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SECOND LAW OF POPULATIONS:

SECOND LAW OF POPULATIONS:. Population growth cannot go on forever. - P. Turchin 2001 (Oikos 94:17-26). !?. The Basic Mathematics of Density Dependence: The Logistic Equation. How does population growth change as numbers in the population change?. We can start with the equation for

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SECOND LAW OF POPULATIONS:

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  1. SECOND LAW OF POPULATIONS: Population growth cannot go on forever - P. Turchin 2001 (Oikos 94:17-26) !?

  2. The Basic Mathematics of Density Dependence:The Logistic Equation How does population growth change as numbers in the population change? We can start with the equation for exponential growth…. dN/dt = rN

  3. Recall the definition of r dN/dt = rN … r is a growth rate, or the difference between birth and death rate So, we can write, dN/dt = (b-d)N For the exponential equation, these birth and death rates are constants… What if they change as a function of population size?

  4. We can rewrite the exponential rate equation Let b’ and d’ represent birth and death rates that are NOT constant through time So, dN/dt = (b’- d’)N Modeling these variables…. The simplest case is to allow them to be linear

  5. Linear relationship: Let b’ = b - aN andLet d’ = d - cN The intercept: b b’ The slope: a N

  6. What happens if... The values of a or c equal zero? This demonstrates that the exponential equation is a special case of the logistic equation….

  7. Rearranging the equation... The Carrying Capacity, K • Definitions • Issues Assumptions of the logistic model

  8. Rate of change and actual population size

  9. What if we do have time lags? Biologically realistic, after all (consider the stage models we’ve been working with) What happens?

  10. Other forms of density dependence • Ricker model Nt+1 = Ntexp[R*(K-Nt)/K] • Beverton-Holt model Nt+1 = Nt * (1 + R) 1+(R/K)*Nt Both of these originally developed for fisheries; many more possibilities exist.

  11. James F. Parnell Gary Kramer The Allee Effect Minimum density required to maintain the population • Defense or vigilance • Foraging efficiency • Mating

  12. SUMMARY • When density affects demographic rates, • “density dependence” • Many ways to model this mathematically: • Logistic (linear) • Ricker • Beverton-Holt • In constant environment, population will • stabilize at carrying capacity

  13. SUMMARY, continued • Carrying capacity has multiple definitions, • biological reality must be considered • Unlike with density-independent models, the • discrete and continuous forms are NOT • equivalent in behavior • Discrete form can exhibit damped oscillations, • stable limit cycles, or chaos

  14. SUMMARY, continued Allee effect: important implications for management and conservation SECOND LAW OF POPULATIONS: they can’t grow forever…

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