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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