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Mathematical Challenges in Spatial Ecology Claudia Neuhauser
he word ecology comes from the Greek are often accompanied by soil erosion that affects oikos, which means “house”, and was water flow and nutrient recycling. There are nu- coined by the German zoologist Ernst merous examples where species invasions drasti- Haeckel in 1870. It is, according to the cally alter ecosystems: for instance, the water hy- Oxford English Dictionary, the “branch of acinth, a plant native to the Amazon and biologyT that deals with organisms’ relations to one considered one of the world’s worst invaders, now another and to the physical environment in which covers many lakes and rivers in the tropics. Land- they live.” Examples of such relations are preda- use changes and species invasions are the two tor-prey interactions and plants competing for re- major causes of species extinctions: it is estimated sources. Such interactions may be influenced by demographic and environmental fluctuations and that human activities have increased extinction occur over a wide range of spatial and temporal rates by a factor of 100 to 1,000. scales. Both empirical work and theoretical work con- Scientific curiosity by itself would be a sufficient tribute to our understanding of how ecosystems impetus for studying ecology, but understanding function and to our ability to successfully manage species relations and interactions is not just an aca- and preserve them. Mathematicians can contribute demic exercise: Humans rely on ecological ser- to this understanding by collaborating with biol- vices for purification of water and air, soil main- ogists on developing models, analyzing models, tenance, pest control, waste management, nutrient and relating theory to empirical work. recycling, and much more; these processes are I will focus on one factor that has become in- controlled by complex interactions of species with creasingly prominent in theoretical and empirical each other and with the environment. ecological studies: namely, space (see Tilman and We are altering the environment at an un- Kareiva, 1997). We live in a spatial world, and the precedented rate: foremost are land-use changes spatial component of ecological interactions has and invasions of nonnative species. These human- been identified as an important factor in how eco- caused alterations of the environment disrupt the functioning of ecosystems, often with devastating logical communities are shaped. Understanding consequences. For instance, land-use changes, such the role of space is challenging both theoretically as the conversion of forests into agricultural land, and empirically. Since it is impossible to cover all aspects of this area in a short article, I will con- Claudia Neuhauser is professor in the Department of centrate on stochastic models in population ecol- Ecology, Evolution, and Behavior at the University of Min- ogy. After describing the most common types of nesota, Twin Cities. Her work was supported in part by spatial models, together with biological examples, National Science Foundation grants DMS-0072262 and DMS-0083468. Her e-mail address is cneuhaus@ I will conclude with some mathematical challenges cbs.umn.edu. in spatial ecology.
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The Role of Models in Ecology In the 1950s, Elton and MacArthur promoted the Models in ecology serve a variety of purposes, idea that a more diverse community is more sta- which range from illustrating an idea to parame- ble. Their conclusion was based on a combination terizing a complex real-world situation. They are of verbal models and observations from natural used to make general predictions, to guide man- and agricultural systems. Their idea was accepted agement practices, and to provide a basis for the until the 1970s, when Robert May investigated development of statistical tools and testable hy- mathematical models of randomly assembled com- potheses. munities that showed a decrease in stability with Models that illustrate ideas are, for instance, diversity—just the opposite of the conclusion of Tilman’s resource model (Tilman, 1982) and the Elton and MacArthur. Although May emphasized Lotka-Volterra competition model. In Tilman’s re- that there was no paradox, since his randomly as- source model, species compete for resources. In sembled communities may not resemble natural the case of competition for just one resource, the systems, the stability-diversity hypothesis was no model predicts that the species with the lowest longer believed to be universally true. May’s con- equilibrium resource requirement will outcompete tribution demonstrated that mathematical reasons all other species. Tilman and collaborators have alone do not suffice and that biological reasons tested the model in freshwater algae, for which sil- need to be sought to understand the relationship icate is the limiting resource, and in grasslands in between diversity and stability. At that time there Minnesota, where nitrogen is the major limiting re- was a marked lack of empirical evidence for or source. The Lotka-Volterra competition model, dis- against the hypothesis; moreover, the word “sta- cussed in detail below, incorporates the observed bility” was used in different ways, making com- reduction in the growth rates of two competing parisons between different statements difficult. species as their densities increase. The model pre- Over the last ten years the stability-diversity de- dicts that coexistence occurs if competition among bate has been revived, and several empirical and individuals of the same species (intraspecific com- theoretical studies have been published that argue petition) is stronger than competition among in- either side. New empirical results and new insights dividuals of different species (interspecific com- from models that address conflicting hypotheses petition). In 1934, Gause tested this model in an might ultimately resolve the debate. experiment on competition between different species of paramecia (freshwater protozoans). Modeling in Spatial Ecology—a Brief Making predictions in management situations