Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems

Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by “moment closure,” neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.     

Intraguild predation with spatially structured interactions

Consumer–resource interactions with intraguild predation (IGP) were studied in a spatial setting (i.e., predators catch prey and individuals reproduce within local neighborhoods only). Pair approximation (a method for deriving ordinary differential equations that approximate the dynamics of a community that interacts in a lattice environment) was used to study the effect of spatially structured species interactions. An individual-based computer simulation was used to extend the study to a case with spatially variable resource densities. The qualitative results of the pair approximation model were similar to those of the corresponding non-spatial model. However, the spatial model predicted coex((istence over a wider range of parameters than the non-spatial model when intraguild prey are nutritionally valuable to intraguild predators. Spatially heterogeneous resource distributions and spatially structured interaction could overturn the qualitative predictions of non-spatial models.     

Dynamics of Epidemiological Models

We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.     

Analytical methods for predicting the behaviour of population models with general spatial interactions

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.     

A versatile ODE approximation to a network model for the spread of sexually transmitted diseases

 We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1) to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2) to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate. Received: 26 August 2001 / Revised version: 15 March 2002 / Published online: 23 August 2002 C.T.B. was supported by the NSF. Key words or phrases: Moment closure approximation – Network model – Pair approximation – Sexually transmitted diseases – Steady/casual partnership network     

Relatedness in spatially structured populations with empty sites: An approach based on spatial moment equations

Taking into account the interplay between spatial ecological dynamics and selection is a major challenge in evolutionary ecology. Although inclusive fitness theory has proven to be a very useful tool to unravel the interactions between spatial genetic structuring and selection, applications of the theory usually rely on simplifying demographic assumptions. In this paper, I attempt to bridge the gap between spatial demographic models and kin selection models by providing a method to compute approximations for relatedness coefficients in a spatial model with empty sites. Using spatial moment equations, I provide an approximation of nearest-neighbour relatedness on random regular networks, and show that this approximation performs much better than the ordinary pair approximation. I discuss the connection between the relatedness coefficients I define and those used in population genetics, and sketch some potential extensions of the theory.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Pair-edge approximation for heterogeneous lattice population models

To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here we evaluate the effectiveness of the pair-edge approximation on a spatially heterogeneous lattice in which some sites are unoccupiable, or "dead". This model has several possible interpretations, including a spatial SIS epidemic model, in which some sites are occupied by immobile host-species individuals while others are empty. We find that, as in the homogeneous model, the pair-edge approximation is significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of "dead" sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveals limitations of pair approximation in the presence of spatial heterogeneity.     

Spread rate for a nonlinear stochastic invasion

Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated using deterministic equations [29]. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic manner). It has been long conjectured [27] that intrinsic stochastic factors associated with interacting individuals can slow the spread of a population or disease, even in a uniform environment. While this conjecture has been borne out by numerical simulations, we are not aware of a thorough analytical investigation. In this paper we analyze the effect of intrinsic stochastic factors when individuals interact locally over small neighborhoods. We formulate a set of equations describing the dynamics of spatial moments of the population. Although the full equations cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower bounds for the expected density of individuals. Analysis of the upper solution gives a bound on the rate of spread of the stochastic invasion process which lies strictly below the rate of spread for the deterministic model. The slow spread is most evident when invaders occur in widely spaced high density foci. In this case spatial correlations between individuals mean that density dependent effects are significant even when expected population densities are low. Finally, we propose a heuristic formula for estimating the true rate of spread for the full nonlinear stochastic process based on a scaling argument for moments. Received: 19 October 1998 / Revised version: 1 September 1999 / Published online: 4 October 2000     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Coupled map lattice approximations for spatially explicit individual-based models of ecology

Spatially explicit individual-based models are widely used in ecology but they are often difficult to treat analytically. Despite their intractability they often exhibit clear temporal and spatial patterning. We demonstrate how a spatially explicit individual-based model of scramble competition with local dispersal can be approximated by a stochastic coupled map lattice. The approximation disentangles the deterministic and stochastic element of local interaction and dispersal. We are thus able to understand the individual-based model through a simplified set of equations. In particular, we demonstrate that demographic noise leads to increased stability in the dynamics of locally dispersing single-species populations. The coupled map lattice approximation has general application to a range of spatially explicit individual-based models. It provides a new alternative to current approximation techniques, such as the method of moments and reaction-diffusion approximation, that captures both stochastic effects and large-scale patterning arising in individual-based models.     

