The effect of habitat fragmentation on cyclic population dynamics: a reduction to ordinary differential equations

Habitat fragmentation is known to be a key factor affecting population dynamics. In a previous study by Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009), the effect of habitat fragmentation on cyclic population dynamics was studied using spatially explicit predator?Cprey models with four different sets of reaction terms. The difficulty with spatially explicit models is that often analytical tractability is lost and the mechanisms behind the behaviour of the models are difficult to analyse. In this study, we employ a simplification procedure based on a Fourier series first-term truncation of the spatially explicit models Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009) to obtain spatially implicit models. These simpler models capture the main features of the spatially explicit models and can be used to explain the dynamics observed by Strohm and Tyson. We find that the spatially implicit models and the spatially explicit models produce similar responses to habitat fragmentation for larger high-quality patch sizes. Additionally, we find that the critical patch size of the spatially implicit models provides an upper bound on the critical patch size of the spatially explicit models. Finally, we derive an approximation of the multi-patch habitat by a single-patch habitat with partial flux boundary conditions which allows for a lower bound on the critical patch size to be calculated.     

The competition-colonization trade-off is dead; long live the competition-colonization trade-off

When applied at the individual patch level, the classic competition-colonization models of species coexistence assume that propagules of superior competitors can displace adults of inferior competitors (displacement competition). But if adults are invulnerable to displacement by propagules (as trees are to seeds), and propagules compete to replace adults that die for reasons independent of the outcome of juvenile competition (a lottery system), a competition-colonization trade-off alone is not able to produce coexistence. However, we show that coexistence is possible if patch density varies spatially, such that it becomes a niche axis. We also show how a dispersal-fecundity trade-off can partition variation in patch density. We discuss the application of these models to empirical systems. An important implication of communities coexisting via variation in patch density is that the amount of habitat loss necessarily interacts with the pattern of loss in affecting extinctions, invasions, and coexistence, in contrast to displacement competition models, for which the spatial pattern of loss is not important or is less important. Finally, with respect to mechanisms promoting coexistence, we suggest that trade-offs between different stages of colonization could be far more common in nature than a trade-off between competitive ability and colonization ability.     

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.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

Spatial dynamics in model plant communities: what do we really know?

A variety of models have shown that spatial dynamics and small-scale endogenous heterogeneity (e.g., forest gaps or local resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real systems. We synthesize results from three different kinds of models: interacting particle systems, moment equations for spatial point processes, and metapopulation or patch models. Studies using all three frameworks agree that spatial dynamics need not enhance coexistence nor slow down dynamics; their effects depend on the underlying competitive interactions in the community. When similar species would coexist in a nonspatial habitat, endogenous spatial structure inhibits coexistence and slows dynamics. When a dominant species disperses poorly and the weaker species has higher fecundity or better dispersal, competition-colonization trade-offs enhance coexistence. Even when species have equal dispersal and per-generation fecundity, spatial successional niches where the weaker and faster-growing species can rapidly exploit ephemeral local resources can enhance coexistence. When interspecific competition is strong, spatial dynamics reduce founder control at large scales and short dispersal becomes advantageous. We describe a series of empirical tests to detect and distinguish among the suggested scenarios.     

Overgrowth competition, fragmentation and sex-ratio dynamics: a spatially explicit, sub-individual-based model

Sessile organisms that compete for access to resources by overgrowing each other may risk the local elimination of one sex or the other, as frequently happens within clumps of the dioecious liverwort Marchantia inflexa. A multi-stage, spatially implicit differential-equation model of M. inflexa growing in an isolated patch, analysed in a previous study, indicated that long-term coexistence of the sexes within such patches may be only temporary. Here we derive a spatially explicit, sub-individual-based model to reconsider this interpretation when much more ecological realism is taken into account, including the process of fragmentation. The model tracks temporally discrete growth increments in continuous space, representing growth architecture and the overgrowth process in significant geometric detail. Results remain generally consistent with the absence of long-term coexistence of the sexes in individual patches of Marchantia. Dynamics of sex-specific growth qualitatively resemble those generated by differential-equation models, suggesting that this much simpler framework may be adequate for multi-patch metapopulation models. Direct competition between fragmenting and non-fragmenting clones demonstrates the importance of fragmentation in overgrowth competition. The results emphasize the need for empirical work on mechanisms of overgrowth and for modeling and empirical studies of life history tradeoffs and sex-ratio dynamics in multi-patch systems.     

