Deterministic Limits to Stochastic Spatial Models of Natural Enemies

Keeling,; Wilson, F. F.; Pacala,

doi:10.2307/3079314

Cited by 3 publications

(5 citation statements)

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“…Among the several theoretical descriptions of population biology systems, the stochastic approach is one capable of revealing the ubiquitous fluctuations observed in these systems [1,2,3,4,5]. Using a stochastic lattice gas model, we study here the dynamics of propagation of an epidemic in a population in which the individuals are separated into three classes determined by their relative states to a given disease: susceptible (S), infected (I) and recovered (R) individuals.…”

Section: Introductionmentioning

confidence: 99%

“…Using a stochastic lattice gas model, we study here the dynamics of propagation of an epidemic in a population in which the individuals are separated into three classes determined by their relative states to a given disease: susceptible (S), infected (I) and recovered (R) individuals. One of the most well known models in this context is the socalled susceptible-infected-recovered (SIR) model [1,2,3,4,5,6,7,8,9,10,11,12,13], a model for an epidemic which occurs during a time interval that is much smaller then the lifetime of the host. It describes the spreading of an epidemic process occurring in a population initially composed by susceptible individuals that become infected by contact with infected individuals.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

Souza

Tomé

2010

Physica A: Statistical Mechanics and its Applications

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We study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed by individuals of three classes: susceptible (S), infected (I), and recovered (R). The stochastic process is defined by local rules and involves the following cyclic process: S→I→R→S (SIRS). The open process S→I→R (SIR) is studied as a particular case of the SIRSprocess. The epidemic process is analyzed at different levels of description: by a stochastic lattice gas model and by a birth and death process. By means of Monte Carlo simulations and dynamical mean-field approximations we show that the SIRS stochastic lattice gas model exhibit a line of critical points separating two phases: an absorbing phase where the lattice is completely full of S individuals and an active phase where S, I and R individuals coexist, which may or may not present population cycles. The critical line, that corresponds to the onset of epidemic spreading, is shown to belong in the directed percolation universality class. By considering the birth and death process we analyse the role of noise in stabilizing the oscillations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

Souza

Tomé

2010

Physica A: Statistical Mechanics and its Applications

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“…that movement rates are of the same magnitude as vital rates. Keeling et al (2002) formulated moment equation models for systems with limited movement using a framework similar to ours. They assumed independent movement and found that the system is stabilized if predators move approximately six times faster than the prey.…”

Section: Discussionmentioning

confidence: 99%

“…For instance, Keeling et al (2002) evaluated the dynamical effects of limited movement rates by decreasing these rates. However, this violates the approximation and at some point, the model will break down.…”

Section: Discussionmentioning

confidence: 99%

Population-level consequences of heterospecific density-dependent movements in predator–prey systems

Sjödin

Brännström

Söderquist

et al. 2014

Journal of Theoretical Biology

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In this paper we elucidate how small-scale movements, such as those associated with searching for food and avoiding predators, affect the stability of predator-prey dynamics. We investigate an individual-based Lotka-Volterra model with density dependent movement, in which the predator and prey populations live in a very large number of coupled patches. The rates at which individuals leave patches depend on the local densities of heterospecifics, giving rise to one reaction norm for each of the two species. Movement rates are assumed to be much faster than demographics rates. A spatial structure of predators and prey emerges which affects the global population dynamics. We derive a criterion which reveals how demographic stability depends on the relationships between the per capita covariance and densities of predators and prey. Specifically, we establish that a positive relationship with prey density and a negative relationship with predator density tend to be stabilizing. On a more mechanistic level we show how these relationships are linked to the movement reaction norms of predators and prey. Numerical results show that these findings hold both for local and global movements, i.e., both when migration is biased towards neighboring patches and when all patches are reached with equal probability.

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“…This led to the development of numerous methods, ranging from simple mean-field approximations to moment closure techniques, for simplifying and analysing individualbased models (Durrett and Levin, 1994a;Rand, 1999;Keeling et al, 2002). In the simplest of individual-based models-where although there are discrete, separate interaction sites there is no spatial structure and interactions are simple births and deaths-a relationship can be shown between the individual-based model and an appropriately chosen deterministic dynamical system.…”

Section: Introductionmentioning

confidence: 99%

From local interactions to population dynamics in site-based models of ecology

Johansson

Sumpter

2003

Theoretical Population Biology

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A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation. r

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Deterministic Limits to Stochastic Spatial Models of Natural Enemies

Cited by 3 publications

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Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

Population-level consequences of heterospecific density-dependent movements in predator–prey systems

From local interactions to population dynamics in site-based models of ecology

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