A moment closure model for sexually transmitted disease transmission through a concurrent partnership network

A moment closure model of sexually transmitted disease spread through a concurrent partnership network is developed. The model employs pair approximations of higher-order correlations to derive equations of motion in terms of numbers of pairs and singletons. The model is derived from an underlying stochastic process of partnership network formation and disease transmission. The model is analysed numerically; and the final size and time evolution are considered for various levels of concurrency, as measured by the concurrency index kappa3 of Kretzschmar and Morris. Additionally, a new way of calculating R0 for spatial network models is developed. It is found that concurrency significantly increases R0 and the final size of a sexually transmitted disease, with some interesting exceptions.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

Lattice population dynamics for plants with dispersing seeds and Vegetative propagation

The population dynamics of plants in a lattice structured habitat are studied theoretically. Plants are assumed to propagate both by producing seeds that scatter over the population and by vegetative reproduction by extending runners, rhizomes, or roots, to neighboring vacant sites. In addtion, the seed production rate may be dependent on the local density in the neighborhood, indicating beneficial or harmful crowding effects. Two sets of population dynamical equation(s) are derived: one based onmean-field approximation and the other based onpair approximation (tracing both global and local densities simultaneously). We examine the accuracy of these approximate dynamics by comparing them with direct computer simulation of the stochastic lattice model. Pair approximation is much more accurate than mean-field approximation. Mean-field approximation overestimates the parameter range for persistence if crowding effects on seed production are harmful or weakly beneficial, but underestimates it if crowding effects are highly beneficial. Dynamics may show bistability (both population persistence and extinction) if the effect of crowding is strongly beneficial. If there is a linear trade-off between seed production and vegetative reproduction, the equilibrium abundance of the population may be maximised by a mixture of seed production and vegetative reproduction, rather than by pure seed production or by pure vegetative reproduction. This result is correctly predicted by pair approximation but not by mean-field approximation.     

Host spatial heterogeneity and the spread of vector-borne infection.

We analyze how spatial heterogeneity in host density affects the advance of vector-borne disease. Infection requires vector infestation. The vector spreads only between hosts occupying the same neighborhood, and the number of hosts varies randomly among neighborhoods. Simulation of a spatially detailed model shows that increasing heterogeneity in host abundance reduces pathogen prevalence. Clumping of hosts can limit the advance of the vector, which inhibits the spread of infection indirectly. Clumping can also increase the chance that the pathogen and vector become physically separated during the initial phase of the epidemic process. The latter limitation on the pathogen's spread, in our simulations, is restricted to small interaction neighborhoods. A mean-field model, which does not maintain spatial correlations between sites, approximates simulation results when hosts are arrayed uniformly, but overestimates infection prevalence when hosts are aggregated. A pair approximation, which includes some of the simulation model's spatial correlations, better describes the vector infestation frequencies across host spatial dispersions.     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

Allelopathy of bacteria in a lattice population: Competition between colicin-sensitive and colicin-producing strains

We analyse the population dynamics of two strains of bacteria (Escherichia coli): one strain produces a toxin (called colicin) that increases the mortality of a colicin-sensitive strain in the neighbourhood, but does not harm the colicin-producing strain itself. On the other hand, in the absence of colicin in the environment, the colicin-sensitive strain enjoys a higher population growth rate. The model is closely related to the evolutionary dynamics of social interaction. It has been established previously that a perfectly mixing population shows bistability; that is, whichever strain dominates initially tends to defeat the other. On the other hand, empirical and computer simulation results in lattice structured populations show neither co-existence nor bistability of the two strains. In this paper, we analyse the lattice model based on pair approximation (forming a system of ordinary differential equations of global densities and local densities), and by the direct c omputer simulation of the spatial stochastic model. Both the pair approximation dynamics and the computer simulation show that, for most regions of parameter values, one strain defeats the other, irrespective of the initial abundance. However, the pair approximation analysis also suggests a relatively narrow parameter region of bistability, which should disappear when the model is considered on a lattice of infinitely large size. The biological implications of the results and the relationship of the present model with other models of the evolution of spite or altruistic behaviours are discussed. This suggests that the dynamics reflected by the spatially explicit lattice model may be sufficient, perhaps even necessary, to support the evolution of colicin.     

From infanticide to parental care: why spatial structure can help adults be good parents

We investigate the evolution of parental care and cannibalism in a spatially structured population where adults can either help or kill juveniles in their neighborhood. We show that spatial structure can reverse the selective pressures on adult behavior, leading to the evolution of parental care, whereas the nonspatial model predicts that cannibalism is the sole evolutionary outcome. Our analysis emphasizes that evolution of such spatially structured populations is best understood at the level of the cluster of invading mutants, and we define invasion fitness as the growth rate of that cluster. We derive an analytical expression for the selective pressures on the trait and show that relatedness and Hamilton's rule are recovered as emergent properties of the spatial ecological dynamics. When adults can also help other adults, the benefits to each class of recipients are weighted by the class reproductive value, a result consistent with that of other models of kin selection. Finally, we advocate a different approach to moment equations and argue that even though the development of moment closure approximations is a necessary line of research, much-needed ecological and evolutionary insight can be gained by studying the unclosed moment equations.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.