Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

Bertacchi, Daniela; Lanchier, Nicolas; Zucca, Fabio

doi:10.1214/10-aap734

Cited by 22 publications

(38 citation statements)

References 36 publications

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“…In either case, the theorem shows that the survival probability decreases to zero as the dispersal range increases to infinity, indicating that long range dispersal promotes extinction of metapopulations subject to a strong Allee effect caused by sexual reproduction. In particular, the effects of dispersal are somewhat opposite for the process with and without sexual reproduction since, as proved in [2], in the presence of long range dispersal, the basic contact process approaches a branching process with critical values for survival significantly smaller than that of the process with nearest neighbor interactions. The paper is laid out as follows.…”

Section: Model Description and Main Resultsmentioning

confidence: 99%

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Survival and extinction results for a patch model with sexual reproduction

Foxall

Lanchier

2016

J. Math. Biol.

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We consider a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we identify a mean-field limit consisting of an infinite set of coupled differential equations. For the mean-field equations, we find explicit conditions for survival and extinction that we call expansion and retreat. Using duality techniques to compare the stochastic model to its mean-field limit, we find that expansion and retreat are also precisely the conditions needed to ensure survival and extinction of the stochastic model when the patch size is large. In addition, we study the dependence of survival on the dispersal range. We find that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to spread in space is to use intermediate dispersal ranges.

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Section: Model Description and Main Resultsmentioning

confidence: 99%

“…• For each (x, w) ∈ I t , at rate a, add to I t the pair of points (x, w 1 ) and (x, w 2 ) where w 1 , w 2 are independent Uniform (0, 1) random variables.…”

Section: The Dual Process Of the Mean-field Equationsmentioning

confidence: 99%

Survival and extinction results for a patch model with sexual reproduction

Foxall

Lanchier

2016

J. Math. Biol.

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“…Nevertheless it is not difficult to provide some partial answers in the case of delay or anticipation of the disturbance. 1.λ (2) µ <λ (1) µ provided that P(t 2 < D 2 < t 1 ) is sufficiently small.…”

Section: Climate Changesmentioning

confidence: 98%

“…implies that a consistent delay of the disturbance reduces the average fitness of the population; on the other hand, Proposition 3.8(2) implies that the average fitness of the population increases if the disturbance arrives so early that most of the population has not yet arrived when the danger is over.Let us discuss now the consequences of a change of p. In the previous sections we showed that p → λ(a, p) is decreasing. This means that if p decreases and there is an instantaneous adaptation of the population to the new conditions (i.e.…”

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confidence: 99%

The timing of life history events in the presence of soft disturbances

Bertacchi

Zucca

Ambrosini

2016

Journal of Theoretical Biology

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We study a model for the evolutionarily stable strategy (ESS) used by biological populations for choosing the time of life-history events, such as migration and breeding. In our model we accounted for both intra-species competition (early individuals have a competitive advantage) and a disturbance which strikes at a random time, killing a fraction 1 − p of the population. Disturbances include spells of bad weather, such as freezing or heavily raining days. It has been shown in [23], that when p = 0, then the ESS is a mixed strategy, where individuals wait for a certain time and afterwards start arriving (or breeding) every day. We remove the constraint p = 0 and show that if 0 < p < 1 then the ESS still implies a mixed choice of times, but strong competition may lead to a massive arrival at the earliest time possible of a fraction of the population, while the rest will arrive throughout the whole period during which the disturbance may occur. More precisely, given p, there is a threshold for the competition parameter a, above which massive arrivals occur and below which there is a behaviour as in [23]. We study the behaviour of the ESS and of the average fitness of the population, depending on the parameters involved. We also discuss how the population may be affected by climate change, in two respects: first, how the ESS should change under the new climate and whether this change implies an increase of the average fitness; second, which is the impact of the new climate on a population that still follows the old strategy. We show that, at least under some conditions, extreme weather events imply a temporary decrease of the average fitness (thus an increasing mortality). If the population adapts to the new climate, the survivors may have a larger fitness. * The first two authors contributed equally to this work.

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“…In Figure 2, (1,2,3,4) is an infection path with type (2, 3, 1) at moment t. Please note that an infection path may be with more than one types. For example, in Figure 2, (1,2,3,4) is also with type (1, 3, 1) at moment t.…”

Section: Subcritical Casementioning

confidence: 99%

Contact Processes with Random Recovery Rates and Edge Weights on Complete Graphs

Xue

Pan

2017

J Stat Phys

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In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with n vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order O(log n) with high probability as n → +∞. In the supercritical case, the process survives at a moment with order exp{O(n)} with high probability as n → +∞. Our proof for the subcritical case is inspired by the graphical method introduce in [6]. Our proof for the supercritical case is inspired by approach introduced in [10], which deal with the case where the contact process is with random vertex weights.

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Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

Cited by 22 publications

References 36 publications

Survival and extinction results for a patch model with sexual reproduction

Survival and extinction results for a patch model with sexual reproduction

The timing of life history events in the presence of soft disturbances

Contact Processes with Random Recovery Rates and Edge Weights on Complete Graphs

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