The influence of patch demographics on metapopulations, with particular reference to successional landscapes

The classical view of metapopulations relates the regional abundance of a species to the balance between the extinction and colonization dynamics of identical local populations. Species in successional landscapes may represent the most appropriate examples of classical metapopulations. However, Levins‐type metapopulation models do not explicitly separate population loss due to successional habitat change from other causes of extinction. A further complication is that the chance of population loss due to successional habitat change may be related to the age of a patch. I developed simple patch occupancy models to include succession and included consideration of patch age structure to address two related questions: what are the implications of changes in patch demographic rates and when is a move to a structured patch occupancy model justified? Age‐related variation in patch demography could increase or decrease the equilibrium fraction of the available habitat occupied by a species when compared to the predictions of an unstructured model. Metapopulation persistence was enhanced when the age class of patches with the highest species occupancy suffered relatively low losses to habitat succession. Conversely, when the age class of patches with the highest species occupancy also had relatively high successional loss rates, extinction thresholds were higher that would be predicted by a simple unstructured model. Hence age‐related variation in patch successional rate introduces biases into the predictions of simple unstructured models. Such biases can be detected from field surveys of the fraction of occupied and unoccupied patches in each age class. Where a bias is demonstrated, unstructured models will not be adequate for making predictions about the effects of changing parameters on metapopulation size. Thinking in successional terms emphasizes how landscapes might be managed to enhance or reduce the patch occupancy by any particular metapopulation     

A hierarchical matrix model to assess the impact of habitat fragmentation on population dynamics: an elasticity analysis

To better understand the role of habitat quality and boundaries on population dynamics at the landscape scale, we develop a model combining a spatially implicit approach, a spatial population Leslie-type model and an implicit model of habitat fragmentation. An original approach of elasticity permits to identify which types of element and boundary influence the most population viability according to the wood fragmentation degree. The studied species is a corridor forest insect sensitive to fragmentation (Abax parallelepipedus, Coleoptera, Carabidae). We show that a single large patch of wood is better than several small patches for the population viability.     

Spatial heterogeneity of mortality and temporal fluctuation in fertility promote coexistence but not vice versa: a random-community approach

Both spatial heterogeneity and temporal fluctuation of the environment are important mechanisms promoting species coexistence, but they work in different manners. We consider many pairs of species with randomly generated survivorship and fertility in the lottery model, and examine how the variability in demographic processes affects the outcome of competition. The results are: [1] Coexistence is easier if habitat difference in mortality is greater, or if year-to-year variation in reproductive rate is larger. But neither habitat-difference in fertility nor temporal variation in mortality promotes coexistence. [2] Mean fertility does not affect the outcome if CV remains constant. In contrast, enhanced mean mortality decreases the fraction of coexisting pairs if the environment fluctuates temporally. [3] We also investigate the effect of limited dispersal of propagules between habitats. Compared with the complete mixing case, the fraction of coexisting pairs is clearly enhanced if the spatial heterogeneity is the major source of environmental variation, but shows slight increase if the temporal fluctuation is dominant. We conclude that spatial heterogeneity is likely to work more effectively in promoting species coexistence than temporal fluctuation, especially when the species suffer relatively high mortality, and disperse their propagules in a limited spatial scale.     

Coexistence of cycling and dispersing consumer species: Armstrong and McGehee in space

Two competing consumer species may coexist using a single homogeneous resource when the more efficient consumer--the one having the lowest equilibrium resource density--has a more nonlinear functional response that generates consumer-resource cycles. We extend this model of nonequilibrium coexistence, as proposed by Armstrong and McGehee, by putting the interaction into a spatial context using two frameworks: a spatially explicit individual-based model and a spatially implicit metapopulation model. We find that Armstrong and McGehee's mechanism of coexistence can operate in a spatial context. However, individual-based simulations suggest that decreased dispersal restricts coexistence in most cases, whereas differential equation models of metapopulations suggest that a low rate of dispersal between subpopulations often increases the coexistence region. This difference arises in part because of two potentially opposing effects on coexistence due to the asynchrony in the temporal dynamics at different locations. Asynchrony implies that the less efficient species is more likely to be favored in some spatial locations at any given time, which broadens the conditions for coexistence. On the other hand, asynchrony and dispersal can also reduce the amplitude of local population cycles, which restricts coexistence. The relative influence of these two effects depends on details of the population dynamics and the representation of space. Our results also demonstrate that coexistence via the Armstrong-McGehee mechanism can occur even when there is little variation in the global densities of either the consumers or the resource, suggesting that empirical studies of the mechanisms should measure densities on several spatial scales.     

Rapid assessment of metapopulation viability under climate and land-use change

In order to predict species response to climate and land-use change, numerically fast and easily applicable assessment tools for species survival are required. We present a set of formulae to calculate the mean lifetime of a metapopulation in a spatially heterogeneous and dynamic landscape subject to habitat patch diminution, loss and/or spatial shift of the habitat network. The formulae require as inputs (i) information about the number, location and size of the habitat patches for several time steps to quantify landscape dynamics in terms of patch destruction, diminution or shifting rates and (ii) data on species traits such as their vulnerability to environmental variation and their dispersal ability to quantify local colonisation and extinction rates. We validate the formulae with a spatially explicit simulation. The analysis is complemented by a protocol for the easy use of the approach and practical application examples. A software implementation is available on request from the authors.     

