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Metapopulations or “Populations of populations” (Levins 1970)

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##### Metapopulations or “Populations of populations” (Levins 1970)

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**Metapopulations or “Populations of populations”(Levins**1970) pop pop pop pop pop pop pop**Assumptions**• Homogeneous patches • No spatial structure • No time lags • Constant c and e • Amount of colonization and extinction is a function of P • Large number of patches (essentially ∞) • No internal dynamics**Equlibria**Internal solution Equilibria Population change (D P) P**General Implications of the Levins’ Model**• If e>c then P = 0 • If c>e then there is a stable internal equilibrium point**Growth of a newly colonized metapopulation**For a very small value of P, the term cP2 in can be ignored leading to an early growth rate of “intrinsic rate of increase” This looks like exponential growth!**With increasing occupation of patches**“carrying capacity” This looks like logistic growth!**Impact of Change in Model Structure**Fewer Patches Smaller Patches rate P P**Mainland-Island Metapopulations**Propagule rain if cm>>c**Mainland-Island Model Equilibrium**Note: the multispecies extension forms the basis of the theory of island biogeography.**The Rescue Effect**“Reduction in the probability of extinction when more population sites are occupied”(Gotelli 1995) eP e(1-P)P Extinction rate**Mainland-Island Model with Rescue Effect**equilibrium: If cm > e then vs**Levins’ Model with Rescue Effect**equilibria: (1) If c> e then (2) If c < e then (3) If c = e then neutral equilibrium**Impact of Habitat Loss**(1- h = 0.25) Let 1-h = represent the fraction of habitat destroyed**Levins’ Model Incorporating Habitat Loss**Equilibrium: Metapopulation extinct is the fraction of occupied patches at steady state**The Levins’ Rule**“A sufficient condition for meta-population survival is that the number of patches remaining after habitat destruction exceeds the number of empty, but suitable, patches prior to patch destruction” (Hanski et al. 1996)**Problems with the Levins’ Rule**vis-á-vis the Rescue Effect For high migration rates where Fis the probability of a subpopulation surviving for one time interval and G is the probability of recolonization when all patches are occupied**Problems with the Levins’ Rule**vis-á-vis the Rescue Effect For high migration rates**Problems with the Levins’ Rule**vis-á-vis the Rescue Effect “A sufficient condition for meta-population survival is that the number of patches remaining after habitat destruction exceeds the number of empty, but suitable, patches prior to patch destruction” (Hanski et al. 1996) e/c**Evidence for a Rescue Effect**With rescue effect No rescue effect**Metapopulation Persistence as a Function of Patch Number**From a stochastic version of the Levins’ model TM expected time to metapopulation extinction TL expected time to local extinction H number of suitable patches (Gurney and Nisbet 1978)**Minimum Viable Metapopulation Size**If we define long term persistence to mean then insures long term persistence, where H is the number of available patches. (Gurney and Nisbet 1978)**Minimum Viable Metapopulation Size**Island type migration Distant dependent migration**Structured Metapopulation Models**Or Internal Dynamics Matter**Structured Metapopulation Models**Structured Model Unstructured Model • Slow local dynamics • Fast local dynamics • Higher migration rates • Low migration rates (1) Local dynamics can be ignored (2) Focus is on extinction rates (1) Migration rates may impact local dynamics (2) Local dynamics cannot be ignored**A “Simple” Structured Model**• An extension of the basic Levins’ model • Equilibrium state depends on emigration (m) and extinction (e) rates • All populations have the same internal dynamics • characterized by logistic growth:**A “Simple” Structured Model**Levins’ Model Strucutured Model a’ = fraction of migrants surviving and landing in a patch b = rate of successful colonization (Hanski and Zhang 1993)**A “Simple” Structured Model**Equilibrium conditions Yuuuch! B=4e/bK**A “Simple” Structured Model**Graphical interpretation no stable equilibrium point 2 stable equilibrium points – one positive 1 stable equilibrium point r = 1 a’ = 0.5 bK = 1**A “Simple” Structured Model**Graphical interpretation r = 1 a’ = 0.5 bK = 1**Spatially Explicit Metapopulation Models**Area effects**Spatially Explicit Metapopulation Models**Distance effects**A Spatially Explicit Metapopulation Model**in Continuous Time Rates are patch specific (Hanski and Gyllenberg 1997)**Expected Equilibrium**Probabilities of Patch Occupation where ei = 1/Ai , Ai is the area of patch i, and (Hanski and Gyllenberg 1997)**A Closer Look at the Function Ci(t)**Ci(t)is the amount of colonization into patch i at time t from all other patches j in the metapopulation c = colonization rate a = distance decay rate for migration dij = distance between patches i and j R = the number of patches in the metapopulation (Hanski and Gyllenberg 1997)**Solving for**Begin by substituting into yielding (Hanski and Gyllenberg 1997)**Solving for**is solved by interation to obtain values for allowing us to determine (Hanski and Gyllenberg 1997)**The Incidence Function Model of Hanski**The incidence is the long term probability that patch i will be occupied. This is a discrete time stochastic model similar in design to the previous, continuous time model. However, it is more flexible in how spatial interactions are entered into the model.**The Incidence Function Model of Hanski**predicted observed