Metapop Equations

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Metapop Equations

  1. 1. Equations Defining Metapopulations
  2. 2. The variables Time: best measured in generations but most convenient for us to measure time in years. Tm=1/emin pe = probability of local Extinction 1" pe = probability of local Persistence pc = probability of local colonization ! x = number of patches Px = probability of regional persistence f = fraction of sites occupied ! i = effect of increasing patch occupancy r = intensity of rescue N e = effective (breeding) population size !
  3. 3. df Levins (1969) equation was basically dt = cf (1" f ) " ef Note that C " E # $ is analogous to G = B " D Bottom line, df = C ! E , and ! dt there are four typical models for estimating C and E: ! ! Extinction Independent Rescue Colonization df df External dt = pc (1" f ) " pe f dt = pc (1" f ) " e(1" f ) f Internal df = if (1" f ) " pe f df = if (1" f ) " e(1" f ) f ! dt ! dt ! !
  4. 4. The method Persistence of one patch over time 1" pe Persistence of one patch over two time periods is: 2 (1" pe )(1" pe ) = (1" pe ) Persistence of one patch over n time periods is: ! n (1" pe ) Persistence of two patches over time is: ! 1" pe1 pe 2 Persistence of many patches over time is: ! x Px = 1" ( pe ) !
  5. 5. Assumptions Patches are homogenous in size, distance from each other, habitat quality, food, CC All patches have same pc and pe over all time periods pc and pe are independent of patch occupation ! ! Instantaneous response to ! No diffusion effect and no spatial structure !
  6. 6. f = fraction of occupied patches (1" f ) = fraction of unoccupied patches C = pc (1" f ) ! rate of colonization in one time period thru immigration. We use it as though it were a probability. C is dependent on patch suitability (area, critical habitat, food, predators, competitors, disease, distance from other occupied patches) & proportion of unoccupied patches. E = pe f rate of extinction in one time period (we must use as though it’s a probability). df dt = pc ( ! f )! pe f 1
  7. 7. One External Source (Propagule Rain) A source that is outside the metapopulation pc is constant df If stable, dt = 0 then solve equation for zero 0 = pc (1" f ) " pe f ˆ pc f = ( pc + pe ) !
  8. 8. Multiple Internal Sources Each occupied internal site produces an excess of propagules that can colonize unoccupied patches i = effect of increasing patch occupancy pc = if because C depends only on patch occp’ncy ! If stable, df dt = 0 then solve equation for zero ! 0 = if (1" f ) " pe f ˆ = 1" pe f i !
  9. 9. Rescue If propagules land in occupied sites, they can "Ne which # pe. If more sites are occupied then more propagules will be available for rescue r = combination of Ne and migration rate pe = r(1" f ) because E depends on breeding pop’n df If stable, dt = 0 then solve equation for zero 0 = pc (1" f ) " r(1" f ) f ˆ = pc f r !
  10. 10. Closed Propagules arise only from w/in the metapop’n & patches rescue each other df If stable, dt = 0 then solve equation for zero 0 = if (1" f ) " r(1" f ) f Oops, can’t solve for f so we must weigh possible results based on likely values of i and r. Barbour & Pugliese ’05 show that there are thresholds, below ! which all solutions indicate total extinction of the metapop’n. Thus, in the end, most closed metapopulations will expire without a stabilizing influence from outside.
  11. 11. Making models realistic All metapop’n models begin with these fundamental equations and then add procedures for modeling the variables and factors affecting the variables. b = per capita birth rate d = per capita death rate ! = P of catastrophic destruction of a patch ! = P of migrant making it to a patch ! (x )= lacunarity (index of l’scape texture) µij = enemy-victim relationship

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