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A simulated mainland-island metapopulation with 25 islands and γ = 0. The solid line is d t /κ and the dashed line is the proportion of occupied islands at time t. The step functions at the bottom of the figure indicates the state of the mainland X n 0,t (dashed line) and the state of the limiting mainland X 0,t (solid line).
Source publication
Stochastic patch occupancy models (SPOMs) are a class of discrete time Markov chains used to model the presence/absence of a population in a collection of habitat patches. This class of model is popular with ecologists due to its ability to incorporate important factors of the habitat patch network such as connectivity and distance between patches...
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Context 1
... the simulation given in Figure 1, we considered a metapopulation comprising 25 islands and set γ = 0, f (x) = x, κ i = 0.1, λ i = 0.4, s i = 0.9 for i = 1, . . . , 25 and s 0 = 0.9. ...
Similar publications
Assessing causes of population decline is critically important to management of threatened species. Stochastic patch occupancy models (SPOMs) are popular tools for examining spatial and temporal dynamics of populations when presence–absence data in multiple habitat patches are available. We developed a Bayesian Markov chain method that extends exis...
Citations
... For systems whose general dynamics strongly depend upon the state of a single node, α t will be large, and so there is little hope in expecting the deterministic process (1.2) to be representative. This is the case for the mainland-island metapopulation model considered in [38], which is instead well-approximated by a semi-deterministic system. Moreover, γ t , Γ t , and δ t provide some essential measures of the regularity of the global rule. ...
We study normal approximations for a class of discrete-time occupancy processes, namely, Markov chains with transition kernels of product Bernoulli form. This class encompasses numerous models which appear in the complex networks literature, including stochastic patch occupancy models in ecology, network models in epidemiology, and a variety of dynamic random graph models. Bounds on the rate of convergence for a central limit theorem are obtained using Stein’s method and moment inequalities on the deviation from an analogous deterministic model. As a consequence, our work also implies a uniform law of large numbers for a subclass of these processes.
... However, it is unclear a priori whether the dynamics of (1.2) and its corresponding occupancy process X t are indeed similar, or in what sense; the extension of the global rule is not unique, P(X i,t = 1) rarely equates with p i,t , and, in terms of total variation at least, p t is never a good approximation of X t (see §2.3 of [4]). In fact, there are occupancy processes whose deterministic approximation is not at all representative of its stochastic counterpart; the mainland-island metapopulation model studied in [33] constitutes such an example. Our first objective here is to discern when the macroscopic dynamics of the deterministic system reflect those of the stochastic model in some reasonable sense. ...
... Perhaps unsurprisingly, for systems whose general dynamics strongly depend upon the state of a single node, α t will be large, and so there is little hope in expecting the deterministic process (1.2) to be representative. This is the case for the mainland-island metapopulation model previously studied by the authors [33], which is instead well-approximated by a semi-deterministic system. The importance of β t → 0 becomes clearer if we restrict our attention to homogeneous systems in the sense that for each i there is a permutation σ on {1, . . . ...
We study deterministic and normal approximations for a class of discrete-time occupancy processes, namely, Markov chains with transition kernels of product Bernoulli form. This class encompasses numerous models which appear in the complex networks literature, including stochastic patch occupancy models in ecology, network models in epidemiology, and a variety of dynamic random graph models. Moment inequalities on the deviation from an analogous deterministic model are presented, alongside bounds on the rate of convergence in law to a normal approximation under the Wasserstein and Kolmogorov metrics, which are obtained using Stein's method. We are able to identify a subclass of weakly interacting processes that exhibit approximate Gaussian behaviour, and we develop a general strong law of large numbers and a central limit theorem for these processes. Applications to epidemic network models, metapopulation models, and convergence of dynamic random graphs, are discussed.