History requires more detailed and realistic models. Ex- Mathematical models have played an important role amples are models that investigate which areas throughout the history of ecology. Early examples need to be preserved to ensure persistence of an are population growth models such as exponen- endangered species and models that predict how tial growth, which Thomas Malthus used in 1798 spatial planting patterns of a genetically modified to argue the consequences of unrestricted popu- crop affect the evolution of pathogen resistance. lation growth, and logistic growth, which was de- Realistic models are essential when experiments veloped by Pierre-François Verhulst in 1835 to are not feasible or when either the temporal or the model density-dependent population growth and spatial scale over which predictions are sought is used by Raymond Pearl and Lowell J. Reed in the very large. For instance, predictions of the effect 1920s to predict the future population size of the of an increase in carbon dioxide in the atmosphere U.S. The 1920s also saw the development of math- on global climate and vegetation over the next one ematical models of multispecies interactions, no- hundred years have to be aided by modeling: we tably the predator-prey and competition models of can manipulate carbon dioxide levels only over Alfred J. Lotka and Vito Volterra. short temporal and small spatial scales; models Before the 1970s ecological population model- then extrapolate to the larger scale. ers typically used ordinary differential equations, Models can help to design experiments, to test seeking equilibria and analyzing stability. The early hypotheses, and to develop new hypotheses and models provided important insights, such as when ideas. New ideas may contradict long-held views species can stably coexist and when predator and and spur new research to resolve controversies; prey densities oscillate over time. conflicting hypotheses may coexist until a syn- A common feature of these early models is thesis is reached. Models are not the ultimate judge that the interactions were based on the mass- in resolving controversies, but they do play an im- action law, an approach that has its conceptual portant role in investigating consequences of al- foundation in modeling chemical reactions. When ternative hypotheses. To illustrate this point, I will the reactants are well mixed and have to collide briefly discuss a long-standing controversy in ecol- in order to react, the mass-action law says that ogy: namely, whether a more diverse community the collision rate (hence the reaction rate) is pro- is more stable, a question that has consequences portional to the product of the concentrations of for the effects of species loss. the reacting molecules.
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In many ecological situations, however, the as- ecology. Although spatial ecology today is still sumption of being well mixed does not hold, and dominated by theoretical investigations, empirical a spatial model with local interactions is more ap- studies that explore the role of space are becom- propriate and can result in predictions that differ ing more common due to technological advances from the well-mixed case. The two most frequently that allow the recording of exact spatial locations. quoted and now classical studies that point to the central role of spatial subdivision are Andrewartha The Mathematical Framework of Spatial and Birch’s 1954 observations of insect populations Models in Ecology that became frequently extinct but persisted glob- Demographic models are the fundamental build- ally due to recolonization from local populations, ing blocks of models in population and commu- and Huffaker’s 1958 laboratory experiment with nity ecology. The simplest demographic models are a predator-prey system of two mites, one that deterministic ones based on ordinary differential feeds on oranges (Eotetranychus sexmaculatus) equations. I will use the logistic model as an ex- and the other a predatory mite (Typhlodromus oc- ample to introduce the different modeling frame- cidentalis) that attacks E. sexmaculatus. Huffaker works. Logistic growth was introduced by Verhulst set up an array of oranges and rubber balls with as a model for population growth with negative different levels of spatial complexity that con- density dependence. Namely, if N(t) denotes the trolled dispersal and demonstrated that a complex, population size at time t, then spatially heterogeneous array promoted coexis- dN N tence, whereas coexistence was impossible in sim- = rN 1 − , pler, spatially homogeneous arrays. Mathematical dt K models later confirmed that spatial subdivision is where r and K are positive parameters; r is the in- important for the persistence of populations. trinsic rate of growth, which is the maximum per Some of the early models include space im- capita growth rate; and K is the carrying capacity, plicitly, such as the Levins model (Levins, 1969), dis- which is the nonzero equilibrium population size. cussed below, which uses the framework of ordi- The easiest way to include space in the logistic nary differential equations to describe the model is the metapopulation framework in which dynamics of a population in a spatially subdivided an infinite number of sites are linked by migration. habitat. Populations may go extinct in patches and In the original formulation all sites are equally ac- may subsequently be recolonized from other oc- cessible, and we say that migration is global (the cupied patches. Space is implicit in this frame- more general framework is discussed below); this work in the sense that recolonization is equally model is spatially implicit, since no explicit spatial likely from all occupied patches, regardless of distances between sites are included. their locations. Models that include space explic- The analogue of the logistic model in the itly, such as reaction-diffusion equations, were metapopulation framework is the Levins model. employed by Skellam in the 1950s to describe the Namely, if u(t) denotes the fraction of occupied invasion of species; they are of the same type as sites, then