Coexistence and Spread of Competitors in Heterogeneous Landscapes

Competition between species is ubiquitous in nature and therefore widely studied in ecology through experiment and theory. One of the central questions is under which conditions a (rare) invader can establish itself in a landscape dominated by a resident species at carrying capacity. Applying the same question with the roles of the invader and resident reversed leads to the principle that “mutual invasibility implies coexistence.” A related but different question is how fast a locally introduced invader spreads into a landscape (with or without competing resident), provided it can invade. We explore some aspects of these questions in a deterministic, spatially explicit model for two competing species with discrete non-overlapping generations in a patchy periodic environment. We obtain threshold values for fragmentation levels and dispersal distances that allow for mutual invasion and coexistence even if the non-spatial competition model predicts competitive exclusion. We obtain exact results when dispersal is governed by a Laplace kernel. Using the average dispersal success, we develop a mathematical framework to obtain approximate results that are independent of the exact dispersal patterns, and we show numerically that these approximations are very accurate.     

Evolution of dispersal in open advective environments

We consider a two-species competition model in a one-dimensional advective environment, where individuals are exposed to unidirectional flow. The two species follow the same population dynamics but have different random dispersal rates and are subject to a net loss of individuals from the habitat at the downstream end. In the case of non-advective environments, it is well known that lower diffusion rates are favored by selection in spatially varying but temporally constant environments, with or without net loss at the boundary. We consider several different biological scenarios that give rise to different boundary conditions, in particular hostile and “free-flow” conditions. We establish the existence of a critical advection speed for the persistence of a single species. We derive a formula for the invasion exponent and perform a linear stability analysis of the semi-trivial steady state under free-flow boundary conditions for constant and linear growth rate. For homogeneous advective environments with free-flow boundary conditions, we show that populations with higher dispersal rate will always displace populations with slower dispersal rate. In contrast, our analysis of a spatially implicit model suggest that for hostile boundary conditions, there is a unique dispersal rate that is evolutionarily stable. Nevertheless, both scenarios show that unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rate can evolve.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

A spatial model for the spread of invading organisms subject to competition

 A spatially explicit integrodifference equation model is studied for the spread of an invading organism against an established competitor. Provided the invader is initially confined to a bounded region, the invasion spreads asymptotically as a travelling wave whose speed depends on the strength of the competitive interaction and on the dispersal characteristics of the invader. Even an inferior, but established, competitor can significantly reduce the invasion speed. The invasion speed is also influenced by the exact shape of the dispersal kernel (especially the thickness of the tail) as well as the mean dispersal distance for each generation. Received 10 April 1996; received in revised form 21 August 1996     

Density dependence slows invader spread in fragmented landscapes

Patchiness is a defining characteristic of most natural and anthropogenic habitats, yet much of our understanding of how invasions spread has come from models of spatially homogeneous environments. Except for populations with Allee effects, an invader's growth rate when rare and dispersal determine its spread velocity; intraspecific competition has little to no influence. How this result might change with landscape patchiness, however, is poorly understood. We used simulation models and their analytical approximations to explore the effect of density dependence on the spread of annual plant invaders moving through heterogeneous landscapes with gaps in suitable habitat. We found that landscape patchiness and discrete invader population size interacted to generate a strong role for density dependence. Intraspecific competition greatly slowed the spread of invasions through patchy landscapes by regulating how rapidly a population could produce enough seeds to surpass habitat gaps. Populations with continuously varying density showed no such effect of density dependence. We adapted a stochastic dispersal model to approximate spread when gap sizes were small relative to the mean dispersal distance and a Markov chain approximation for landscapes with large gaps. Our work suggests that ecologists must consider reproduction at both low and high densities when predicting invader spread.     

Spreading speeds as slowest wave speeds for cooperative systems

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.     

Spatial interplay of plant competition and consumer foraging mediate plant coexistence and drive the invasion ratchet

Indirect effects may play an important role in structuring plant communities. Using a spatially explicit model of consumer foraging and plant competition, we demonstrate how the relationship between the spatial area over which plants compete and the spatial scale of consumer behaviour can determine the outcome of competition when one plant species provides a refuge for mobile consumers (i.e. refuge-mediated apparent competition). Once an initial population of the invader is established, complete invasion may be inevitable because of an ever-advancing invasion front ratchets forward driven by a feeding front of mobile consumers. Because the spatial extent of apparent competition determines the area available for colonization, consumers may also dictate the rate at which an invasion occurs. We find that, as long as refuge-mediated apparent competition is sufficiently localized, invasion is possible even in systems characterized by low overall levels of consumer pressure. Moreover, we show that a stable equilibrium can result in which both resident and invading plants coexist, suggesting that spatial heterogeneity created by refuge-mediated apparent competition may be important in mediating coexistence in plant communities. The spatial interplay of consumer behaviour and plant competition may be an underappreciated mechanism affecting the composition, diversity and spatial pattern of plant communities.     

