1. Population Growth and Regulation BIOL400 31 August 2015

2. Population  Individuals of a single species sharing time and space  Ecologists must define limits of populations they study Almost no population is closed to immigration and emigration Almost no population is closed to immigration and emigration

3. Exponential Population Growth  Equations: N t = N 0 e rt N t = N 0 e rt dN/dt = rN dN/dt = rN  Model terms: r = per-individual rate of change (= b – d) r = per-individual rate of change (= b – d) = intrinsic capacity for increase, given the = intrinsic capacity for increase, given the environmental conditions environmental conditions N = population size, at time t N = population size, at time t e = 2.718 e = 2.718

4. Fig. 8.13 p. 131

5. Exponential Population Growth  Assumptions of the Model: Constant per-capita rate of increase, regardless of how high N gets Constant per-capita rate of increase, regardless of how high N gets Continuous breeding Continuous breeding

6. Geometric Population Growth  Model modified for discrete annual breeding  N t = N 0 t  = e r is the annual rate of increase in N is the annual rate of increase in N  Example: N 0 = 1000, = 1.10 N 1 = 1100N 2 = 1210N 1 = 1100N 2 = 1210 N 3 = 1331N 4 = 1464N 3 = 1331N 4 = 1464 N 5 = 1611N 10 = 2594N 5 = 1611N 10 = 2594 N 25 = 10,834N 100 = 13,780,612N 25 = 10,834N 100 = 13,780,612 Q: Can you spot the oversimplification of nature here?

7. Fig. 8.10 p. 129

8. Fig. 9.1 p. 144 R 0 = per-generation multiplicative rate of increase

9. Logistic Population Growth  Equations: N t = K/(1 + e a-rt ) N t = K/(1 + e a-rt ) K = karrying kapacity of the environmentK = karrying kapacity of the environment a positions curve relative to origina positions curve relative to origin dN/dt = Nr[(K-N)/K] dN/dt = Nr[(K-N)/K]  Assumption: Growth rate will slow as N approaches K

10. Fig. 9.4 p. 146

11. Table p. 148 As N increases, per-capita rate of increase declines, but the absolute rate of increase always peaks at ½ K absolute rate of increase always peaks at ½ K

12. Data from Populations in the Field

13. Fig. 9.8 p. 150  Cormorants in Lake Huron  Low numbers due to toxins  Increase is not strongly sigmoid

14. Fig. 9.9 p. 150  Ibex in Switzerland  Reintroduced after elimination via hunting  Roughly sigmoid (=logistic) but with big decline in 1960s

15. Fig. 9.10 p. 151  Whooping cranes of single remaining wild population 15 in 1941, now over 200 15 in 1941, now over 200  r increased in 1950s  Every mid-decade, there is a mini-crash Apparently related to predation cycles Apparently related to predation cycles

16. Fig. 9.15 p. 154  Cladocerans Predominant lake zooplankton Predominant lake zooplankton  No constant K; big swings seasonally

17. Can We Improve Our Models?  1) Theta logistic model  2) Time-lag logistic model  3) Stochastic models  4) Population projection matrices

18. Theta Logistic Model  New term, , defines curve relating growth rate to N  dN/dt = Nr[(K-N)/K]  Fig. 9.12 p. 152

19. Fig. 9.13 p. 152

20. Time-Lag Models  Logistic model in which population growth rate depends not on present N, but on N one (or more) time periods prior  Assumes population’s demographic response to density may be delayed

21. Fig. 9.14 p. 153  With time lag, stable ups and downs may occur

22. Fig. 11.14 p. 170  Water fleas show stable approach to K at 18  C  Time lag effect occurs at 25  C Daphnia store energy to use when food resources collapse Daphnia store energy to use when food resources collapse

23. Stochastic Models  Predict a range of possible population projections, with calculation of the probability of each Fig. 9.17 p. 156

24. Population Projection Matrices  Use matrix algebra to project population growth, based on fecundity and age- specific survivorship Fig. 9.18A p. 157  Application: Determining whether changes in one aspect or another of the life history of an organism have the greater impact on r (calculate “elasticity” of each life-history parameter)

25. Fig. 9.19 p. 159

26. HANDOUT—Biek et al. 2002

27. Survivorship in a Population  Three types of curves are recognized following Pearl (1928)  Examination of the survivorship of various species shows that most have a mixed pattern Fig. 8.6 p. 124

28. Fig. 8.8 p. 126

29. Life Table  Used to project population growth  Can be used to determine R 0, from which r or can be calculated  1) Vertical (=Static): useful if there is long-term stability in age-specific mortality and fecundity  2) Cohort: data taken from a population followed over time (ideally, a cohort followed until all have died) Observing year-year survivorship, or Observing year-year survivorship, or Collecting data on age at death Collecting data on age at death

30. Table 8.5 p. 128

31. Table 8.3 p. 122