Fisher’s 1937 model for the spread of a novel al- du lele. Though these models include space, they still = λu(1 − u) − u. dt do not allow for spatial correlations, since local populations are effectively infinite. The colonization rate is equal to the parameter λ In the late 1960s and early 1970s, F. Spitzer in times the product of the fraction of occupied sites the United States and R. L. Dobrushin in the Soviet and the fraction of vacant sites. Time is scaled so Union began to develop a framework for spatial sto- that the rate at which sites become vacant equals 1, chastic models known as interacting particle sys- and a nontrivial equilibrium exists if λ is greater tems. These are stochastic processes that evolve than 1. Deterministic models with global disper- on the d-dimensional integer lattice. Each site on sal, called mean-field models, originated in physics. the lattice is in one of a finite or countable num- The idea is to replace complicated local interactions ber of different states, and sites change their states by an “effective field” produced by all other par- according to rules that depend on the states of ticles and to use the mass-action law to describe neighboring sites. Originally, interacting particle the dynamics. systems were motivated by attempts to describe Models at the next level of complexity explicitly phenomena in physics, but it soon became evi- include distances between sites. The simplest ex- dent that this framework is useful in biology. Some amples are reaction-diffusion equations, such as of these models can be viewed as spatial general- Fisher’s equation, which was introduced to ecology izations of classical ecological models such as the by Skellam. It describes logistic growth in an ex- logistic model and the Lotka-Volterra competition plicitly spatial setting by model. ∂u ∂2u During the 1970s the availability of comput- = D + λu(1 − u) − u, ers greatly expanded the use of spatial models in ∂t ∂x2
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where u = u(x, t) is the population density at lo- in a one-dimensional model that offspring are dis- cation x at time t. The parameter D is called the persed uniformly over the interval [−1, 1]. When diffusion coefficient, and the reaction term =1, the neighborhood consists of the two near- λu(1 − u) − u has the same interpretation as in est sites, and when =1/2, the neighborhood con- the Levins model. sists of the nearest and the next nearest sites. As Closely related to reaction-diffusion equations decreases further, more sites are included in the are integro-differential equations. The analogue neighborhood, each site being equally accessible. of Fisher’s equation in this framework is Convergence to the above integro-differential equa- tion can be shown under mild assumptions on the ∂u(x, t) = dispersal kernel. ∂t The conceptual difference between the reac- − u(x, t)+λ(1 − u(x, t)) k(x, y)u(y,t) dy. tion-diffusion equation and the integro-differen- tial equation is that in the former, individuals take The function k(x, y) is a probability density that de- many small steps in a random-walk-like fashion be- scribes colonization of a vacant site at x from an tween reproductive events, like zooplankton in a occupied site at y; it usually depends only on the water column, whereas in the latter individuals distance between x and y. take only one big step right after they are born, like Finally, interacting particle systems allow the in- plant seeds. Both the partial differential equation clusion of local interactions. As mentioned above, and the integro-differential equation reduce to the an interacting particle system is a stochastic Levins model when the initial distribution is con- d process with state space EZ , where E is some fi- stant in space. nite (or countably infinite) set. The state at time t In the following I will describe the framework is a configuration that assigns each site in Zd a of interacting particle systems in more detail. I will value in E. The dynamics of the model are de- also mention two other frameworks that are pop- scribed by rate functions that indicate how a site ular in ecological modeling but less often used in changes its state based on the current states of sites mathematical studies. in a local neighborhood. The rate functions define exponentially distributed times, and we say that a Interacting Particle Systems change occurs at rate µ if the waiting time T for General Framework and Basic Techniques this event is exponentially distributed with mean Interacting particle systems are continuous-time − d 1/µ , that is, P(T>t)=e µt . The interacting- Markov processes on EZ , where E is either a finite particle analogue of logistic growth in space is the set or an infinite but countable set. (A Markov contact process (see Liggett, 1985). In this model process is a stochastic process in which the future each site on the d-dimensional integer lattice is ei- state depends only on the current state and the past ther vacant (state 0) or occupied (state 1). A site becomes irrelevant.) The temporal evolution is becomes occupied at a rate equal to λ times the given by rate functions that allow the process to fraction of occupied sites within a given neigh- evolve in continuous time. The contact process borhood (for instance, all sites within distance R), described above is just one of many examples and an occupied site becomes vacant at rate 1. studied over the last thirty years. See Liggett (1985) The above models are related in ways that are for the foundation of interacting particle systems, now well understood and that have been made a description of analytical techniques, and some mathematically rigorous both for the contact of the basic models and results, and Durrett (1988, process and for many other models (Durrett and 1995) for graphical methods and a summary of Neuhauser, 1994; for other examples see Durrett, some more recent work. 1995). For example, the reaction-diffusion equation Few analytical methods are available to analyze follows from the contact process by scaling space these models. The two most important techniques and introducing fast stirring in the following