Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species

Patch occupancy theory predicts that a trade-off between competition and dispersal should lead to regional coexistence of competing species. Empirical investigations, however, find local coexistence of superior and inferior competitors, an outcome that cannot be explained within the patch occupancy framework because of the decoupling of local and spatial dynamics. We develop two-patch metapopulation models that explicitly consider the interaction between competition and dispersal. We show that a dispersal-competition trade-off can lead to local coexistence provided the inferior competitor is superior at colonizing empty patches as well as immigrating among occupied patches. Immigration from patches that the superior competitor cannot colonize rescues the inferior competitor from extinction in patches that both species colonize. Too much immigration, however, can be detrimental to coexistence. When competitive asymmetry between species is high, local coexistence is possible only if the dispersal rate of the inferior competitor occurs below a critical threshold. If competing species have comparable colonization abilities and the environment is otherwise spatially homogeneous, a superior ability to immigrate among occupied patches cannot prevent exclusion of the inferior competitor. If, however, biotic or abiotic factors create spatial heterogeneity in competitive rankings across the landscape, local coexistence can occur even in the absence of a dispersal-competition trade-off. In fact, coexistence requires that the dispersal rate of the overall inferior competitor not exceed a critical threshold. Explicit consideration of how dispersal modifies local competitive interactions shifts the focus from the patch occupancy approach with its emphasis on extinction-colonization dynamics to the realm of source-sink dynamics. The key to coexistence in this framework is spatial variance in fitness. Unlike in the patch occupancy framework, high rates of dispersal can undermine coexistence, and hence diversity, by reducing spatial variance in fitness.     

Competition and coexistence in regional habitats

Habitat heterogeneity plays a key role in the dynamics and structures of communities. In this article, a two-species metapopulation model that includes local competitive dynamics is analyzed to study the population dynamics of two competing species in spatially structured habitats. When local stochastic extinction can be ignored, there are, as in Lotka-Volterra equations, four outcomes of interspecific competition in this model. The outcomes of competition depend on the competitive intensity between the competing pairs. An inferior competitor and a superior competitor, or two strongly competing species, can never stably coexist, whereas two weak competitors (even if they are very similar species) may coexist over the long term in such environments. Local stochastic extinction may greatly affect the outcomes of interspecific competition. Two competing species can or cannot stably coexist depending not only on the competitive intensity between the competing pairs but also on their precompetitive distributions. Two weak competitors that have similar precompetitive distributions can always regionally coexist. Two strongly competing species that competitively exclude each other in more stable habitats may be able to stably coexist in highly heterogenous environments if they have similar precompetitive distributions. There is also a chance for an inferior competitor to coexist regionally or even to exclude a superior competitor when the superior competitor has a narrow precompetitive distribution and the inferior competitor has a wide precompetitive distribution.     

Competition and species coexistence in a metapopulation model: Can fast asymmetric migration reverse the outcome of competition in a homogeneous environment?

We investigate whether asymmetric fast migration can modify the predictions of classical competition theory and, in particular revert species dominance. We consider a model of two species competing for an implicit resource on a habitat divided into two patches. Both patches are connected through constant migration rates and in each patch local dynamics are driven by a Lotka-Volterra competition system.Local competition is asymmetric with the same superior competitor in both patches. Migration is asymmetric, species dependent and fast in comparison to local competitive interactions. The species and patches are taken to be otherwise similar: in both patches we assume the same carrying capacities for both species, and the same growth rates and pair-wise competition coefficients for each species.We show that global dynamics can be described by a classical Lotka-Volterra competition model. We found that by modifying the ratio of intraspecific migration rates for both species all possible combinations of global species relative dominance can be achieved. We find specific conditions for which the local superior competitor is globally excluded. This is to our knowledge the first study showing that fast asymmetric migration can lead to inferior competitor dominance in a homogeneous environment. We conclude that disparity of temporal scales between migration and local dynamics may have important consequences for the maintenance of biodiversity in spatially structured populations.     

Movement toward better environments and the evolution of rapid diffusion

We study a reaction-diffusion-advection model for two ecologically equivalent competitors with different dispersal strategies inhabiting a spatially heterogeneous environment. The competitors represent different phenotypes of the same species. One is assumed to disperse by simple diffusion, the other by diffusion together with directed movement toward more favorable environments. We show that under suitable conditions on the underlying spatial domain, the competitor that moves toward more favorable environments may have a competitive advantage even if it diffuses more rapidly than the other competitor. This is in contrast with the case in which both competitors disperse by pure diffusion, where the competitor that diffuses more slowly always has the advantage. We determine competitive advantage by examining the invasibility, i.e. stability or instability, of steady states with only one competitor present. The mathematical approach is a perturbation analysis of principal eigenvalues.     

