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 We can start with the equation for exponential growth…. How does population growth change as numbers in the population change? 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 and Let d’ = d - cN N b’ The intercept: b The slope: a

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 Beverton-Holt model Both of these originally developed for fisheries; many more possibilities exist. N t+1 = N t exp[R*(K-N t )/K] N t+1 = N t * (1 + R) 1+(R/K)*N t

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

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 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 Carrying capacity has multiple definitions, biological reality must be considered

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