way. are duality and coupling, but whether these meth- The integer lattice is replaced by a lattice with ods are available depends on the model. Duality sites distance apart, and neighboring sites ex- can be loosely described as a technique that allows − change their contents at rate 2/2. One can show one to trace the history of a finite number of sites that in the limit as → 0, the probability that a site back to time 0, which then allows one to determine is occupied converges to a function satisfying the their state at the current time based on their his- reaction-diffusion equation with D =1/2. tory and the configuration at time 0. This is akin An integro-differential equation can be ob- to identifying one’s ancestors. tained from the contact process through a simi- Coupling is a technique that allows comparison lar rescaling procedure. Namely, replace the inte- of two processes. For example, coupling can be ger lattice by a lattice with sites distance apart used to show that if the contact process survives and discretize the dispersal kernel so that as for some value of λ, then it will do so for all greater tends to 0, dispersing offspring have access to an values. Namely, one can couple two contact ever-growing number of sites. For example, assume processes with different birth rates in such a way
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that whenever the process with the smaller birth retains the desired properties for all times provided rate has an occupied site, so has the process with the sites in the oriented site percolation process the larger birth rate, provided both processes start do percolate. with the same initial configuration. The number of models that have been studied Another powerful technique is rescaling. This is quite large. In the following, I will focus on a class method, developed by Bramson and Durrett (see of single-species and multispecies demographic Durrett, 1995), involves a comparison between the models that describe competitive interactions process of interest and oriented site percolation. Ori- within and between species. The emphasis in sin- ented site percolation with density p is a process gle-species models is on whether a nontrivial sta- { ∈ 2 } on (z,k) Z : z + k is even in which every site, tionary distribution exists in which the species independently of all others, is open with proba- has a positive density and what patterns result − bility p and closed with probability (1 p). If p is from local interactions. In multispecies models sufficiently large, there is a positive probability that there is an additional aspect, namely, whether there is an infinite open path starting at (0, 0), and species can coexist and what the mechanisms of we say that the system percolates. (An open path coexistence are. from (x, 0) to (y,k) is a sequence of points z0 =(x, 0),z1,z2,...,zk =(y,k) such that for Survival of a Single Species in Space 0 ≤ j ≤ k − 1 , either z = z +(−1, 1) or j+1 j The simplest single-species population model is the z = z +(1, 1).) The basic idea of the rescaling ar- j+1 j contact process introduced above, which describes gument is to show that for appropriate p>0 the growth when the birth rate is density dependent process under consideration, when viewed on suit- but the death rate is density independent. The able length and time scales, dominates an oriented site percolation model in which sites are open with density dependence in the birth dynamics enters probability p (Figure 1). because only vacant sites can be colonized. After more than twenty-five years of study, the contact process is quite well understood. There exists a crit- ical birth parameter, greater than 1 (recall that 1 is the critical value for the nonspatial model), which depends on both the spatial dimension and the type of neighborhood, such that if the birth rate exceeds this threshold, the process survives, and otherwise the process dies out. The density of oc- Time cupied sites is a nondecreasing function of the birth parameter. The nontrivial stationary distri- bution (if it exists) has positive correlations. Though the correlations are weak, they can be seen in the nearest-neighbor case, where occupied sites appear clumped. If deaths also depend on the density, the cor- responding model is known as the annihilating branching process. Individuals die at a rate equal Figure 1. The rescaled lattice: Each box represents a space-time to the fraction of occupied neighbors, and the box. The dot in the center of each box represents a site in the birth process is the same as in the contact process. associated oriented site percolation process; open sites are Individuals give birth at rate λ, and their offspring indicated by full circles, closed sites by open circles. Open disperse within a given neighborhood, establish- paths are indicated by the arrows connecting open sites. ing themselves only on vacant sites. If the disper- sal neighborhood is the same as the neighborhood that determines mortality, then a nontrivial sta- For instance, to show that the contact process tionary equilibrium exists for all λ>0. It is the survives (that is, there exists a nontrivial station- λ/(1 + λ) ary distribution with a positive density of occupied product measure with density ; that is, sites), one needs to show that at any time the num- sites are independently occupied with probability ber of individuals in sufficiently large space-time λ/(1 + λ). Despite local dispersal, no spatial cor- boxes exceeds a certain threshold with probabil- relations build up at equilibrium (for references, ity close to 1. The space-time boxes are coupled see Neuhauser, 1999). to sites in an oriented site percolation process, so Multispecies Models that whenever a site is open in the oriented site per- Multispecies models can be built from single- colation process, the number of occupied sites in species models by incorporating interactions be- the contact process at any time in the corre- tween species. I will focus on competitive interac- sponding space-time box exceeds a certain thresh- tions and discuss different mechanisms for old. One can then show that the contact process coexistence.