Spatial Assortment of Mixed Propagules Explains the Acceleration of Range Expansion

Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread. To date, no overarching formula exists that can be applied to all three types of range expansion. We investigated how propagule pressure, i.e., the initial number of individuals and their composition in terms of dispersal ability, affects the spread of a population. A system of integrodifference equations was then used to model the spatiotemporal dynamics of the population. We studied the dynamics of dispersal ability as well as the instantaneous and asymptotic rate of spread. We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate. The instantaneous rate of spread was found to be fully determined by the growth and dispersal abilities of the population at the advancing edge of the invasion. We derived a formula for the asymptotic rate of spread under different scenarios of propagule pressure. The results suggest that data collected from the core of the invasion may underestimate the spreading rate of the population. Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.     

Extraordinary sex ratios: cultural effects on ecological consequences

We model sex-structured population dynamics to analyze pairwise competition between groups differing both genetically and culturally. A sex-ratio allele is expressed in the heterogametic sex only, so that assumptions of Fisher's analysis do not apply. Sex-ratio evolution drives cultural evolution of a group-associated trait governing mortality in the homogametic sex. The two-sex dynamics under resource limitation induces a strong Allee effect that depends on both sex ratio and cultural trait values. We describe the resulting threshold, separating extinction from positive growth, as a function of female and male densities. When initial conditions avoid extinction due to the Allee effect, different sex ratios cannot coexist; in our model, greater female allocation always invades and excludes a lesser allocation. But the culturally transmitted trait interacts with the sex ratio to determine the ecological consequences of successful invasion. The invading female allocation may permit population persistence at self-regulated equilibrium. For this case, the resident culture may be excluded, or may coexist with the invader culture. That is, a single sex-ratio allele in females and a cultural dimorphism in male mortality can persist; a low-mortality resident trait is maintained by father-to-son cultural transmission. Otherwise, the successfully invading female allocation excludes the resident allele and culture and then drives the population to extinction via a shortage of males. Finally, we show that the results obtained under homogeneous mixing hold, with caveats, in a spatially explicit model with local mating and diffusive dispersal in both sexes.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

Integrodifference equations in patchy landscapes

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton–Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.     

Generalizing Levins metapopulation model in explicit space: models of intermediate complexity

A recent study [Harding and McNamara, 2002. A unifying framework for metapopulation dynamics. Am. Nat. 160, 173-185] presented a unifying framework for the classic Levins metapopulation model by incorporating several realistic biological processes, such as the Allee effect, the Rescue effect and the Anti-rescue effect, via appropriate modifications of the two basic functions of colonization and extinction rates. Here we embed these model extensions on a spatially explicit framework. We consider population dynamics on a regular grid, each site of which represents a patch that is either occupied or empty, and with spatial coupling by neighborhood dispersal. While broad qualitative similarities exist between the spatially explicit models and their spatially implicit (mean-field) counterparts, there are also important differences that result from the details of local processes. Because of localized dispersal, spatial correlation develops among the dynamics of neighboring populations that decays with distance between patches. The extent of this correlation at equilibrium differs among the metapopulation types, depending on which processes prevail in the colonization and extinction dynamics. These differences among dynamical processes become manifest in the spatial pattern and distribution of “clusters” of occupied patches. Moreover, metapopulation dynamics along a smooth gradient of habitat availability show significant differences in the spatial pattern at the range limit. The relevance of these results to the dynamics of disease spread in metapopulations is discussed.     

<Emphasis Type="Italic">PRUNUS</Emphasis>: a spatially explicit demographic model to study plant invasions in stochastic,heterogeneous environments

To model the invasion of Prunus serotina invasion within a real forest landscape we built a spatially explicit, non-linear Markov chain which incorporated a stage-structured population matrix and dispersal functions. Sensitivity analyses were subsequently conducted to identify key processes controlling the spatial spread of the invader, testing the hypothesis that the landscape invasion patterns are driven in the most part by disturbance patterns, local demographical processes controlling propagule pressure, habitat suitability, and long-distance dispersal. When offspring emigration was considered as a density-dependent phenomenon, local demographic factors generated invasion patterns at larger spatial scales through three factors: adult longevity; adult fecundity; and the intensity of self-thinning during stand development. Three other factors acted at the landscape scale: habitat quality, which determined the proportion of the landscape mosaic which was potentially invasible; disturbances, which determined when suitable habitats became temporarily invasible; and the existence of long distance dispersal events, which determined how far from the existing source populations new founder populations could be created. As a flexible “all-in-one” model, PRUNUS offers perspectives for generalization to other plant invasions, and the study of interactions between key processes at multiple spatial scales.     

