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Published byRegina Morgan Modified over 9 years ago
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Metapopulations Objectives –Determine how e and c parameters influence metapopulation dynamics –Determine how the number of patches in a system affects the probability of local extinction and probability of regional extinction –Compare ‘propagule rain’ vs. ‘internal colonization’ metatpopulation dynamics –Evaluate how the ‘rescue effect’ affects metapopulation dynamics
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Metapopulations There are relatively few examples where the entire population resides within a single patch Most species are patchily distributed across space Hence it becomes a population of populations…metapopulations
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Metapopulations Metapopulation theory (Levin 1969, 1970) describes a network of patches, some occupied some not, where subpopulations are interacting (“winking”) The classic model is then based upon presence-absence, not a demographic model like source-sink dynamics
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Fragmentation & Heterogeneity
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Metapopulations Let’s define extinction and colonization mathematically Extinction p e thus persistence is 1-p e Colonization is p i with vacancy is 1-p i We can consider the fate of a single patch over time or the entire metapopulation over time
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Metapopulations For a given patch, the likelihood of persistence for n years is simply p n = (1 – p e ) n E.g. if a patch has a probability of persistence = 0.8 in a given year, the probability for 3 years = 0.8 3 = 0.512 If we had 100 patches, approximately 52 would persist and 48 would go extinct
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Metapopulations To consider the fate of the entire metapopulation (i.e. the probability of extinction of the entire population) If all patches have the same probability of extinction, it is simply p e x For example, if p e =0.5 across 6 patches then P x = 1-(p e ) x or 0.5 6 = 0.0156 or 1.5%
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Metapopulations Now that we have defined e and c, let us consider the basic metapopulation model where f is the fraction of patches occupied in the system (e.g. 5/25 = 0.2) If f is the fraction of patches occupied, then 1-f is the fraction empty, the we can compute I as I = p i ( 1 - f ) df / dt = I -E
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Metapopulations Focusing on E, the rate at which occupied patches go extinct E should depend upon the number of patches occupied as well as the extinction probability (p e ) Substituting our new values of I and E E = p e f df/dt = p i (1 – f) - p e f
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Metapopulations This is called the propagule rain model or an island-mainland model, because the colonization rate does NOT depend on patch occupancy patterns-it is assumed that colonists are available to populate and empty patch and they can come from within or outside the metapopulation df/dt = p i (1 – f) - p e f
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Metapopulations At equilibrium the fraction of patches is constant, although the exact combination is dynamic The equilibrium fraction can be derived by setting the ‘rate’ to 0 dt/dt = 0 = p i – p i f –p e f and then f = p i / (p i + p e )
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Metapopulations There are many important assumptions (as with any model), the most important being all patches are created equal; p e and p i are constant over time and apply to patches irrespective of population size and finally, spatial arrange and proximity are not important to p e or p i You probably can imagine how when f is high, p i is probably large
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Metapopulations This type of model is called the internal colonization model because colonization rates depend on current status (f) of the metapopulation system Extinction of a patch may depend on the fraction of patches occupied in the system When f is high, there are many potential colonists and p e decreases This is termed the rescue effect
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Metapopulations Rescue effect and Internal Colonization Model
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Metapopulations Objectives –Determine how e and c parameters influence metapopulation dynamics –Determine how the number of patches in a system affects the probability of local extinction and probability of regional extinction –Compare ‘propagule rain’ vs. ‘internal colonization’ metatpopulation dynamics –Evaluate how the ‘rescue effect’ affects metapopulation dynamics
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