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α 21 Lotka-Volterra type competition. The classical, nonspatial, two-species Lotka-Volterra competition model is given by Species 1 wins Founder control dN1 N1 N2 = r1N1 1 − − α12 , dt K1 K1 K /K dN2 N2 N1 2 1 = r2N2 1 − − α21 , dt K2 K2
where Ni(t) is the abundance of species i at time t. The positive parameters ri and Ki denote the in- Coexistence Species 2 wins trinsic rates of growth and the carrying capacities, respectively, just as in the logistic model. The in- teraction between the two species is given by the nonnegative parameters αij, which describe the ef- fect of species j on species i. The minus sign in α K1/K2 12 front of the αij’s indicates that the interaction is competitive: the presence of either species reduces the density of the other. Besides Gause’s experi- Figure 2. The phase diagram in the α12 − α21 plane for the ment mentioned earlier, there have been many ex- nonspatial Lotka-Volterra model with interspecific periments that measure the strength of interspe- competition. The horizontal and vertical dashed lines cific competition, such as the study by Rees et al. represent the boundaries of regions where the behavior of a four-species community of annual plants that differs: coexistence, species 1 wins, species 2 wins, and grow in a sand dune habitat in Norfolk, UK, and a founder control. study by Freckleton et al. of a community of range- model means that for any >0, there exists a suf- land annuals in Australia. ficiently large box such that with probability at least Analysis of the model (see Figure 2) reveals that (1 − ) both types will be in the box for sufficiently if α
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spatial models. The reason is that even though the α 21 Spatial-conjecture mean-field model has two locally stable equilibria and the outcome of competition depends on initial densities, one of the two equilibria turns out to be Species 1 wins Spatial-rigorous stronger when using the partial differential equa- tion or integro-differential equation approach. That is, as long as the stronger species occurs somewhere at a sufficiently high density over a sufficiently K2/K1 long interval, that species will expand its range via traveling waves and will eventually take over. Competition-Colonization Trade-Off. The com- petition-colonization trade-off is another mecha- Species 2 wins nism that is frequently invoked to explain coexis- tence, for instance in the Minnesota grasslands coexistence studied by Tilman’s group. Some grass species al- α 12 locate more biomass to their roots, which makes K1/K2 them better competitors for nitrogen, while oth- ers allocate more to seeds, which makes them bet- Figure 3. The phase diagram in the α12 − α21 plane for the ter colonizers; and the different species coexist. The spatial Lotka-Volterra model with interspecific competition. trade-off can be illustrated in a model in which Rigorous results for the spatially explicit model show that the species are hierarchically ordered. This model is coexistence and the founder control regions are smaller than in known as the grass-bushes-trees model in mathe- the nonspatial case (solid lines); we conjecture that the founder matics (for references, see Durrett, 1995) and the control region disappears in the spatial model (broken line). hierarchical model in ecology (for references, see the article by Lehman and Tilman in Tilman and equilibrium density. For the spatially explicit model Kareiva, 1997). In the two-species mean-field ver- with local interactions, it is impossible to maintain sion, the model is given by the following system an arbitrarily small density of one species when the of ordinary differential equations: neighborhood size is kept fixed: a single individ- ual of one type within a neighborhood of individ- du1 = λ1u1(1 − u1) − u1, uals of the other type has density equal to the rec- dt iprocal of the neighborhood size and thus cannot du2 = λ2u2(1 − u1 − u2) − u2 − λ1u1u2, be made arbitrarily small. The smaller the neigh- dt borhood size, the more pronounced this effect is. where u (t) is the density of species i at time t. Furthermore, spatial segregation of species oc- i Species 1 behaves like a contact process, and there- curs as interspecific competition increases relative fore for species 1 to survive, λ needs to exceed to intraspecific competition, resulting in cluster for- 1 the critical value (which is 1 for the mean-field mation of like species. Using duality, we were able model). Species 2 behaves like a contact process to give lower bounds on cluster sizes and conjec- in the absence of species 1 but is replaced by off- tured that the clusters will not continue to grow spring of species 1 if they land on sites that are forever if the αij’s are less than 1. This is differ- occupied by species 2. Offspring of species 2 can ent from the voter model (αij =1), where the clus- land only on vacant sites, whereas offspring of ters continue to grow. The spatial segregation of species for parameter values within the coexis- species 1 can land on vacant sites or on sites oc- tence region of the nonspatial model again de- cupied by species 2. Species 1 is thus considered pends on neighborhood size. Namely, for equal the superior competitor. The principal conclusion competition parameters that are less than 1, we can from this model is that the inferior competitor choose a neighborhood size large enough so that (species 2) can coexist with the superior competi- coexistence is possible. But if the neighborhood tor (species 1) only if its birth rate exceeds the size is fixed and the competition parameters ap- square of the birth rate of species 1, which makes proach 1 from below, then spatial segregation oc- species 2 the superior colonizer. Rescaling shows curs. that the spatial version of this model has the same Finally, using coupling, we found an expansion qualitative properties. of the parameter region where one species excludes Spatial Heterogeneity. If factors such as temper- the other, but the winner depends on the compe- ature, pH, light, and soil moisture vary over small tition parameters rather than on initial densities. enough scales, then species can coexist by spe- In other words, the founder control region shrinks; cializing on particular conditions. Biologically this we conjecture that the founder control region dis- is a very important factor for coexistence. From a appears altogether. The loss of the founder control mathematical point of view, however, this type of region appears to be a common phenomenon in coexistence is less interesting, since one essentially
