Logistic Equation

1. Logistic Equation dN/dt = rmN(K-N) = rmN(1-N/K) K

3. What are the assumptions of the logistic equation? 1) population started with a stable age distribution - already discussed 2) density is measured equally - i.e. how is density measured? This can be a problem when the biomass of different life stages are different, also size/individual is different under different conditions. 3) depressive effects of density on the rate of increase operates instantaneously without time lags.

4. Time Lags dN/dt = rmN (K - Nt-tau) K

5. Time Lags tau

6. Time Lags Key point - oscillations can be intrinsic in the system and not driven by external environmental changes.

7. Nicholson’s Blowflies

8. Nicholson’s Blowflies

9. Nicholson’s Blowflies

10. Nicholson’s Blowflies The recruitment of new individuals into the population was determined at the egg-laying stage by the influence of food supply on per capita fecundity. As a result, fluctuations were significantly reduced. The time lag in density-dependent response was reduced.

11. Daphnia

12. Daphnia

13. Population Projection N0,t N1,t Population Vector Nt = N2,t . . Nn,t

14. Population Projection The number of new organisms (age 0) at time t+1 will be the contribution of fertility from all aged mothers. N0,t+1 = foN0 + f1N1 + f2N2 + f3N3 + … + fkNk The number of organisms age 1 at time t+1 = s0N0 The number of organisms age 2 at time t+1 = s1N1 The number of organisms age 3 at time t+1 = s2N2

15. f0 f1 f2 f3 f4 …… fk-1 fk s0 0 0 0 0 …… 0 0 0 s1 0 0 0 …… 0 0 0 0 s3 0 0 …… 0 0 0 0 0 s4 0 …… 0 0 . . . . . . . . . . . . . . 0 0 0 0 0 sk-1 0

16. See example in class MNt = Nt+1 M

20. Stage Model