Coexistence of competitors in metacommunities due to spatial variation in resource growth rates; does R * predict the outcome of competition?

Simple mathematical models are used to investigate the coexistence of two consumers using a single limiting resource that is distributed over distinct patches, and that has unequal growth rates in the different patches. Relatively low movement rates or high demographic rates of an inefficient resource exploiter allow it to coexist at a stable equilibrium with a more efficient species whose ratio of movement to demographic rates is lower. The range of conditions allowing coexistence depends on the between‐patch heterogeneity in resource growth rates, but this range can be quite broad. The between‐patch movement of the more efficient consumer turns patches with high resource growth rates into sources, while low‐growth‐rate patches effectively become sinks. A less efficient species can coexist with or even exclude the more efficient species from the global environment if it is better able to bias its spatial distribution towards the source patches. This can be accomplished with density independent dispersal if the less efficient species has a lower ratio of per capita between‐patch movement rate to demographic rates. Conditions that maximize the range of efficiencies allowing coexistence of two species are: a relatively high level of heterogeneity in resource growth conditions; high dispersal (or low demographic rates) of the superior competitor; and low dispersal (or high demographic rates) of the inferior competitor. Global exclusion of the more efficient competitor requires that the inferior competitor have sufficient movement to also produce a source‐sink environment.     

A toad more traveled: the heterogeneous invasion dynamics of cane toads in Australia

To predict the spread of invasive species, we need to understand the mechanisms that underlie their range expansion. Assuming random diffusion through homogeneous environments, invasions are expected to progress at a constant rate. However, environmental heterogeneity is expected to alter diffusion rates, especially by slowing invasions as populations encounter suboptimal environmental conditions. Here, we examine how environmental and landscape factors affect the local invasion speeds of cane toads (Chaunus [Bufo] marinus) in Australia. Using high-resolution cane toad data, we demonstrate heterogeneous regional invasion dynamics that include both decelerating and accelerating range expansions. Toad invasion speed increased in regions characterized by high temperatures, heterogeneous topography, low elevations, dense road networks, and high patch connectivity. Regional increases in the toad invasion rate might be caused by environmental conditions that facilitate toad reproduction and movement, by the evolution of long-distance dispersal ability, or by some combination of these factors. In any case, theoretical predictions that neglect environmental influences on dispersal at multiple spatial scales may prove to be inaccurate. Early predictions of cane toad range expansion rates that assumed constant diffusion across homogeneous landscapes already have been proved wrong. Future attempts to predict range dynamics for invasive species should consider heterogeneity in (1) the environmental factors that determine dispersal rates and (2) the mobility of invasive populations because dispersal-relevant traits can evolve in exotic habitats. As an invasive species spreads, it is likely to encounter conditions that influence dispersal rates via one or both of these mechanisms.     

Spatial patterns of coexistence of competing species in patchy habitat

The single-species spatially realistic patch occupancy metapopulation model is, in this study, extended to a metacommunity of many competing species. Competition is assumed to reduce the local carrying capacity (effective patch area), which in turn increases local extinction rates and reduces colonization rates because of smaller population sizes. Each species is described by three parameters: pre-competitive abundance (equilibrium incidence of patch occupancy, which reflects the rate of colonization in relation to extinction rate), the spatial range of migration, and competitive ability. The model ignores spatio–temporal correlations caused by interspecific interactions, because in metacommunities of unequal competitors inhabiting heterogeneous landscapes, correlations in the occurrence of species are driven more by patch heterogeneity than by competition. The model allows the calculation of multispecies equilibria in patchy habitats without simulations. In general, the number of coexisting species in the metacommunity increases with decreasing strength of competition, increasing rate of colonization, and decreasing range of migration. Habitat heterogeneity in the form of spatial variation in patch areas tends to facilitate coexistence. Poor competitors may coexist with superior competitors in the patch network if the former have higher colonization rates (competition–colonization trade-off). When migration distances are short, competition leads to spatial pattern formation: Species tend to have restricted spatial distributions in the network, but contrary to intuitive expectations, often the distributions of many species are nested. Having more dispersive species enhances both local and global diversity, whereas more local migration decreases local but increases global diversity.     