An integrative framework of coexistence mechanisms in competitive metacommunities

Species distribution in a metacommunity varies according to their traits, the distribution of environmental conditions and connectivity among localities. These ingredients contribute to coexistence across spatial scales via species sorting, patch dynamics, mass effects and neutral dynamics. These mechanisms however seldom act in isolation and the impact of landscape configuration on their relative importance remains poorly understood. We present a new model of metacommunity dynamics that simultaneously considers these four possible mechanisms over spatially explicit landscapes and propose a statistical approach to partition their contribution to species distribution. We find that landscape configuration can induce dispersal limitations that have negative consequences for local species richness. This result was more pronounced with neutral dynamics and mass effect than with species sorting or patch dynamics. We also find that the relative importance of the four mechanisms varies not only among landscape configurations, but also among species, with some species being mostly constrained by dispersal and/or drift and others by sorting. Changes in landscape properties might lead to a shift in coexistence mechanisms and, by extension, to a change in community composition. This confirms the importance of considering landscape properties for conservation and management. Our results illustrate the idea that ecological communities are the results of multiple mechanisms acting at the same time and complete our understanding of spatial processes in competitive metacommunities.     

Dynamics of species coexistence: maintenance of a plant-ant competitive metacommunity

The metacommunity approach is an adequate framework to study coexistence between interacting species at different spatial scales. However, empirical evidence from natural metacommunities necessary to evaluate the predictive power of theoretical models of species coexistence remains sparse. We use two African ant species, Cataulacus mckeyi and Petalomyrmex phylax , symbiotically associated with the myrmecophyte Leonardoxa africana africana , to examine spatio-temporal dynamics of species coexistence and to investigate which environmental and life-history parameters may contribute to the maintenance of species diversity in this guild of symbiotic ants. Using environmental niche partitioning as a conceptual framework, we combined data on habitat variation, social structure of colonies, and population genetics with data from a colonisation experiment and from observation of temporal dynamics. We propose that the dynamics of ant species colonisation and replacement at local and regional scales can be explained by a set of life history traits for which the two ants exhibit hierarchies, coupled with strong environmental differences between the different patches in the level of environmental disturbances. The role of the competition–colonisation tradeoff is discussed and we propose that interspecific tradeoffs for traits related to dispersal and to reproduction are also determinant for species coexistence. We therefore suggest that species-sorting mechanisms are predominant in the dynamics of this metacommunity, but we also emphasise that there may be many ways for two symbionts in competition for the same host to coexist. The results speak in favour of a more complete integration of the various metacommunity models in a single theoretical framework.     

Eco-evolutionary maintenance of diversity in fluctuating environments

Growing evidence suggests that temporally fluctuating environments are important in maintaining variation both within and between species. To date, however, studies of genetic variation within a population have been largely conducted by evolutionary biologists (particularly population geneticists), while population and community ecologists have concentrated more on diversity at the species level. Despite considerable conceptual overlap, the commonalities and differences of these two alternative paradigms have yet to come under close scrutiny. Here, we review theoretical and empirical studies in population genetics and community ecology focusing on the ‘temporal storage effect’ and synthesise theories of diversity maintenance across different levels of biological organisation. Drawing on Chesson's coexistence theory, we explain how temporally fluctuating environments promote the maintenance of genetic variation and species diversity. We propose a further synthesis of the two disciplines by comparing models employing traditional frequency-dependent dynamics and those adopting density-dependent dynamics. We then address how temporal fluctuations promote genetic and species diversity simultaneously via rapid evolution and eco-evolutionary dynamics. Comparing and synthesising ecological and evolutionary approaches will accelerate our understanding of diversity maintenance in nature.     