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deals with noninteracting communities, each particle systems primarily by placing individuals specialized to its own habitat type, and coexis- on continuous space, such as the real plane, rather tence is trivial provided dispersal allows exchange than restricting them to a regular lattice. To obtain of migrants. biologically sensible results, one needs to introduce Spatial heterogeneity generated by species dy- a local carrying capacity that limits the number of namics is mathematically more interesting and individuals locally. Questions of interest are sim- also biologically important. For instance, light gaps ilar to the ones I discussed above within the frame- in forests are created through tree fall. These gaps work of interacting particle systems. represent small-scale spatial heterogeneity in light The point-process framework is particularly availability and can allow species that differ in suitable for modeling plant populations. Several their light requirements to coexist. A light- forest models employ this framework (for refer- tolerant species that reproduces on a fast time-scale ences, see the article by Pacala and Levin in Tilman and disperses over short distances can coexist and Kareiva, 1997). Point processes model dy- with a shade-tolerant species by temporarily ex- namics at the level of an individual. Each individ- ploiting the gaps before the shade-tolerant species ual is represented by a point with a unique loca- dominates the location again. This mechanism is tion and any other attributes one wishes to track called successional niche (Pacala and Rees, 1998). (species type, genotype, and so forth). The advan- We saw in the spatial Lotka-Volterra competi- tage of models based on individuals is that they tion model that competitive interactions between allow calibration using field data. For example, species can generate spatial segregation, a form of Pacala and coworkers calibrated the forest model spatial heterogeneity. This can facilitate coexis- SORTIE by collecting field data for fecundity, sur- tence of a large number of species by allowing dif- vivorship, mortality, dispersal, and so on for every ferent communities to exist side by side in patches tree species in the simulation model. The simula- without ever competing with species that live in tion model was used, for instance, to predict bio- another type of patch. mass and species diversity under different dis- Spatial heterogeneity does not always facilitate turbance regimes. coexistence. Indeed, habitat fragmentation, which Spatial point distributions describe the spatial results in a spatially heterogeneous landscape, is patterns of point processes. As in the case of in- implicated as a major cause of species extinction. teracting particle systems, local dispersal of indi- For instance, vast areas of Minnesota consisted of viduals can result in spatial correlations. Since it is continuous prairie before European settlement. generally impossible to obtain the probability dis- Now a large proportion of the original prairie has tribution that describes the spatial pattern in equi- been converted either to urban areas or to farm librium, much more modest goals are pursued, like land, resulting in a sparse patchwork of prairie frag- finding the average number of individuals per unit ments. Many of the fragments are so isolated that area or the covariance structure of the spatial pat- they have ceased to fulfill their ecological function. tern. But even this is difficult. Pacala and collabo- The effects of habitat fragmentation have been studied both theoretically and empirically. The hi- rators have developed a nonrigorous approxima- erarchical model has been explored quite exten- tion method called the moment closure method. sively in the context of habitat fragmentation where (This method had been used before in physics.) a fixed proportion of habitat is assumed to be de- The method works only for a certain class of rate stroyed. The mean-field model exhibits somewhat functions, and even then it does not always give the surprising behavior: namely, the best competitor correct answer. It amounts to deriving equations for is affected most severely by habitat destruction and the mean vector of species densities and the asso- goes extinct first as the fraction of destroyed habi- ciated covariance matrix. These equations contain tat increases (for references, see the article by higher-order moments and are therefore not closed. Lehman and Tilman in Tilman and Kareiva, 1997). The equations can be closed and then solved nu- However, this effect depends crucially on the rel- merically either by neglecting higher-order central ative dispersal ranges of the competitors, and one moments or by approximating them by lower-order can construct situations in a spatially explicit moments (the latter works, for instance, in the case model that produce the opposite conclusion where global dispersal results in a Poisson distri- (Neuhauser, 1998). bution). This method allows one to study spatially homogeneous models that are at the onset of ex- The Point-Process Framework hibiting spatial correlations, such as when off- Point processes are stochastic processes in which spring are dispersed over intermediate distances. points are distributed in d-dimensional real space (Dispersal over very short distances has the po- and the temporal dynamics are described by rate tential to result in large spatial correlations, whereas functions. Each point represents an individual, dispersal over long distances can be well approxi- and the rate functions determine the demographic mated by deterministic partial differential equations processes. Point processes differ from interacting or integro-differential equations, similarly to the de-
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terministic approximations for interacting particle maintain the metapopulation: patches that are too systems, which do not exhibit spatial correlations.) isolated are never colonized. Interacting particle systems and point processes Theoretical results on the persistence of such are not very different qualitatively. For instance, metapopulations are difficult to obtain due to the the same mechanisms mentioned above that allow complex spatial arrangement of patches. Hanski for coexistence in interacting particle systems are and Ovaskainen (2000) have made some progress present in the corresponding point-process model. in the case of deterministic models. They defined This suggests at least some degree of robustness a quantity, the metapopulation capacity, that de- with respect to the modeling framework. How- pends solely on the spatial arrangement and the ever, point processes are much harder to analyze area of patches and that allows one to determine than interacting particle systems, which explains whether the population will persist. Theoretical re- why studies of point processes rely primarily on sults of persistence of stochastic