1. Background knowledge expected Population growth models/equations exponential and geometric logistic Refer to 204 or 304 notes Molles Ecology Ch’s 10 and 11 Krebs Ecology Ch 11 Gotelli - Primer of Ecology (on reserve)

2. Habitat loss Pollution Overexploitation Exotic spp Small fragmented isolated popn’s Inbreeding Genetic Variation Reduced N Demographic stochasticity Env variation Catastrophes Genetic processes Stochastic processes The ecology of small populations

3. How do ecological processes impact small populations? Stochasticity and population growth Allee effects and population growth Outline for this weeks lectures

4. Immigration + Emigration - Birth (Natality) + Death (Mortality) - N t+1 = N t +B-D+I-E Population N t Demography has four components

5. Exponential population growth (population well below carrying capacity, continuous reproduction closed pop’n) Change in population at any time dN = (b-d) N = r N where r =instantaneous rate of increase dt ∆t ∆N Cumulative change in population N t = N 0 e rt N 0 initial popn size, N t pop’n size at time t e is a constant, base of natural logs

6. Trajectories of exponential population growth r > 0 r = 0 r < 0 N t Trend

7. Geometric population growth (population well below carrying capacity, seasonal reproduction) N t+1 = N t +B-D+I-E ∆N = N t+1 - N t = N t +B-D+I-E - N t = B-D+I-E Simplify - assume population is closed; I and E = 0 ∆N = B-D If B and D constant, pop’n changes by r d = discrete growth factor N t+1 = N t +r d N t = N t (1+ r d ) Let 1+ r d =, the finite rate of increase N t+1 = N t N t = t N 0

8. DISCRETE vs CONTINUOUS POP’N GROWTH Reduce the time interval between the teeth and the Discrete model converges on continuous model = e r or Ln ( ) = r Following are equivalent r > 0 > 1 r = 0 = 1 r< 0 < 1 Trend

9. Geometric population growth (population well below carrying capacity, seasonal reproduction) N t+1 = (1+r dt ) N t = (1+r dt ) (1+r dt-1 ) N t-1 = (1+r dt ) (1+r dt-1 ) (1+r dt-2 ) N t-2 = (1+r dt ) (1+r dt-1 ) (1+r dt-2 ) (1+r dt-3 ) N t-3 Add data N t-3 = 10 r dt = 0.02 r dt-1 = - 0.02 r dt-2 = 0.01 r dt-3 = - 0.01 What is the average growth rate?

10. Geometric population growth (population well below carrying capacity, seasonal reproduction) What is average growth rate? = (1+0.02) + (1-0.02) + (1+0.01) + (1-0.01) = 1 4 Arithmetic mean Predict N t+1 given N t-3 was 10

11. Geometric population growth (population well below carrying capacity, seasonal reproduction) What is average growth rate? Geometric mean = [(1+0.02) (1-0.02) (1+0.01) (1-0.01)] 1/4 = 0.999875 KEYPOINT Long term growth is determined by the geometric not the arithmetic mean Geometric mean is always less than the arithmetic mean Calculate N t+1 using geometric mean N t+1 = 4 x 10 (0.999875) 4 x10 = 9.95 N t+1 = (1+0.02) (1-0.02) (1+0.01) (1-0.01) 10 = 9.95

12. DETERMINISTIC POPULATION GROWTH For a given N o, r or r d and t The outcome is determined Eastern North Pacific Gray whales Annual mortality rates est’d at 0.089 Annual birth rates est’d at 0.13 r d =0.13-0.89 = 0.041 = 1.04 1967 shore surveys N = 10,000 Estimated numbers in 1968 N 1 = N 0 = ? Estimated numbers in 1990 N 23 = 23 N 0 = (1.04) 23. 10,000 = 24,462

13. DETERMINISTIC POPULATION GROWTH For a given N o, r or r d and t The outcome is determined

14. Population growth in eastern Pacific Gray Whales - fitted a geometric growth curve between 1967-1980 - shore based surveys showed increases till mid 90’s In US Pacific Gray Whales were delisted in 1994

15. Mean r \ SO what about variability in r due to good and bad years? ENVIRONMENTAL STOCHASTICITY leads to uncertainty in r acts on all individuals in same way b-d Bad 0 Good Variance in r =  2 e = ∑r 2 - (∑r) 2 N N

16. Population growth + environmental stochasticity Ln N t Deterministic 1+r= 1.06,  2 e = 0 1+r= 1.06,  2 e = 0.05 Expected Expected rate of increase is r-  2 e /2

17. Predicting the effects of greater environmental stochasticity Onager (200kg) Israel - extirpated early 1900’s - reintroduced 1982-7 - currently N > 100 RS varies with Annual rainfall Survival lower in droughts

18. Global climate change (GCC) is expected to ----> changes in mean environmental conditions ----> increases in variance (ie env. stochasticity) mean drought < 41 mm Pre-GCC Post-GCC Mean rainfall is the same BUT Variance and drought frequency is greater in “post GCC” Data from Negev

19. Simulating impact on populations via rainfall impact on RS Variance in rainfall LowHigh Number of quasi-extinctions = times pop’n falls below 40

20. Simulating impact on populations adding impact on survival CONC’n Environmental stochasticity can influence extinction risk

21. But what about variability due to chance events that act on individuals Chance events can impact the breeding performance offspring sex ratio and death of individuals ---> so population sizes can not be predicted precisely Demographic stochasticity

22. Dusky seaside sparrow subspecies non-migratory salt marshes of southern Florida decline DDT flooding habitat for mosquito control Habitat loss - highway construction 1975 six left All male Dec 1990 declared extinct

26. Extinction rates of birds as a function of population size over an 80-year period 0 30 60 1 10 100 1000 10,000 ** * * * * ** 10 breeding pairs – 39% went extinct 10-100 pairs – 10% went extinct 1000>pairs – none went extinct * Population Size (no. pairs) % Extinction Jones and Diamond. 1976. Condor 78:526-549

27. random variation in the fitness of individuals (  2 d ) produces random fluctuations in population growth rate that are inversely proportional to N demographic stochasticity =  2 d /N expected rate of increase is r -  2 d /2N Demographic stochasticity is density dependant

28. How does population size influence stochastic processes? Demographic stochasticity varies with N Environmental stochasticity is typically independent of N Long term data from Great tits in Whytham Wood, UK

29. Partitioning variance Species  2 d  2 e Swallow0.180.024 Dipper0.270.21 Great tit0.570.079 Brown bear0.160.003 in large populations N >>  2 d /  2 e Environmental stochasticity is more important Demographic stochasticity can be ignored N crit = 10 *  2 d /  2 e (approx N crit = 100)

30. Stochasticity and population growth N 0 = 50 = 1.03 Simulations - = 1.03,  2 e = 0.04,  2 d = 1.0 N* =  2 d /4 r - (  2 e /2) N* Unstable eqm below which pop’n moves to extinction

31. Environmental stochasticity -fluctuations in repro rate and probability of mortality imposed by good and bad years -act on all individuals in similar way -Strong affect on in all populations Demographic stochasticity -chance events in reproduction (sex ratio,rs) or survival acting on individuals - strong affect on in small populations Catastrophes -unpredictable events that have large effects on population size (eg drought, flood, hurricanes) -extreme form of environmental stochasticity SUMMARY so far Stochasticity can lead to extinctions even when the mean population growth rate is positive

32. Key points Population growth is not deterministic Stochasticity adds uncertainty Stochasticity is expected to reduce population growth Demographic stochasticity is density dependant and less important when N is large Stochasticity can lead to extinctions even when growth rates are, on average, positive