterogeneity relative to the organism's characteristic dispersal distance. When organisms tend to disperse only a short distance, population persistence is dominated by local conditions in high quality patches, but when dispersal distance is relatively large, poor quality habitat exerts a greater influence.     

Integrodifference equations in the presence of climate change: persistence criterion,travelling waves and inside dynamics

To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. In the case of persistence, we separate the travelling wave and general solutions into spatially distinct neutral fractions to study the inside dynamics. It is shown that each neutral genetic fraction rearranges itself spatially so as to asymptotically achieve the profile of the travelling wave. To measure the genetic diversity of the population density we calculate the Shannon diversity index and related indices, and use these to illustrate how diversity changes with underlying parameters.

     

The evolution of an 'intelligent' dispersal strategy: biased, correlated random walks in patchy landscapes

Theoretical work exploring dispersal evolution focuses on the emigration rate of individuals and typically assumes that movement occurs either at random to any other patch or to one of the nearest‐neighbour patches. There is a lack of work exploring the process by which individuals move between patches, and how this process evolves. This is of concern because any organism that can exert control over dispersal direction can potentially evolve efficiencies in locating patches, and the process by which individuals find new patches will potentially have major effects on metapopulation dynamics and gene flow. Here, we take an initial step towards filling this knowledge gap. To do this we constructed a continuous space population model, in which individuals each carry heritable trait values that specify the characteristics of the biased correlated random walk they use to disperse from their natal patch. We explore how the evolution of the random walk depends upon the cost of dispersal, the density of patches in the landscape, and the emigration rate. The clearest result is that highly correlated walks always evolved (individuals tended to disperse in relatively straight lines from their natal patch), reflecting the efficiency of straight‐line movement. In our models, more costly dispersal resulted in walks with higher correlation between successive steps. However, the exact walk that evolved also depended upon the density of suitable habitat patches, with low density habitat evolving more biased walks (individuals which orient towards suitable habitat at quite large distances from that habitat). Thus, low density habitat will tend to develop individuals which disperse efficiently between adjacent habitat patches but which only rarely disperse to more distant patches; a result that has clear implications for metapopulation theory. Hence, an understanding of the movement behaviour of dispersing individuals is critical for robust long‐term predictions of population dynamics in fragmented landscapes.     

Integrodifference model for blowfly invasion

We propose a stage-structured integrodifference model for blowflies’ growth and dispersion taking into account the density dependence of fertility and survival rates and the non-overlap of generations. We assume a discrete-time, stage-structured, model. The spatial dynamics is introduced by means of a redistribution kernel. We treat one and two dimensional cases, the latter on the semi-plane, with a reflexive boundary. We analytically show that the upper bound for the invasion front speed is the same as in the one-dimensional case. Using laboratory data for fertility and survival parameters and dispersal data of a single generation from a capture-recapture experiment in South Africa, we obtain an estimate for the velocity of invasion of blowflies of the species Chrysomya albiceps. This model predicts a speed of invasion which was compared to actual observational data for the invasion of the focal species in the Neotropics. Good agreement was found between model and observations.     

Spontaneous Patchiness in a Host-Parasitoid Integrodifference Model

Observations of a host-parasitoid interaction in which victims are significantly less motile than their exploiters suggest the possibility of stable spatial pattern in a fairly homogeneous environment. Findings of pattern formation in continuous-time models are not fully able to account for this behavior. Those findings often rely on questionable biological conditions, and more fundamentally, the continuous nature of time in such models does not reflect the reality of the observed interaction. In this paper, we introduce a discrete-time spatial model of the interaction. The final state of our model is often a striking spatial pattern, similar to those observed. We analyze the model, describe its transient behavior, and find the conditions under which these spatial patterns occur, as well as an estimate of maximum possible patch size under those conditions. We also discuss the existence of such conditions in the natural system.     

