On moment closures for population dynamics in continuous space

Murrell, David J.; Dieckmann, Ulf; Law, Richard

doi:10.1016/j.jtbi.2004.04.013

Cited by 156 publications

(169 citation statements)

References 31 publications

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“…Further, our methods enable quantification of uncertainties in resulting parameter estimates, critical information for comparing and integrating results of multiple studies. An important caveat, as mentioned above, is that the approximation used in the moment equation is not uniquely defined and the error may be dependent on the model parameter (Murrell et al 2004), a clear drawback for inferential analysis that should be more carefully addressed in future work.…”

Section: Discussionmentioning

confidence: 99%

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Fitting Ecological Process Models to Spatial Patterns Using Scalewise Variances and Moment Equations

Detto

Muller‐Landau

2013

The American Naturalist

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. 7, 2012; Accepted October 29, 2012; Electronically published February 19, 2013 Online enhancement: zip file with appendixes, supplemental figure. abstract: Ecological spatial patterns are structured by a multiplicity of processes acting over a wide range of scales. We propose a new method, based on the scalewise variance-that is, the variance as a function of spatial scale, calculated here with wavelet kernel functions-to disentangle the signature of processes that act at different and similar scales on observed spatial patterns. We derive exact and approximate analytical solutions for the expected scalewise variance under different individual-based, spatially explicit models for sessile organisms (e.g., plants), using moment equations. We further determine the probability distribution of independently observed scalewise variances for a given expectation, including complete spatial randomness. Thus, we provide a new analytical test of the null model of spatial randomness to understand at which scales, if any, the variance departs significantly from randomness. We also derive the likelihood function that is needed to estimate parameters of spatial models and their uncertainties from observed patterns. The methods are demonstrated through numerical examples and case studies of four tropical tree species on Barro Colorado Island, Panama. The methods developed here constitute powerful new tools for investigating effects of ecological processes on spatial point patterns and for statistical inference of process models from spatial patterns.

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Section: Discussionmentioning

confidence: 99%

“…Fourier scale (m) Fourier scale (m) (19) and (20) may fail for moderate overdispersion (Murrell et al 2004) and high density (Raghib et al 2011). Moreover, the simplified model (eq.…”

Section: A) This Model Now Involves All the Parameters And The Mean mentioning

confidence: 99%

Fitting Ecological Process Models to Spatial Patterns Using Scalewise Variances and Moment Equations

Detto

Muller‐Landau

2013

The American Naturalist

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“…Unfortunately, spatial moment equations are mathematically intractable, because they form an infinite hierarchy with each moment depending on higher-order moments. Consequently, an extensive body of literature (11,19) has focused on developing moment closures to approximate the dynamics of second spatial moments, such as the term R(t), or more generally the function g (⌬x, t, ⌬t), by which we denote the covariance between two samples of the population density, taken at times t and t ϩ ⌬t with a spatial lag ⌬x (see Methods). Although moment closure methods can be used to supplement simulation studies, they rely fundamentally on the ultimately arbitrary choice of the closure.…”

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confidence: 99%

Space and stochasticity in population dynamics

Ovaskainen

Cornell

2006

Proc. Natl. Acad. Sci. U.S.A.

106

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Organisms interact with each other mostly over local scales, so the local density experienced by an individual is of greater importance than the mean density in a population. This simple observation poses a tremendous challenge to theoretical ecology, and because nonlinear stochastic and spatial models cannot be solved exactly, much effort has been spent in seeking effective approximations. Several authors have observed that spatial population systems behave like deterministic nonspatial systems if dispersal averages the dynamics over a sufficiently large scale. We exploit this fact to develop an exact series expansion, which allows one to derive approximations of stochastic individual-based models without resorting to heuristic assumptions. Our approach makes it possible to calculate the corrections to mean-field models in the limit where the interaction range is large, and it provides insight into the performance of moment closure methods. With this approach, we demonstrate how the buildup of spatiotemporal correlations slows down the spread of an invasion, prolongs time lags associated with extinction debt, and leads to locally oscillating but globally stable coexistence of a host and a parasite.interaction kernel ͉ perturbation theory ͉ spatial models ͉ stochastic models T he law of mass action, also called the mean-field assumption, was first introduced by Waage and Guldberg in 1864 (1), who related the rates of chemical reactions to the proportional amounts of the reacting substances. Volterra (2) brought the concept to ecology by interpreting Lotka's model (3) of autocatalytic reaction in terms of predator-prey interactions. Since then, most mathematical models in ecology have been derived by using the mean-field assumption, i.e., by replacing local densities by global densities. In recent years, theoretical ecologists have attempted to overcome the assumption by constructing models that adequately capture spatial and stochastic population processes (4-10). One approach is moment closure (11) and the related methods of pair approximations (12, 13) and corrected mean-field models (14). These methods have been very widely used, and they have even provided the basis for advising policy concerning real epidemics (15). However, the assumptions of moment closure are justified by heuristic rather than mathematical arguments. Another approach is to derive stochastic equations by linearizing around a mean-field model (16,17). This method has proved useful for the study of spatial autocorrelations and synchrony but is incapable of studying the impact of fluctuations, e.g., to mean densities. Here we present a previously undescribed theory that combines elements from both approaches and is able to capture the nonlinear effects of fluctuations without resorting to ad hoc assumptions. Our approach makes it possible to calculate exactly the corrections to mean-field models in the limit where the interaction range is large, and it provides insight into the performance of moment closure methods.We develop our theory in t...

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“…In view of this, k (n) μ are also called (factorial) moment functions, cf. e.g., [11]. Recall that 0 -the set of all finite γ ∈ defined in (1.3)-is an element of B( ).…”

Section: Definition 21mentioning

confidence: 99%

Evolution of states in a continuum migration model

Kondratiev

Kozitsky

2017

Anal.Math.Phys.

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The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in R d in which the constituents appear (immigrate) with rate b(x) and disappear, also due to competition. For this model, we prove the existence of the evolution of states μ 0 → μ t such that the moments μ t (N n ), n ∈ N, of the number of entities in compact ⊂ R d remain bounded for all t > 0. Under an additional condition, we prove that the density of entities and the second correlation function remain point-wise bounded globally in time.

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On moment closures for population dynamics in continuous space

Cited by 156 publications

References 31 publications

Fitting Ecological Process Models to Spatial Patterns Using Scalewise Variances and Moment Equations

Fitting Ecological Process Models to Spatial Patterns Using Scalewise Variances and Moment Equations

Space and stochasticity in population dynamics

Evolution of states in a continuum migration model

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