... However, due to its coupled nonlinearity, deriving the exact solution of the basic in-host model (even in the non-delay and constant coefficients case) remains a long-standing open research problem. On the other hand, Markov chain models have proven to be useful in exploring many biological stochastic processes ranging from disease transmission and metapopulation evolution to gene induction [16][17][18][19][20]. From a dynamical system perspective, analysis of such models means to consider the Kolmogorov (master) equations, i.e., a system of linear ODEs, which characterize the evolution of the probability of the process being in a given state at a given time. ...
We present the analysis of a time-inhomogeneous Markov chain model based on the in vivo viral infection dynamics. The exact solution of a general in-host model with time-dependent rates is obtained by using the Lie-theoretic approach. The results provide both an improvement in numerical efficiency and the potential for analytical solution of other biological processes without clear symmetry.
... In order to address this question, we study the limiting behaviour of the metapopulation when the number of patches is large. Using an analysis similar to our earlier work [38,39,41], we show large metapopulations display a deterministic limit and asymptotic independence of local populations. All proofs are given in the Appendix. ...
We study a variant of Hanski's incidence function model that accounts for the
evolution over time of landscape characteristics which affect the persistence
of local populations. In particular, we allow the probability of local
extinction to evolve according to a Markov chain. This covers the widely
studied case where patches are classified as being either suitable or
unsuitable for occupancy. Threshold conditions for persistence of the
population are obtained using an approximating deterministic model that is
realized in the limit as the number of patches becomes large.
... If one of more patches were to remain large, then we would expect the approximation to be poor. An example of the type of behaviour to be expected in this case is given in McVinish and Pollett [16]. ...
In this paper, we study the relationship between certain stochastic and
deterministic versions of Hanski's incidence function model and the spatially
realistic Levins model. We show that the stochastic version can be well
approximated in a certain sense by the deterministic version when the number of
habitat patches is large, provided that the presence or absence of individuals
in a given patch is influenced by a large number of other patches. Explicit
bounds on the deviation between the stochastic and deterministic models are
given.
... The approach taken in this paper continues the line of investigation started in [23,24]. Our analysis begins with a SPOM similar to the incidence function model of [17] as the SPOM of interest. ...
... We note that the approach used in [23,24] to study the limiting behaviour of similar metapopulation models can not be applied here since the colonisation probability for patch i is a nonlinear function of λ i . A different approach is required. ...
... Proof The sequence of Markov chains {X n } ∞ n=1 can be constructed on the same probability space as in the final step of the proof of Theorem 1 in [24]. For this we need an infinite array of independent standard uniform random variables U j,t , j = 1, 2 . . . ...
Metapopulation models have been used to better understand the conditions necessary for the persistence of the metapopulation. In this paper, we study a stochastic patch occupancy model that incorporates variation in quality and connectivity of the habitat patches. Two important assumptions are imposed in our analysis. Firstly, the distance between patches has a special form. This amounts to assuming that migrating individuals follow certain pathways. Secondly, the area of the habitat patches is assumed to scale with the number of patches in the metapopulation. Under these assumptions, a deterministic limit is obtained as the number of patches goes to infinity. Using the deterministic limiting process, a condition for persistence of the metapopulation is derived.
Populations often exist, either by choice or by external pressure, in a fragmented way, referred to as a metapopulation. Typically, the dynamics accounted for within metapopulation models are assumed to be static. For example, patch occupancy models often assume that the colonisation and extinction rates do not change, while spatially structured models often assume that the rates of births, deaths and migrations do not depend on time. While some progress has been made when these dynamics are changing deterministically, less is known when the changes are stochastic. It can be quite common that the environment a population inhabits determines how these dynamics change over time. Changes to this environment can have a large impact on the survival probability of a population and such changes will often be stochastic. The typical metapopulation model allows for catastrophes that could eradicate most, if not all, individuals on an entire patch. It is this type of phenomenon that this article addresses. A Markov process is developed that models the number of individuals on each patch within a metapopulation. An approximation for the original model is presented in the form of a piecewise-deterministic Markov process and the approximation is analysed to present conditions for extinction.
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.
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.