Dispersal and life history strategies in epiphyte metacommunities: alternative solutions to survival in patchy,dynamic landscapes

Host trees for obligate epiphytes are dynamic patches that emerge, grow and fall, and metacommunity diversity critically depends on efficient dispersal. Even though species that disperse by large asexual diaspores are strongly dispersal limited, asexual dispersal is common. The stronger dispersal limitation of asexually reproducing species compared to species reproducing sexually via small spores may be compensated by higher growth rates, lower sensitivity to habitat conditions, higher competitive ability or younger reproductive age. We compared growth and reproduction of different groups of epiphytic bryophytes with contrasting dispersal (asexual vs. sexual) and life history strategies (colonists, short- and long-lived shuttle species, perennial stayers) in an old-growth forest stand in the boreo-nemoral region in eastern Sweden. No differences were seen in relative growth rates between asexual and sexual species. Long-lived shuttles had lower growth rates than colonists and perennial stayers. Most groups grew best at intermediate bark pH. Interactions with other epiphytes had a small, often positive effect on growth. Neither differences in sensitivity of growth to habitat conditions nor differences in competitive abilities among species groups were found. Habitat conditions, however, influenced the production of sporophytes, but not of asexual diaspores. Presence of sporophytes negatively affected growth, whereas presence of asexual diaspores did not. Sexual species had to reach a certain colony size before starting to reproduce, whereas no such threshold existed for asexual reproduction. The results indicate that the epiphyte metacommunity is structured by two main trade-offs: dispersal distance vs. reproductive age, and dispersal distance vs. sensitivity to habitat quality. There seems to be a trade-off between growth and sexual reproduction, but not asexual. Trade-offs in species traits may be shaped by conflicting selection pressures imposed by habitat turnover and connectivity rather than by species interactions. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Saddle-Point Approximations,Integrodifference Equations,and Invasions

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.     

Estimating species' absence, colonization and local extinction in patchy landscapes: an application of occupancy models with rodents

Making an inference on the absence of a species in a site is often problematic, due to detection probability being, in most cases, <1. Inference is more complicated if detection probability, together with distribution patterns, vary during the year, since the possibility of inferring a species absence, at reasonable costs, may be possible only in certain periods. Our aim here is to show how such challenging situations can be by tackled by applying some recently developed occupancy models combined with sample size (number of repeated surveys) estimation. We thus analysed the distribution of two rodents Myodes glareolus and Mus musculus domesticus in a fragmented landscape in central Italy pointing out how it is possible to identify true absences, non-detections, extinctions/colonizations and determine seasonal values of detection probability.     

The arrangement of resources in patchy landscapes: effects on distribution, survival, and resource acquisition of chironomids

The spatial arrangement of resources in patchy habitats influences the distribution of individuals and their ability to acquire resources. We used Chironomus riparius, a ubiquitous aquatic insect that uses leaf particles as an important resource, to ask how the dispersion of resource patches influences the distribution and resource acquisition of mobile individuals in patchy landscapes. Two experiments were conducted in replicated laboratory landscapes (38×38 cm) created by arranging sand and leaf patches in a 5×5 grid so that the leaf patches were either aggregated or uniformly dispersed in the grid. One-day-old C. riparius larvae were introduced into the landscapes in one of three densities (low, medium, high). In experiment 1, we sampled larvae and pupae by coring each patch in each landscape 3, 6, 12, or 24 days after adding larvae. In experiment 2, emerging adults were collected daily for 42 days from each patch in each landscape. In aggregated landscapes, individuals were aggregated in one patch type or the other during a particular developmental stage, but the ”preferred” type changed depending on developmental stage and initial density. Adult emergence was lower by about 30% in all aggregated landscapes. In dispersed landscapes, individuals used both types of patch throughout their life cycles at all initial densities. Thus, patch arrangement influences the distribution of mobile individuals in landscapes, and it influences resource acquisition even when average resource abundance is identical among landscapes. Regardless of patch arrangement, high initial density caused accumulation of early instars in edge patches, 75% mortality of early instars, a 25% increase in development time, and a 60% reduction in adult emergence. Because mortality was extremely high among early-instar larvae in high-density treatments, we do not have direct evidence that the mechanism by which patch arrangement operates is density dependent. However, the results of our experiments strongly suggest that dispersion of resource patches across a landscape reduces local densities by making non-resource patches available for use, thereby reducing intraspecific competition. Received: 20 July 1999 / Accepted: 28 January 2000