Theoretical models of species' borders: single species approaches

The range of potential mechanisms limiting species' distributions in space is nearly as varied and complex as the diversity of life itself. Yet viewed abstractly, a species' border is a geographic manifestation of a species' demographic responses to a spatially and temporally varying world. Population dynamic models provide insight into the different routes by which range limits can arise owing to gradients in demographic rates. In a metapopulation context, for example, range limits may be caused by gradients in extinction rates, colonization rates or habitat availability. We have consider invasion models in uniform and heterogeneous environments as a framework for understanding non-equilibrium range limits, and explore conditions under which invasions may cease to spread leaving behind a stationary range limit. We conclude that non-equilibrial range dynamics need further theoretical and empirical attention.     

The community context of species' borders: ecological and evolutionary perspectives

Species distributional limits may coincide with hard dispersal barriers or physiological thresholds along environmental gradients, but they may also be influenced by species interactions. We explore a number of models of interspecific interactions that lead to (sometimes abrupt) distribution limits in the presence and absence of environmental gradients. We find that gradients in competitive ability can lead to spatial segregation of competitors into distinct ranges, but that spatial movement tends to broaden the region of sympatry between the two species, and that Allee effects tend to sharpen these boundaries. We generalize these simple models to include metapopulation dynamics and other types of interactions including predator–prey and host–parasite interactions. We derive conditions for range limits in each case. We also consider models that include coevolution and gene flow and find that character displacement along environmental gradients can lead to stable parapatric distributions. We conclude that it is essential to consider coevolved species interactions as a potential mechanism limiting species distributions, particularly when barriers to dispersal are weak and environmental gradients are gradual.     

Allee effects, invasion pinning, and species' borders

All species' ranges are the result of successful past invasions. Thus, models of species' invasions and their failure can provide insight into the formation of a species' geographic range. Here, we study the properties of invasion models when a species cannot persist below a critical population density known as an "Allee threshold." In both spatially continuous reaction-diffusion models and spatially discrete coupled ordinary-differential-equation models, the Allee effect can cause an invasion to fail. In patchy landscapes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of parameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally unstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss their implications for spatially complex dynamics, including critical phenomena, in ecological landscapes. Our results suggest caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed: under a wide range of plausible ecological conditions, species' ranges may be limited by an Allee effect. Several example systems appear to fit our general model.     

Competition and species packing in patchy environments

In models of competition in which space is treated as a continuum, and population size as continuous, there are no limits to the number of species that can coexist. For a finite number of sites, N, the results are different. The answer will, of course, depend on the model used to ask the question. In the Tilman-May-Nowak ordinary differential equation model, the number of species is asymptotically C log N with most species packed in at the upper end of the competitive hierarchy. In contrast, for metapopulation models with discrete individuals and stochastic spatial systems with various competition neighborhoods, we find a traditional species area relationship CN(a), with no species clumping along the phenotypic gradient. The exponent a is larger by a factor of 2 for spatially explicit models. In words, a spatial distribution of competitors allows for greater diversity than a metapopulation model due to the effects of recruitment limitation in their competition.     

Species borders: ecological and evolutionary perspectives

Recent ecological studies on species borders have used a number of approaches to establish causation for specific environmental factors and to identify the traits involved. These include interspecific comparisons, detailed investigations of marginal populations, and experimental manipulation. Species borders continue to be largely ignored in evolutionary biology, although some work suggests that marginal populations may often be relatively better-adapted to unfavourable conditions but perform poorly under most other conditions.     

The effect of branching angle on adaptive growth in patchy environments

In clonal plants, the genetic individual (genet) develops via the production of multiple physiological individuals (ramets). The branching angle between the ramets can significantly influence the shape of the developing clone. We investigated the adaptive significance of this phenomenon by means of a spatially explicit dynamic model of clonal growth. We studied the effect of the branching angle on the efficiency of filling habitat patches, varying the sizes and shapes of the patches. Two growth forms were compared: the Narrow Range (NR) versus Wide Range (WR) form. In the NR plant, the branching angle was always acute, while in the WR plant, both acute and wide angles could occur. We hypothesized that the NR plant would be less successful, as narrower branching constrained the plant’s ability to turn. The simulations revealed an opposite trend: the NR plant occupied more space in most of the simulated habitats. However, the effect was weak in general. We conclude that the branching angle between ramets is likely to be a neutral trait in terms of natural selection.     