Species Invasiveness in Biological Invasions: A Modelling Approach

The study of invasiveness, the traits that enable a species to invade a habitat, and invasibility, the habitat characteristics that determine its susceptibility to the establishment and spread of an invasive species, provide a useful conceptual framework to formulate the biological invasion problem in a modelling context. Another important aspect is the complex interaction emerging among the invader species, the noninvader species already present in the habitat, and the habitat itself. Following a modelling approach to the biological invasion problem, we present a spatially explicit cellular automaton model (Interacting Multiple Cellular Automata (IMCA)). We use field parameters from the invader Gleditsia triacanthos and the native Lithraea ternifolia in montane forests of central Argentina as a case study to compare outputs and performance of different models. We use field parameters from another invader, Ligustrum lucidum, and the native Fagara coco from the same system to run the cellular automaton model. We compare model predictions with invasion values from aerial photographs. We discuss in detail the importance of factors affecting species invasiveness, and give some insights into habitat invasibility and the role of interactions between them. Finally, we discuss the relevance of mathematical modelling for studying and predicting biological invasions. The IMCA model provided a suitable context for integrating invasiveness, invasibility, and the interactions. In the invasion system studied, the presence of an invader's juvenile bank not only accelerated the rate of invasion but was essential to ensure invasion. Using the IMCA model, we were able to determine that not only adult survival but particularly longevity of the native species influenced the spread velocity of the invader, at least when a juvenile bank is present. Other factors determining velocity of invasion detected by the IMCA model were seed dispersal distance and age of reproductive maturity. We derived relationships between species' adult survival, fecundity and longevity of both theoretical and applied relevance for biological invasions. Invasion velocities calculated from the aerial photographs agreed well with predictions of the IMCA model.     

Disappearing refuges in time and space: how environmental change threatens species coexistence

Understanding the impacts of environmental changes on species survival is a major challenge in ecological research, especially when shifting from single- to multispecies foci. Here, we apply a spatially explicit two-species simulation model to analyze the effects of geographic range shifting and habitat isolation on different coexistence mechanisms. The model explicitly considers dispersal, local competition, and growth on a single resource. Results highlight that both range shifting and habitat isolation severely impact coexistence. However, the strength of these impacts depends on the underlying coexistence mechanisms. Neutrally coexisting species are particularly sensitive to habitat isolation, while stabilized coexistence through overcompensatory density regulation is much more sensitive to range shifting. We conclude that, at the community level, the response to environmental change sensitively depends on the underlying coexistence mechanisms. This suggests that predictions and management recommendations should consider differences between neutral versus stabilized community structures whenever possible.     

Predicting dispersal and recruitment of <Emphasis Type="Italic">Miconia calvescens</Emphasis> (Melastomataceae) in Australian tropical rainforests

Miconia calvescens (Melastomataceae) is a serious invader in the tropical Pacific, including the Hawaiian and Tahitian Islands, and currently poses a major threat to native biodiversity in the Wet Tropics of Australia. The species is fleshy-fruited, small-seeded and shade tolerant, and thus has the potential to be dispersed widely and recruit in relatively intact rainforest habitats, displacing native species. Understanding and predicting the rate of spread is critical for the design and implementation of effective management actions. We used an individual-based model incorporating a dispersal function derived from dispersal curves for similar berry-fruited native species, and life-history parameters of fecundity and mortality to predict the spatial structure of a Miconia population after a 30 year time period. We compared the modelled population spatial structure to that of an actual infestation in the rainforests of north Queensland. Our goal was to assess how well the model predicts actual dispersion and to identify potential barriers and conduits to seed movement and seedling establishment. The model overpredicts overall population size and the spatial extent of the actual infestation, predicting individuals to occur at a maximum 1,750 m from the source compared with the maximum distance of any detected individual in the actual infestation of 1,191 m. We identify several characteristic features of managed invasive populations that make comparisons between modelled outcomes and actual infestations difficult. Our results suggest that the model’s ability to predict both spatial structure and spread of the population will be improved by incorporating a spatially explicit element, with dispersal and recruitment probabilities that reflect the relative suitability of different parts of the landscape for these processes.     