metapopulations simulations or approximation methods. are lacking, though they are quite important, in par- ticular when dealing with small populations where The Metapopulation Framework demographic stochasticity affects extinction of Many populations live in patchy habitats due to subpopulations. habitat fragmentation or to naturally occurring Most metapopulation work deals with single spatial heterogeneities. The metapopulation frame- populations. However, populations do not live in work, introduced into ecology by Levins in 1969, isolation. A habitat can contain tens to hundreds is ideally suited to deal with such situations. Sub- of species which interact in complex ways. Though sequently this modeling framework has been it is not feasible to investigate models at this level greatly expanded, in particular due to Ilkka Han- of complexity by specifying all possible interactions ski’s empirical and theoretical work on butterfly between a large number of species, it would be populations in Finland (for a summary, see Han- worthwhile to study communities of a small num- ski, 1999). ber of species that live in a patchy environment. Metapopulations are spatially implicit or explicit Different species have different life histories re- patch models. Each patch is either occupied or va- sulting in different dispersal strategies. How this cant; occupied patches become extinct at a rate that affects persistence of the entire community is not may depend on population size or patch area, and well understood. Much work is needed to under- vacant patches become colonized by migrants stand such metacommunities. from other patches. The dynamics within a patch can be deterministic or stochastic. Frequently, in- The Future of Spatial Ecology stead of modeling individuals, one tracks the prob- I discussed spatial models that deal with a single ability of patch occupancy, and one uses area as species or with competition between species and a proxy for population size to determine extinc- for which various results have been proved math- tion and colonization rates of patches. ematically rigorously. We have a good under- The metapopulation framework has the flexi- standing of what types of spatial patterns emerge bility to accommodate any spatial arrangement in single-species models and also what mecha- and thus provides a more realistic framework than nisms produce coexistence or mutual exclusion of interacting particle systems. This is particularly im- different species and which of these mechanisms portant if one attempts to parameterize a real require an explicitly spatial component. Even population, as Hanski and coworkers have done in though little is known rigorously about how robust their butterfly studies in the Åland islands in Fin- the results are with respect to specific dynamics, land. The butterflies are the Glanville fritillary, a there is a general sense that the results are robust species that occurs in dry meadows that contain and that we have identified the basic mechanisms. the larval host plants. There are a large number of Determining the relative importance of the dif- (mostly small) suitable patches, scattered through- ferent mechanisms in natural populations, how- out the landscape. Local butterfly populations are ever, remains a challenge. small, with a high turnover rate. For many years In the following I wish to point to challenges in Hanski and coworkers have collected data, such as areas in ecology that I did not discuss above and population sizes, extinction rates, and recolo- to topics that would benefit from a mathematically nization rates. Their study is unique in both spa- rigorous investigation. tial and temporal extent. By combining empirical Other Areas and theoretical work, they demonstrated that de- A large part of the theoretical ecology literature mographic stochasticity and inbreeding cause fre- deals with predator-prey or host-pathogen interac- quent local extinctions but that the metapopula- tions. Hassell, May, and collaborators (see the arti- tion as a whole persists because of recolonization cle by Hassell and Wilson in Tilman and Kareiva, and asynchrony in local dynamics. Their study 1997) have developed a sizable body of work on cou- also points to the importance of sufficiently large pled map lattices, primarily in the context of host- and closely connected networks of patches to parasitoid models. (Coupled map lattices are
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spatially explicit, deterministic models with local dy- tools are available for analyzing systems of differ- namics.) In addition, some stochastic and spatially ential or difference equations, but little is available explicit versions of predator-prey and host- for more complex models. One example of a tool pathogen models have been analyzed mathemat- that can be readily used for spatially explicit sto- ically rigorously (see Durrett, 1995). Most models chastic models is the moment closure method men- involve just pairwise interactions. Less well un- tioned above. This approximation method has pro- derstood are models that deal with multiple hosts, vided insight into the effects of local interactions multiple predators/pathogens, and more complex in space, but has the drawback that it does not al- interactions, such as a predator feeding on a prey ways yield correct answers. A better understand- species that in turn feeds on some other host. ing of this method and the development of other Ecological models often neglect genetics. Start- methods are needed. ing with Wright’s work, there is a large literature Computer simulations play an important role in on genetics models that include space either spatial ecology. It is often easy to write the code implicitly or explicitly. The mating structure can for the spatial model, but the parameter space is create genetic variation due to reduced gene flow frequently too large to do an exhaustive search to between locations, resulting in a patchy distribu- identify regions of qualitatively different behavior. tion of genotypes. In addition, different genotypes In this case analytical approximations can prove might be favored in different parts of the range. very valuable. One way to proceed is to use the pro- This variation can have important consequences, gression of models outlined above as a guide. in particular when individuals are suddenly faced Starting with a spatially explicit interacting parti- with new ecological and evolutionary conditions, cle system, one can first look at the correspond- such as rapid climate change or reduction in range ing reaction-diffusion or integro-differential equa- due to habitat destruction. tion that results from introducing either fast Robustness stirring or long-range dispersal. Since these mod- Results from theoretical models are based on spe- els are