Patterns of parasitism by insect parasitoids in patchy environments

Abstract. 1. This paper shows how the different spatial patterns of per cent parasitism in patches of different host density can be explained within a single model framework that takes into account the parasitoid's aggregative response, and the factors limiting the degree of host exploitation within patches.
2. Two contrasting laboratory examples are presented in which the distribution of searching parasitoids and the resulting levels of parasitism in different patches are both known for a range of parasitoid densities.
3. A model is described predicting the number of hosts parasitized per patch, in which the number of parasitoids searching is determined from a simple expression allowing different degrees of aggregation.
4. The model generates patterns of parasitism encompassing the two laboratory examples and a wide range of examples from the field.
5. The importance of density dependent spatial distributions of parasitism to population stability is briefly discussed.     

Searching efficiency of the lady beetleCoccinella septempunctata larvae in uniform and patchy environments

Experiments were conducted on the searching behavior and searching efficiency of the lady beetleCoccinella septempunctata bruckii Mulsant under conditions of various prey distributions and prey densities. The larvae changed their searching behavior before and after feeding. Before feeding the larvae moved quickly and the searching paths were nearly linear. But after feeding the speed decreased and turning angle increased. The speed and turning angle reverted gradually and recovered the initial pattern 95 s after feeding. The searching efficiency differed depending on the prey distribution. At low prey density, searching was most efficient when prey were distributed uniformly. But at middle and high prey densities, searching was most efficient when prey items were highly aggregated. The observed searching behavior of 4th instarC. septempunctata larvae was likely to be optimal considering the natural distribution of colonies of their prey, aphids.     

Evolutionary stability of ideal free dispersal strategies in patchy environments

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244-251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt's (Am Nat 140:1010-1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

Populations in variable environments: the effect of variability in a species' primary resource

Mechanistic models for herbivore populations responding to rainfall-driven pasture are used to explore the effect of temporal variability in a primary resource on the abundance and distribution of a species. If the numerical response of the herbivore to pasture is a convex function, then gains made over time intervals with above average rainfall do not compensate for losses incurred when rainfall is below average. Populations therefore fare worse when rainfall is variable compared with when rainfall is reliable. It is demonstrated that this result is independent of the distribution of rainfall. Sensitivity of a species to variability, and hence the limit to its distribution in variable environments, is directly proportional to the difference between population growth rate under ideal conditions and the estimated rate of decline as the species' resource tends to zero. When density dependence is included in the numerical response, the average abundance of a species declines with increasing variability in its primary resource. However, a model for the dynamics of pasture and rabbits (Oryctolagus cuniculus) and red foxes (Vulpes vulpes) in southern Australia, is used to illustrate that trophic interactions can reverse the effect of variability: in the absence of foxes, the mean abundance of rabbits declines with variability as expected, but in the full model the mean abundance of rabbits increases.     

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.     

Environmental variability uncovers disruptive effects of species' interactions on population dynamics

How species respond to changes in environmental variability has been shown for single species, but the question remains whether these results are transferable to species when incorporated in ecological communities. Here, we address this issue by analysing the same species exposed to a range of environmental variabilities when (i) isolated or (ii) embedded in a food web. We find that all species in food webs exposed to temporally uncorrelated environments (white noise) show the same type of dynamics as isolated species, whereas species in food webs exposed to positively autocorrelated environments (red noise) can respond completely differently compared with isolated species. This is owing to species following their equilibrium densities in a positively autocorrelated environment that in turn enables species–species interactions to come into play. Our results give new insights into species'' response to environmental variation. They especially highlight the importance of considering both species'' interactions and environmental autocorrelation when studying population dynamics in a fluctuating environment.     

Transmission dynamics for vector-borne diseases in a patchy environment

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross–MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number \(\mathcal{R }_0\) is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of \(\mathcal{R }_0\) . It is shown that the disease free equilibrium is asymptotically stable if \(\mathcal{R }_0<1\) , and it is unstable if \(\mathcal{R }_0>1\) ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for \(\mathcal{R }_0\) are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for \(\mathcal{R }_0\) in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, \(\mathcal{R }_0\) is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.     