Evolutionarily labile species interactions and spatial spread of invasive species

Both exotic and native species have been shown to evolve in response to invasions, yet the impacts of rapidly evolving interactions between novel species pairs have been largely ignored in studies of invasive species spread. Here, I use a mathematical model of an interacting invasive predator and its native prey to determine when and how evolutionary lability in one or both species might impact the dynamics of the invader's spatial advance. The model shows that evolutionarily labile invaders continually evolve better adapted phenotypes along the moving invasion front, offering an explanation for accelerating spread and spatial phenotype clines following invasion. I then analytically derive a formula to estimate the relative change in spread rate due to evolution. Using parameter estimates from the literature, this formula shows that moderate heritabilities and selection strengths are sufficient to account for changes in spread rates observed in historical and ongoing invasions. Evolutionarily labile native species can slow invader spread when genes flow from native populations with exposure to the invader into native populations ahead of the invasion front. This outcome is more likely in systems with highly diffuse native dispersal, net directional movement of natives toward the invasion front, or human inoculation of uninvaded native populations.     

Competition between similar invasive species: modeling invasional interference across a landscape

As the number of biological invasions increases, interactions between different invasive species will become increasingly important. Several studies have examined facilitative invader–invader interactions, potentially leading to invasional meltdown. However, if invader interactions are negative, invasional interference may lead to lower invader abundance and spread. To explore this possibility, we develop models of two competing invaders. A landscape simulation model examines the patterns created by two such species invading into the same region. We then apply the model to a case study of Carduus nutans L. and C. acanthoides L., two economically important invasive weeds that exhibit a spatially segregated distribution in central Pennsylvania, USA. The results of these spatially-explicit models are generally consistent with the results of classic Lotka–Volterra competition models, with widespread coexistence predicted if interspecific effects are weaker than intraspecific effects for both species. However, spatial segregation of the two species (with lower net densities and no further spread) may arise, particularly when interspecific competition is stronger than intraspecific competition. A moving area of overlap may result when one species is a superior competitor. In the Carduus system, our model suggests that invasional interference will lead to lower levels of each species when together, but a similar net level of thistle invasion due to the similarity of intra- and interspecific competition. Thus, invasional interference may have important implications for the distribution and management of invasive species.     

Emerging infectious diseases and animal social systems

Emerging infectious diseases threaten a wide diversity of animals, and important questions remain concerning disease emergence in socially structured populations. We developed a spatially explicit simulation model to investigate whether—and under what conditions—disease-related mortality can impact rates of pathogen spread in populations of polygynous groups. Specifically, we investigated whether pathogen-mediated dispersal (PMD) can occur when females disperse after the resident male dies from disease, thus carrying infections to new groups. We also examined the effects of incubation period and virulence, host mortality and rates of background dispersal, and we used the model to investigate the spread of the virus responsible for Ebola hemorrhagic fever, which currently is devastating African ape populations. Output was analyzed using regression trees, which enable exploration of hierarchical and non-linear relationships. Analyses revealed that the incidence of disease in single-male (polygynous) groups was significantly greater for those groups containing an average of more than six females, while the total number of infected hosts in the population was most sensitive to the number of females per group. Thus, as expected, PMD occurs in polygynous groups and its effects increase as harem size (the number of females) increases. Simulation output further indicated that population-level effects of Ebola are likely to differ among multi-male–multi-female chimpanzees and polygynous gorillas, with larger overall numbers of chimpanzees infected, but more gorilla groups becoming infected due to increased dispersal when the resident male dies. Collectively, our results highlight the importance of social system on the spread of disease in wild mammals.     

Spatial processes that maintain biodiversity in plant communities

The composition of communities of sessile organisms, and the change in species diversity with time, is a spatially explicit phenomenon. Three spatial factors clearly affect diversity: (1) the structure and heterogeneity of the landscape that limits species immigration and ultimate community size; (2) neighborhood interactions that determine colonization and extinction rates and influence residence times of local populations; and (3) disturbances that open spatially contiguous areas for recolonization by less abundant species. The importance of these three factors was first reviewed and then examined with a spatially explicit, multi-species model of plant dispersal, competition and establishment, with an assumption of neutrality (all species had equivalent life histories) that reduced the initial dimensionality of the problem. The simulations assumed that the probability of immigration was a linear function of mainland abundance and distance to islands, similar to the equilibrium theory of island biogeography and the unified neutral theory of biodiversity. The rate of increase in species richness was not constant across island sizes, declining as island area became very large. This pattern was explained by the spatial dynamics of colonization and establishment, a non-random process that cannot be explained by passive sampling alone. Simulations showed that population establishment depended critically on rare long-distance dispersal events while population persistence was achieved by the formation of aggregated species distributions that developed through restricted dispersal and local competitive interactions. Nevertheless, species richness always declined to a single species in the absence of disturbances, while up to 40 species could persist to 10,000 years when spatially dependent mortality was added. Further explorations with spatially explicit models will be required to fully appreciate the consequence of land use change and altered disturbance regimes on patterns of species distribution and the maintenance of diversity.