more tractable, one can find regions in the cific dynamics within a specific framework. This parameter space where the behavior changes qual- is particularly true for spatially explicit stochastic itatively. Often, the spatially explicit model has models; proofs break down when the dynamics interesting behavior close to where the corre- change ever so slightly. For instance, the birth rate sponding deterministic models have phase tran- in the contact process is proportional to the num- sitions; boundaries of those subspaces may be a ber of occupied sites in the neighborhood. Many good starting point for examining the interacting results for the contact process use duality, which particle system (though by no means the only crucially relies on this exact linearity of the rates. place where one should look for interesting be- But many results, like the existence of a critical havior). value and positive correlations, should hold for Empirical Studies more general rate functions. Likewise, the dy- A frequent objective of empirical studies is to de- namics are crucial in the proof of spatial segrega- scribe and explain spatial patterns and to deter- tion of the two competing species in the Lotka- mine the mechanisms that underlie the observed Volterra model in one and two spatial dimensions patterns. A mathematical model can help identify when interspecific competition becomes more im- such mechanisms, but caution is in order since dif- portant relative to intraspecific competition, but ferent mechanisms can result in similar patterns. the same phenomenon should occur for other rate Building models that distinguish between the dif- functions as well. Understanding this type of ro- ferent mechanisms is necessary to guide empiri- bustness is essential if we want to make general cal studies to determine which mechanisms likely statements. cause the observed patterns. This is a challenging Robustness is needed also with respect to the task that requires both the creation and the analy- modeling framework. For instance, deterministic sis of new models. Only a thorough understand- spatial models often exhibit complex patterns that ing of the theoretical properties of these models are not robust with respect to stochastic pertur- will inform empirical studies. bations. Though these patterns are intriguing, and Currently, most spatial studies focus on just one their study has given rise to a whole field that species. In the future there will be an increased studies emerging phenomena, it is doubtful need for multispecies spatial models that can be whether patterns that rely on specific determinis- parameterized by field data. It will thus be neces- tic dynamics and that are not robust to stochastic sary to develop both robust models that can be pa- perturbations are relevant to natural systems where rameterized and statistical methods that will allow environmental and demographic stochasticity are the analysis of data from multispecies studies. important factors. Finally, close collaboration between mathe- Tools maticians and experimental biologists is needed for The development of analytical tools that are ac- mathematical studies to have an impact on the un- cessible to biologists is of high importance. Such derstanding of how ecosystems function. This poses
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a different kind of challenge: communication be- tween the two groups. To effectively communicate, About the Cover biologists and mathematicians need to speak a This month’s cover is associated with the ar- common language. This typically means that math- ticle in this issue by Claudia Neuhauser on spa- ematicians need to make an effort to translate tial models. It is based on data calculated by their results into plain English and to explain why Linda Buttel and Rick Durrett of Cornell Uni- they might be relevant to a biologist, and biologists versity. Durrett, who has been a collaborator need to make an effort to explain their terminol- of Neuhauser, writes, “It shows the behaviour ogy and to learn to identify situations in which of a model of the spread of a disease like mathematics can be useful. measles that you can only acquire once. Black sites are individuals susceptible to the dis- References ease, yellow sites are infected, and red sites [1] H. G. ANDREWARTHA and L. C. BIRCH, The Distribution are those who have had the disease and are and Abundance of Animals, University of Chicago now immune. This sequence of snapshots in Press, 1954. time illustrates a result proved by Ted Cox and [2] R. DURRETT, Lecture Notes on Particle Systems and Per- me in 1988—when the epidemic does not die colation, Wadsworth, Pacific Grove, CA, 1988. out, it expands linearly and has an asymp- [3] ——— , Ten Lectures on particle systems, Lectures on Probability Theory (Saint-Flour, 1993) , Lecture Notes totic shape, which is a convex set with the in Mathematics, vol. 1608, Springer, Berlin, 1995. same point symmetries as the square lattice.” [4] R. DURRETT and C. NEUHAUSER, Particle systems and re- Other images like this one can be found in action-diffusion equations, Ann. of Prob. 22 (1994), Durrett’s article “Stochastic spatial models” in 289–333. volume 41 of the SIAM Review (1999), pages [5] I. HANSKI, Metapopulation Ecology, Oxford Univer- 677–718. sity Press, 1999. One of the papers by Durrett and Neuhauser [6] I. HANSKI and O. OVASKAINEN, The metapopulation ca- is “Epidemics with recovery in D =2”, Ann. pacity of a fragmented landscape, Nature 404 (2000), Appl. Probab. 1 (1991), pages 189–206. 755–758. [7] C. B. HUFFAKER, Experimental studies on predation: —Bill Casselman ([email protected]) Dispersion factors and predator-prey oscillations, Hilgardia 27 (1958), 343–383. [8] R. LEVINS, Some demographic and genetic conse- quences of environmental heterogeneity for bio- logical control, Bull. Entomol. Soc. Amer. 15 (1969), 237–240. [9] T. M. LIGGETT, Interacting Particle Systems, Springer, New York, 1985. [10] C. NEUHAUSER, Habitat destruction and competitive co- existence in spatially explicit models with local in- teractions, J. Theoret. Biol. 193 (1998), 445–463. [11] ——— , The role of explicit space in plant competi- tion models, Perplexing Problems in Probability, Birkhäuser, Boston, 1999, pp. 355–372. [12] C. NEUHAUSER and S. W. PACALA, An explicitly spatial version of the Lotka-Volterra model with interspe- cific competition, Ann. Appl. Probab. 9 (1999), 1226–1259. [13] S. W. PACALA and M. REES, Field experiments that test alternative hypotheses explaining successional di- versity, Amer. Naturalist 152 (1998), 729–737. [14] D. TILMAN, Resource Competition and Community Structure, Princeton University Press, 1982. [15] D. TILMAN and P. KAREIVA, Spatial Ecology, Princeton University Press, 1997.
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