Morphological response to competition for light in the clonal Trifolium repens (Fabaceae)

? Premise of the study: Plant communities in temperate zones are dominated by clonal plants that can plastically modify their growth characteristics in response to competition. Given that plants compete with one another, and the implications this has for species coexistence, we conducted a study to assess how clonal species morphologically respond to competition for light depending on its intensity and heterogeneity, which are determined by the competitor species. ? Methods: We assessed the morphological response to competition for light of the clonal species Trifolium repens L. by measuring its growth performance, and vertical and horizontal growth traits. We used five competitive environments, i.e., one without competitor and four differing by their competitor species creating different conditions of competition intensity and heterogeneity. ? Key results: The morphological response of Trifolium repens to competition for light depended on the competitor identity. Competition intensity and heterogeneity, determined by competitor identity, had an interactive effect on most traits. The increase in petiole elongation and specific leaf area due to increased competition intensity was observed only at low to intermediate competition heterogeneity. Competition heterogeneity promoted the elongation of clone connections allowing space exploration. ? Conclusions: Our results demonstrated that the intensity and heterogeneity of competition, which depended on competitor identity, are of primary importance in determining the plastic response of Trifolium repens. This emphasizes that it is important to consider the fine-scale spatial distribution of individuals when studying their interactions within plant communities.     

Diffusion-mediated persistence in three-species competition models with heteroclinic cycles.

We consider a model composed of two patches. One patch has three competing species forming a heteroclinic cycle within the path. The other is a refuge for one of the three species, which can diffuse between the two patches. The remaining two competitors are confined to the competitive patch and cannot diffuse. A new heteroclinic cycle can exist in the model, and the underlying cycle in the competitive patch cannot appear with a positive diffusion rate. It is proved that the model can be made persistent under appropriate diffusion conditions even if the underlying heteroclinic cycle is an attractor in the competitive patch and the patch is not persistent without the refuge. Further it is shown that the model with a specific structure is globally stable if the underlying cycle is a repeller.     

Alternative stable states and regional community structure

Many models of local species interactions predict the occurrence of priority effects due to alternative stable equilibria (ASE). However, few empirical examples of ASE have been shown. One possible explanation for the disparity is that local ASE are difficult to maintain regionally in patch dynamic models. Here we examine two possible mechanisms for regional coexistence of species engaged in local ASE. Biotically generated heterogeneity (e.g., habitat modification that favors further invasion by conspecifics) results in regional exclusion of one species at equilibrium. In contrast, abiotic heterogeneity due to spatial variation in resource supply ratios generates local-scale ASE and ensures regional coexistence with sufficiently broad environmental gradients. Abiotic heterogeneity can result in a species that is the dominant competitor over some of its range being excluded if the area where it is dominant is too small. Biotic heterogeneity can lead to alternative stable landscapes or regional priority effects, while abiotic heterogeneity results in regional determinism. Broad environmental gradients in resource supply favor regional coexistence of species that exhibit local ASE.     

Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes

Mounting theoretical evidence suggests that demographic stochasticity, environmental heterogeneity and biased movement of organisms individually affect the dynamics of biological invasions and range expansions. Studies of species spread in heterogeneous landscapes have traditionally characterized invasion velocities as functions of the mean resource density throughout the landscape, thus neglecting higher‐order moments of the spatial resource distribution. Here, we show theoretically that different spatial arrangements of resources lead to different spread velocities even if the mean resource density throughout the landscape is kept constant. Specifically, we find that increasing the resource autocorrelation length causes a reduction in the speed of species spread. The model shows that demographic stochasticity plays a key role in the slowdown, which is strengthened when individuals can actively move towards resources. We then experimentally corroborated the theoretically predicted reduction in propagation speed in microcosm experiments with the protist Euglena gracilis by comparing spread in landscapes with different resource autocorrelation lengths. Our work identifies the resource autocorrelation length as a key modulator and a simple measure of landscape susceptibility to biological invasions, which needs to be considered for predicting invasion dynamics within naturally heterogeneous environmental corridors.     

Predator's invasion into an isolated patch with spatially heterogeneous prey distribution

The invasion success of a diffusing predator which changes its diffusion coefficient depending on whether the prey exists or not is investigated. The prey is assumed to be immobile and distributed in an isolated patch. The isolated patch consists of two kinds of region: prey-existing zone and prey-vacant zone. We discuss what relation a heterogeneity of prey distribution has with the predator's invasion success into the patch. Its spatial heterogeneity appears to affect significantly the predator's invasion. In an Appendix we briefly treat an analogous problem involving two competing species.     

