The limiting behaviour of a stochastic patch occupancy model

Theoretical Population Biology

Chan

2016

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ABSTRACT. We study a variant of Hanski's incidence function model that allows habitat patch characteristics to vary over time following a Markov process. The widely studied case where patches are classified as either suitable or unsuitable is included as a special case. For large metapopulations, we determine a recursion for the probability that a given habitat patch is occupied. This recursion enables us to clarify the role of landscape dynamics in the survival of a metapopulation. In particular, we show that landscape dynamics affects the persistence and equilibrium level of the metapopulation primarily through its effect on the distribution of a local population's life span. IntroductionA metapopulation is a collection of local populations of a single focal species occupying spatially distinct habitat patches. Much of the research on metapopulations has focussed on identifying and quantifying extinction risks, with mathematical modelling playing an important role. Levins [33] proposed the first model of a metapopulation, which, despite its many simplifying assumptions, provided a number of important insights [21]. The importance of spatial features such as landscape heterogeneity and patch connectivity to metapopulation persistence was demonstrated in subsequent research [22,50,45,24] and by connections to interacting particle systems [34,17,18]. Of particular relevance to the current work is the Incidence Function Model (IFM) [22], which relates the colonisation and local extinction probabilities to landscape characteristics. This model has been used which breeds only in burned trees [53]. In these examples, the landscape dynamics are driven by secondary succession, and this is often the case regardless of whether the focal species is part of a seral community or the climax community.Some authors [9,63,28] have attempted to deal with landscape dynamics by incorporating the time elapsed since the patch was created through the local extinction probability. A more widely used and studied approach incorporates landscape dynamics by allowing each patch to alternate between being suitable or unsuitable for supporting a local population. In its simplest form all patches are treated equally [31,55,65,54]. A more general form used by DeWoody et al. [14] and Xu et al. [66] incorporates differences between patches in area and extinction rates. These studies demonstrate an important relationship between the time scale of metapopulation dynamics and landscape dynamics.When the habitat life span is too short, the metapopulation is unable to become established [31,14]. Furthermore, ignoring landscape dynamics leads to inaccurate persistence criteria. In general, the persistence criteria is optimistic [55,66], though not necessarily for species which are able to react to habitat destruction [54].The main problem we see with the suitable/unsuitable classification is that it is too coarse. Treating patches as being unsuitable may be reasonable following a destructive event, but this approach is unable to handle typical en...

Section: Introductionmentioning

confidence: 58%

“…This type of phase structure has previously been used in [1,13,25,39]. We note that if observations were instead taken after the colonisation phase, then the model would display the rescue effect [22].…”

Section: Model Descriptionmentioning

confidence: 96%

A metapopulation model with Markovian landscape dynamics

Theoretical Population Biology

Chan

2016

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“…In other words, we conjecture that this metapopulation will persist for a long time if r(A) > 1 but will go extinct quickly if r(A) ≤ 1, a property called the extinction threshold. In support of this conjecture, note that if the extinction and colonisation events were assumed to occur in distinct alternating phases as in [18,20], then stability of the fixed points could be established using similar arguments to those used to prove Theorem 3 in [20].…”

Section: 1] Satisfy Assumption (H)mentioning

confidence: 78%

The Limiting Behaviour of Hanski's Incidence Function Metapopulation Model

2014

J. Appl. Probab.

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Hanski's incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.

“…More generally, the presence-absence assumption has simplified modelling, data collection and analysis for a number of metapopulations [7,8,9,10,11,12,13,14]. However, this assumption is not always adequate, for example in stock dynamics where more detail is required [15].…”

Section: Introductionmentioning

confidence: 99%

A model for a spatially structured metapopulation accounting for within patch dynamics

Smith

Mathematical Biosciences

2014

We develop a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics. Using a deterministic approximation derived from a functional law of large numbers, we develop conditions for extinction and persistence of the metapopulation in terms of the birth, death and migration parameters. Interestingly, we observe the Allee effect in a metapopulation comprising two patches of greatly different sizes, despite there being decreasing patch specific per-capita birth rates. We show that the Allee effect is due to way the migration rates depend on the population density of the patches.