1. Performance of small populations

2. Distribution of population sizes
Number of cases
Scorzonera hispanica

3. Processes in small populations
Deterministic
Loss of pollinators, problems with finding mates
Stochastic
Effect of random fluctuations

4. Deterministic effects
Demographic – effects directly on population dynamics
Genetic – effects via genetic diversity

5. Demographic effects
Allee effect

6. Allee effect
Limited possibility to find a partner due to low population density

7. Limitation by number of pollinators
Ågren, J. (1996), Lythrum salicaria

8. Limitation by pollen availability
Testing limitations by pollen availability
Separation from plant vitality and habitat quality

9. Ågren, J. (1996), Lythrum salicaria

10. Stochastic effect
Demographic
Environmental
Disturbance

11. Demographic stochasticity

12. Demographic stochasticity
Application of probabilities to limited sample (mortality, reproduction….)
Problem only in small populations

13. 50% survival probability

14. Data
Probabilities of transitions in the life-cycle
0.5
0.9
0.3 4

15. Lathyrus vernus

16. Data
Killer whale

17. Population transition matrix
Seedling vegetative flowering
Seedling
Vegetative
Flowering
Population growth rate
Stable stage distribution

18. Modeling demographic stochasticity?
Matrix multiplication

19. Application of probabilities to each individuals
Average
Good luck
Bad luck
Ineffective calculation

20. Alternative Use knowledge on distribution of the probabilities
Transition probability – multinomial distribution
Reproduction – binomial or Poisson distribution

21. Poisson distribution

22. Demographic stochasticity or changing condition?
Killer whales
1: yearling
2: juvenile
3: mature females
4: post reproductive females
Demographic stochasticity or changing condition?

23. Killer whale
Only 1 offspring per female, number has binomial distribution
Other multinomial distribution

24. Killer whale
Populations with 4 individuals

25. Population size
Killer whale

26. Projection of population size in time

27. Demographic or environmental stochasticity?
How probable it is that there is no reproduction over 5 years?
Time without reproduction – killer whale

28. Extinction of lineages
Branching process models

29. Environmental stochasticity

30. Environmental stochasticity
Variation of demography in time
Combination of good and bad year

31. Data Variability in transition probabilities over time
Combination of spatial and temporal variation

32. Models of environmental stochasticity
Independent between years
Markov chains – dependencies between years
Autoregressive techniques – continuous changes in quality

33. Independent performance between years
Random combination of different qualities

34. Variation in transition probabilities
Whole matrices vs. single transitions
Covariance

35. Scorzonera hispanica

36. Thomas 1990, simulace

37. Variation vs. mean
Variation– probability of survival in small populations
Mean – long-term trends

38. Population growth rate in changing environment
Not possible to average – depends on stable stage distribution
Simulation – multiple combinations of all matrices

39. Long term population growth rate

40. Demo. and env. stochasticity

41. Extinction Demo. stochasticity – integer numbers
Env. stochasticita – fractions

42. Disturbance

43. Effect of disturbance
Relationship between disturbance size and frequency
Problems with quantification

44. Disturbance
Song et al. 2001

45. Without disturbance
With disturbance
Mangel & Tier 1994, simul