Comparable ecological dynamics underlie early cancer invasion and species dispersal, involving self-organizing processes

Occupancy of new habitats through dispersion is a central process in nature. In particular, long-distance dispersal is involved in the spread of species and epidemics, although it has not been previously related with cancer invasion, a process that involves cell spreading to tissues far away from the primary tumour.Using simulations and real data we show that the early spread of cancer cells is similar to the species individuals spread and we suggest that both processes are represented by a common spatio-temporal signature of long-distance dispersal and subsequent local proliferation. This signature is characterized by a particular fractal geometry of the boundaries of patches generated, and a power-law scaled, disrupted patch size distribution. In contrast, invasions involving only dispersal but not subsequent proliferation (“physiological invasions”) like trophoblast cells invasion during normal human placentation did not show the patch size power-law pattern. Our results are consistent under different temporal and spatial scales, and under different resolution levels of analysis.We conclude that the scaling properties are a hallmark and a direct result of long-distance dispersal and proliferation, and that they could reflect homologous ecological processes of population self-organization during cancer and species spread. Our results are significant for the detection of processes involving long-range dispersal and proliferation like cancer local invasion and metastasis, biological invasions and epidemics, and for the formulation of new cancer therapeutical approaches.     

Body size mediated coexistence of consumers competing for resources in space

Body size is a major phenotypic trait of individuals that commonly differentiates co-occurring species. We analyzed inter-specific competitive interactions between a large consumer and smaller competitors, whose energetics, selection and giving-up behaviour on identical resource patches scaled with individual body size. The aim was to investigate whether pure metabolic constraints on patch behaviour of vagile species can determine coexistence conditions consistent with existing theoretical and experimental evidence. We used an individual-based spatially explicit simulation model at a spatial scale defined by the home range of the large consumer, which was assumed to be parthenogenic and semelparous. Under exploitative conditions, competitive coexistence occurred in a range of body size ratios between 2 and 10. Asymmetrical competition and the mechanism underlying asymmetry, determined by the scaling of energetics and patch behaviour with consumer body size, were the proximate determinant of inter-specific coexistence. The small consumer exploited patches more efficiently, but searched for profitable patches less effectively than the larger competitor. Therefore, body-size related constraints induced niche partitioning, allowing competitive coexistence within a set of conditions where the large consumer maintained control over the small consumer and resource dynamics. The model summarises and extends the existing evidence of species coexistence on a limiting resource, and provides a mechanistic explanation for decoding the size-abundance distribution patterns commonly observed at guild and community levels.     

Evaluating methods to quantify spatial variation in the velocity of biological invasions

Invading species rarely spread homogeneously through a landscape and invasion patterns typically display irregular frontal boundaries as the invasion progresses through space. Those irregular patterns are generally produced by local environmental factors that may slow or accelerate movement of the frontal boundary. While there is an abundant literature on species distribution modelling methods that quantify local suitability for species establishment, comparatively few studies have examined methods for measuring the local velocity of invasions that can then be statistically analysed in relation to spatially variable environmental factors. Previous studies have used simulations to compare different methods for estimating the overall rate of spread of an invasion. We adopted a similar approach of simulating invasions that resemble two real case‐studies, both in terms of their spatial resolution (i.e. considering the size of one cell as one km) and their spatial extent (> 600 000 km²). Simulations were sampled to compare how different methods used to measure local spread rate, namely the neighbouring, nearest distance and Delaunay methods, perform for spatio‐temporal comparisons. We varied the assessment using three levels of complexity of the spatio‐temporal pattern of invasion, three sample sizes (500, 1000 and 2000 points), three different spatial sampling patterns (stratified, random, aggregated), three interpolation methods (generalized linear model, kriging, thin plate spline regression) and two spatio‐temporal modelling structures (trend surface analysis and boundary displacement), resulting in a total of 486 different scenarios. The thin plate spline regression interpolation method, in combination with trend surface analysis, was found to provide the most robust local spread rate quantification as it was able to reliably accommodate different sampling conditions and invasion patterns. This best approach was successfully applied to two case‐studies, the invasion of France by the horse‐chestnut leafminer Cameraria ohridella and by the bluetongue virus, generally in agreement with previously published values of spread rates. Potential avenues for further research are discussed.     

