Connecting deterministic and stochastic metapopulation models

Barbour, A. D.; McVinish, Ross; Pollett, P. K.

doi:10.1007/s00285-015-0865-4

Cited by 9 publications

(11 citation statements)

References 29 publications

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“…In previous work [2], we have demonstrated that deterministic metapopulation models provide good approximations to their stochastic counterparts, at least over finite time horizons, provided that the colonization pressure at a patch results from the sum of the effects of many other patches -this is equivalent to the assumption in Ovaskainen and Cornell [12] that the colonization kernel has long range. However, if the landscape is not uniform, even the equilibrium state of the deterministic system is unknown.…”

Section: Introductionmentioning

confidence: 94%

“…However, in many circumstances, the processes may remain for long periods in an apparent stochastic equilibrium, a quasi-stationary distribution. In [2], it is shown that the stochastic processes can be well approximated by corresponding deterministic systems, at least over bounded time intervals. Thus, if the stochastic processes have initial conditions corresponding to any equilibrium of the deterministic systems, they remain close to this equilibrium over bounded time intervals, with asymptotically high probability.…”

Section: The Equilibrium Of a Metapopulationmentioning

confidence: 99%

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Local approximation of a metapopulation’s equilibrium

2018

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We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset [Formula: see text] of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to [Formula: see text], the equilibrium occupation probability in Levins's model, at any point [Formula: see text] not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.

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Section: Introductionmentioning

confidence: 94%

Section: The Equilibrium Of a Metapopulationmentioning

confidence: 99%

Local approximation of a metapopulation’s equilibrium

2018

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“…Demonstrating this behaviour usually involves showing a law of large numbers holds so that the transition rates of the individual particles are well approximated by some deterministic process. A propagation of chaos result was established for the occupancy process in [3,9], where the independent site approximation was coupled to the occupancy process and the two processes shown to be close over finite time intervals.…”

Section: Propagation Of Chaos and Stochastic Orderingmentioning

confidence: 99%

Dominating occupancy processes by the independent site approximation

McVinish¹

2021

Preprint

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Occupancy processes are a broad class of discrete time Markov chains on {0, 1} n encompassing models from diverse areas. This model is compared to a collection of n independent Markov chains on {0, 1}, which we call the independent site model.We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.

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“…The form of (2) suggests that it will be important to understand the evolution of linear functionals of the state. Under certain conditions, the spin system is well approximated by a deterministic system ρ(t) = (ρ i (t)) n i=1 in the sense that sup φ∈Φ | n i=1 φ i (η i (t) − ρ i (t))| is small for sufficiently regular Φ ⊂ R n [4]. In particular, this deterministic system is given by the solution to…”

Section: The Basic Modelmentioning

confidence: 99%

“…To enable the application of standard techniques, as in [4,23], we compare η(t) with an independent site approximation ω(t) with transition rates…”

Section: Appendix A: Independent Site Approximationmentioning

confidence: 99%

Fast approximate simulation of finite long-range spin systems

McVinish¹,

Hodgkinson²

2019

Preprint

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Tau leaping is a popular method for performing fast approximate simulation of certain continuous time Markov chain models typically found in chemistry and biochemistry. This method is known to perform well when the transition rates satisfy some form of scaling behaviour. In a similar spirit to tau leaping, we propose a new method for approximate simulation of spin systems which approximates the evolution of spin at each site between sampling epochs as an independent two-state Markov chain. When combined with fast summation methods, our method offers considerable improvement in speed over the standard Doob-Gillespie algorithm. We provide a detailed analysis of the error incurred for both the number of sites incorrectly labelled and for linear functions of the state.

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Connecting deterministic and stochastic metapopulation models

Cited by 9 publications

References 29 publications

Local approximation of a metapopulation’s equilibrium

Local approximation of a metapopulation’s equilibrium

Dominating occupancy processes by the independent site approximation

Fast approximate simulation of finite long-range spin systems

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