Patch dynamics integrate mechanisms for savanna tree–grass coexistence

Many mechanisms have been suggested to explain the coexistence of woody species and grasses in savannas. However, evidence from field studies and simulation models has been mixed. Patch dynamics is a potentially unifying mechanism explaining tree–grass coexistence and the natural occurrence of shrub encroachment in arid and semi-arid savannas. A patch-dynamic savanna consists of a spatial mosaic of patches. Each patch maintains a cyclical succession between dominance of woody species and grasses, and the succession of neighbouring patches is temporally asynchronous. Evidence from empirical field studies supports the patch dynamics view of savannas. As a basis for future tests of patch dynamics in savannas, several hypotheses are presented and one is exemplarily examined: at the patch scale, realistically parameterized simulation models have generated cyclical succession between woody and grass dominance. In semi-arid savannas, cyclical successions are driven by precipitation conditions that lead to mass recruitment of shrubs in favourable years and to simultaneous collapse of shrub cohorts in drought years. The spatiotemporal pattern of precipitation events determines the scale of the savanna vegetation mosaic in space and time. In a patch-dynamic savanna, shrub encroachment is a natural, transient phase corresponding to the shrub-dominated phase during the successional cycle. Hence, the most promising management strategy for encroached areas is a large-scale rotation system of rangelands. In conclusion, patch dynamics is a possible scale-explicit mechanism for the explanation of tree–grass coexistence in savannas that integrates most of the coexistence mechanisms proposed thus far for savannas.     

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.     

Contrasting support for alternative models of genomic variation based on microhabitat preference: species‐specific effects of climate change in alpine sedges

Deterministic processes may uniquely affect codistributed species’ phylogeographic patterns such that discordant genetic variation among taxa is predicted. Yet, explicitly testing expectations of genomic discordance in a statistical framework remains challenging. Here, we construct spatially and temporally dynamic models to investigate the hypothesized effect of microhabitat preferences on the permeability of glaciated regions to gene flow in two closely related montane species. Utilizing environmental niche models from the Last Glacial Maximum and the present to inform demographic models of changes in habitat suitability over time, we evaluate the relative probabilities of two alternative models using approximate Bayesian computation (ABC) in which glaciated regions are either (i) permeable or (ii) a barrier to gene flow. Results based on the fit of the empirical data to data sets simulated using a spatially explicit coalescent under alternative models indicate that genomic data are consistent with predictions about the hypothesized role of microhabitat in generating discordant patterns of genetic variation among the taxa. Specifically, a model in which glaciated areas acted as a barrier was much more probable based on patterns of genomic variation in Carex nova, a wet‐adapted species. However, in the dry‐adapted Carex chalciolepis, the permeable model was more probable, although the difference in the support of the models was small. This work highlights how statistical inferences can be used to distinguish deterministic processes that are expected to result in discordant genomic patterns among species, including species‐specific responses to climate change.     

The effective size of a metapopulation living in a heterogeneous patch network

I analyze stochastic patch occupancy models (SPOMs), which record habitat patches as empty or occupied. A problem with SPOMs has been that if the spatial structure of a heterogeneous habitat patch network is taken into account, the computational effort needed to analyze a SPOM grows as a power of 2n, where n is the number of habitat patches. I propose a computationally feasible approximation method, which approximates the behavior of a heterogeneous SPOM by an "ideal" metapopulation inhabiting a network of identical and equally connected habitat patches. The transformation to the ideal metapopulation is based on weighting the individual patch occupancies by the dynamic values of the habitat patches, which may be calculated from the deterministic mean-field approximation of the original SPOM. Conceptually, the method resembles the calculation of the effective size of a population in the context of population genetics. I demonstrate how the method may be applied to SPOMs with flexible structural assumptions and with spatially correlated and temporally varying parameter values. I apply the method to a real habitat patch network inhabited by the Glanville fritillary butterfly, illustrating that the metapopulation dynamics of this species are essentially driven by temporal variability in the environmental conditions.     

Continuum or discrete patch landscape models for savanna birds? Towards a pluralistic approach

Conceptualising landscapes as a mosaic of discrete habitat patches is fundamental to landscape ecology, metapopulation theory and conservation biology. An emerging question in ecology is: when is the discrete patch model more appropriate than alternative and conceptually appealing models such as the continuum model? There is limited empirical testing of the utility of alternative landscape models compared to the discrete patch model for a range of species. In this paper, we constructed three alternative sets of models for testing the effect of landscape structure on diversity and abundance of a suite of woodland birds in a savanna landscape of northern Australia: the null model (only site‐scale habitat variables, landscape context not important), the continuum model, and the discrete patch model. We utilised high‐spatial resolution satellite images to quantify spatial gradients in tree cover density (the continuum model), and to then aggregate the fine‐scale heterogeneity in tree cover into discrete patches of trees, with grass cover forming the “matrix” (the discrete patch‐model). We then evaluated the relative importance of the alternative models using generalised linear models and an information theoretic approach. We found that the importance of the models varied among species, with no single model dominant. Species that move between open grassy areas and woody shelter responded well to the continuum model, reflecting the importance of gradients in density of forage (grasses) and cover (trees), while the discrete model performed best for species that forage in all vegetation strata, and nest predominantly in dense woody vegetation. This finding supports a pluralistic approach, highlighting the need for adopting and testing more than one landscape model in savanna landscapes, and in other landscapes that do not have a well defined patch structure.