A flow network model for animal movement on a landscape with application to invasion

Animal movement, whether for foraging, mate-seeking, predator avoidance, dispersal, or migration, is a fundamental aspect of ecology that shapes spatial abundance distributions, genetic compositions, and dynamics of populations. A variety of movement models have been used for predicting the effects of natural or human-caused landscape changes, invading species, or other disturbances on local ecology. Here we introduce the flow network—a general modeling framework for population dynamics and movement in a metapopulation representing a network of habitat sites (nodes). Based on the principles of physical transport phenomena such as fluid flow through pipes (Pouiselle’s Law) and analogously, the flow of electric current across a circuit (Ohm’s Law), the flow network provides a novel way of modeling movement, where flow rates are functions of relative node pressures and the resistance to movement between them. Flow networks offer the flexibility of incorporating abiotic and biotic conditions that affect either pressures, resistance, or both. To illustrate an application of the flow network, we present a theoretical invasion scenario. We consider the effects of spatial structure on the speed of invasion by varying the spatial regularity of node arrangement. In the context of invasion, we model management actions targeting nodes or edges, and consider the effects on speed of invasion, node occupation, and total abundance. The flow network approach offers the flexibility to incorporate spatial heterogeneity in both rates of flow and site pressures and offers an intuitive approach to connecting population dynamics and landscape features to model movement.     

Variation in the degree of specialization can maintain local diversity in model communities

We hypothesize that the continuum between generalist and specialist adaptations is an important general trade-off axis in the maintenance of local diversity, and we explore this idea with a simple model in which there are patch types to which species arrive as propagules and compete. Each patch type is defined by a competitive ranking of all species. A highly specialist species is the top competitor in one patch type but has a relatively low average ranking across different patch types, while a generalist species has a high average rank across patch types but is not the top competitor in any patch type. We use random dispersal and vary the fecundity of all species together to vary total propagule density and therefore recruitment limitation and density-dependent mortality. When fecundity is very high, each patch becomes occupied by its specialist species and generalists go extinct, so the number of species at equilibrium is equal to the number of patch types. If fecundity is very low, generalists dominate and specialists go extinct. There is a range of fecundity levels in which specialists, generalists, and intermediates coexist, and the number of species is substantially greater than the number of patch types. While coexistence of specialists and generalists has been considered a problem in evolutionary ecology, our results suggest to the contrary that this trade-off contributes to the maintenance of local diversity.     

Dispersal-mediated coexistence of competing predators

Models of metapopulations have often ignored local community dynamics and spatial heterogeneity among patches. However, persistence of a community as a whole depends both on the local interactions and the rates of dispersal between patches. We study a mathematical model of a metacommunity with two consumers exploiting a resource in a habitat of two different patches. They are the exploitative competitors or the competing predators indirectly competing through depletion of the shared resource. We show that they can potentially coexist, even if one species is sufficiently inferior to be driven extinct in both patches in isolation, when these patches are connected through diffusive dispersal. Thus, dispersal can mediate coexistence of competitors, even if both patches are local sinks for one species because of the interactions with the other species. The spatial asynchrony and the competition-colonization trade-off are usual mechanisms to facilitate regional coexistence. However, in our case, two consumers can coexist either in synchronous oscillation between patches or in equilibrium. The higher dispersal rate of the superior prompts rather than suppresses the inferior. Since differences in the carrying capacity between two patches generate flows from the more productive patch to the less productive, loss of the superior by emigration relaxes competition in the former, and depletion of the resource by subsidized consumers decouples the local community in the latter.     

Persistence and periodic orbits of a three-competitor model with refuges.

We consider a model composed of four patches. One patch has three competing species forming a heteroclinic cycle within the patch. The remaining patches are refuges for the three competitors, and each species can diffuse between the competitive patch and its refuge. It is proved that the model can be made persistent by the introduction of the refuges for the competitors even if the isolated competitive patch has an attracting heteroclinic cycle. Further it is shown that Hopf bifurcation is possible when we change the value of the diffusion constant and periodic orbits may exist in a specific case.     

Invasion in a heterogeneous world: resistance,coexistence or hostile takeover?

We review and synthesize recent developments in the study of the invasion of communities in heterogeneous environments, considering both the invasibility of the community and impacts to the community. We consider both empirical and theoretical studies. For each of three major kinds of environmental heterogeneity (temporal, spatial and invader-driven), we find evidence that heterogeneity is critical to the invasibility of the community, the rate of spread, and the impacts on the community following invasion. We propose an environmental heterogeneity hypothesis of invasions, whereby heterogeneity both increases invasion success and reduces the impact to native species in the community, because it promotes invasion and coexistence mechanisms that are not possible in homogeneous environments. This hypothesis could help to explain recent findings that diversity is often increased as a result of biological invasions. It could also explain the scale dependence of the diversity–invasibility relationship. Despite the undoubted importance of heterogeneity to the invasion of communities, it has been studied remarkably little and new research is needed that simultaneously considers invasion, environmental heterogeneity and community characteristics. As a young field, there is an unrivalled opportunity for theoreticians and experimenters to work together to build a tractable theory informed by data.