Performance of small populations

Distribution of population sizes Number of cases Scorzonera hispanica

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

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

Demographic effects Allee effect

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

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

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

Ågren, J. (1996), Lythrum salicaria

Stochastic effect Demographic Environmental Disturbance

Demographic stochasticity

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

50% survival probability

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

Lathyrus vernus

Data Killer whale

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

Modeling demographic stochasticity? Matrix multiplication

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

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

Poisson distribution

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

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

Killer whale Populations with 4 individuals

Population size Killer whale

Projection of population size in time

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

Extinction of lineages Branching process models

Environmental stochasticity

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

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

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

Independent performance between years Random combination of different qualities

Variation in transition probabilities Whole matrices vs. single transitions Covariance

Scorzonera hispanica

Thomas 1990, simulace

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

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

Long term population growth rate

Demo. and env. stochasticity

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

Disturbance

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

Disturbance Song et al. 2001

Without disturbance With disturbance Mangel & Tier 1994, simul