The shape of the spatial kernel and its implications for biological invasions in patchy environments

Ecological and epidemiological invasions occur in a spatial context. We investigated how these processes correlate to the distance dependence of spread or dispersal between spatial entities such as habitat patches or epidemiological units. Distance dependence is described by a spatial kernel, characterized by its shape (kurtosis) and width (variance). We also developed a novel method to analyse and generate point-pattern landscapes based on spectral representation. This involves two measures: continuity, which is related to autocorrelation and contrast, which refers to variation in patch density. We also analysed some empirical data where our results are expected to have implications, namely distributions of trees (Quercus and Ulmus) and farms in Sweden. Through a simulation study, we found that kernel shape was not important for predicting the invasion speed in randomly distributed patches. However, the shape may be essential when the distribution of patches deviates from randomness, particularly when the contrast is high. We conclude that the speed of invasions depends on the spatial context and the effect of the spatial kernel is intertwined with the spatial structure. This implies substantial demands on the empirical data, because it requires knowledge of shape and width of the spatial kernel, and spatial structure.     

Search mechanism of a stream grazer in patchy environments: the role of food abundance

Summary The search behavior of the grazing stream insect Baetis tricaudatus (Ephemeroptera: Baetidae) was examined in field and laboratory experiments. Regardless of food abundance in experimental habitats, nymphs spent significantly more time in food patches than predicted if they had moved randomly with respect to patches. A significant reduction in movement rate within patches relative to movement rate between patches largely accounted for these results. The movement pattern within patches was highly systematic and in agreement with predictions of optimal foraging theory since food was uniformly distributed within patches. Between-patch search movements were affected by food abundance in the most recently grazed patch. Search intensity after departure from a patch was positively related to food abundance in the patch while movement rate after patch departure was inversely related to patch food level. These effects produced between-patch movement patterns that were suboptimal in the experimental habitats because they resulted in revisitation of previously depleted patches. However, differences between experimental and natural habitats in the spatial occurrence of patch types suggest that Baetis between-patch search behavior may be adaptive in natural habitats.     

Within-population variation in localized and integrated responses of Trifolium repens to biotically patchy environments

Summary Genets of Trifolium repens (white clover) were collected from three patches of old permanent pasture dominated by Agrostis capillaris, Holcus lanatus or Lolium perenne. Plants derived from the genets were grown with plants of one grass species present on one side of each T. repens, and a different grass species on the other side, in all combinations of two of the three grasses. Different modules (a node with its associated internode, leaf, and axillary bud) on the same clover plant responded independently to the microenvironment provided by their own neighbouring grasses. In contrast, all apical meristems on the plant reacted similarly, showing a unified response and integrating the effects of the different microenvironments experienced by the whole clover plant. This is consistent with what is known both physiologically about the nutrition of meristems and modules, and ecologically about the exploratory growth habit of the species. Averaged over all associated grasses, there was no significant variation in the final dry weight of the different clover genets but these differed in their growth habit response to different grasses. In response to Agrostis as a neighbour, each meristem of T. repens rapidly produced many small modules. New modules were produced more slowly and were larger when Holcus or Lolium was the neighbour. The same pattern of differences occurred among clovers sampled from different backgrounds. Either genetic differences paralleled plastic responses, or plastic changes in phenotype that developed in response to different neighbours in the field persisted in the greenhouse. Plants taken from backgrounds of different grass species showed different responses to growing with those grass species. The differences were manifest primarily in a positive leading diagonal effect of Holcus or not-Holcus. They were the result primarily of differences in the dry weight per module and the probability of development of the axillary bud into a branch. This confirms earlier results, and implicates the central importance of branching as a means of local response to the microenvironment.     

Orientation and polarity in collectively migrating cell structures: statics and dynamics

Collective cell migration is often characterized by the spontaneous onset of multicellular protrusions (known as fingers) led by a single leader cell. Working with epithelial Madin-Darby canine kidney monolayers we show that cells within the fingers, as compared with the epithelium, are well oriented and polarized along the main finger direction, which suggests that these cells actively migrate. The cell orientation and polarity decrease continuously from the tip toward the epithelium over a penetration distance of typically two finger lengths. Furthermore, laser photoablation experiments at various locations along these fingers demonstrate that the cells in the fingers are submitted to a tensile stress whose value is larger close to the tip. From a dynamical point of view, cells entering a finger gradually polarize on timescales that depend upon their particular initial position. Selective laser nanosurgery of the leader lamellipodium shows not only that these structures need a leader to progress, but that this leader itself is the consequence of a prior self-organization of the cells forming the finger. These results highlight the complex interplay between the collective orientation within the fingers and the mechanical action of the leader.