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I. Review of Logistic Population Model

I. Review of Logistic Population Model. N t = 2, R = 0.15, K = 450. A. Discrete equation. - Built in time lag = 1 - Nt+1 depends on Nt. I. Review of Logistic Population Model. B. Density Dependence. Review of Logistic Population Model C. Assumptions.

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I. Review of Logistic Population Model

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  1. I. Review of Logistic Population Model Nt = 2, R = 0.15, K = 450 A. Discrete equation - Built in time lag = 1 - Nt+1 depends on Nt

  2. I. Review of Logistic Population Model B. Density Dependence

  3. Review of Logistic Population ModelC. Assumptions • No immigration or emigration • No age or stage structure to influence births and deaths • No genetic structure to influence births and deaths • No time lags in continuous model

  4. K Review of Logistic Population ModelC. Assumptions • Linear relationship of per capita growth rate and population size (linear DD)

  5. Review of Logistic Population ModelC. Assumptions • Linear relationship of per capita growth rate and population size (linear DD) • Constant carrying capacity – availability of resources is constant in time and space • Reality?

  6. I. Review of Logistic Population Model Discrete equation Nt = 2, r = 1.9, K = 450 Damped Oscillations r <2.0

  7. I. Review of Logistic Population Model Discrete equation Nt = 2, r = 2.5, K = 450 Stable Limit Cycles 2.0 < r < 2.57 * K = midpoint

  8. I. Review of Logistic Population Model Discrete equation Nt = 2, r = 2.9, K = 450 • Chaos • r > 2.57 • Not random • change • Due to DD • feedback and time • lag in model

  9. Review of Logistic Population ModelD. Deterministic vs. Stochastic Models Nt = 1, r = 2, K = 100 * Parameters set deterministic behavior same

  10. Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 Review of Logistic Population ModelD. Deterministic vs. Stochastic Models * Stochastic model, r and K change at random each time step

  11. Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 Review of Logistic Population ModelD. Deterministic vs. Stochastic Models * Stochastic model

  12. Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 Review of Logistic Population ModelD. Deterministic vs. Stochastic Models * Stochastic model

  13. Environmental StochasticityA. Defined • Unpredictable change in environment occurring in time & space • Random “good” or “bad” years in terms of changes in r and/or K • Random variation in environmental conditions in separate populations • Catastrophes = extreme form of environmental variation such as floods, fires, droughts • High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

  14. Environmental StochasticityA. Defined • Unpredictable change in environment occurring in time & space • Random “good” or “bad” years in terms of changes in r and/or K • Random variation in environmental conditions in separate populations • Catastrophes = extreme form of environmental variation such as floods, fires, droughts • High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

  15. Environmental StochasticityA. Defined • Unpredictable change in environment occurring in time & space • Random “good” or “bad” years in terms of changes in r and/or K • Random variation in environmental conditions in separate populations • Catastrophes = extreme form of environmental variation such as floods, fires, droughts • High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

  16. Environmental StochasiticityB. Examples – variable fecundity Relation Dec-Apr rainfall and number of juvenile California quail per adult (Botsford et al. 1988 in Akcakaya et al. 1999)

  17. Environmental StochasiticityB. Examples - variable survivorship Relation total rainfall pre-nesting and proportion of Scrub Jay nests to fledge (Woolfenden and Fitzpatrik 1984 in Akcakaya et al. 1999)

  18. Environmental StochasiticityB. Examples – variable rate of increase Muskox population on Nunivak Island, 1947-1964 (Akcakaya et al. 1999)

  19. Environmental StochasiticityC. Incorporating into Logistic Model Random variable with mean and variance

  20. II. Environmental StochasiticityC. Incorporating into Logistic Model • Randomize r and/or K for each time step • Using Excel, =NORMINV(RAND( ), mean, sd) function provides random variable based on normal distribution with specified mean & variance

  21. Nt = 2, r = 0.15, SD = 0.1; K = 100 Environmental StochasiticityC. Incorporating into Logistic Model • Random r, K is constant * Stochastic model behavior

  22. Nt = 2, r = 0.15, SD = 0.1; K = 100 Environmental StochasiticityC. Incorporating into Logistic Model • Random r, K is constant

  23. Nt = 2, r = 0.15, SD = 0.1; K = 100 Environmental StochasiticityC. Incorporating into Logistic Model • Random r, K is constant

  24. Environmental StochasiticityC. Incorporating into Logistic Model • Random r, K is constant General Trend: • Population grows erratically at smaller population sizes, stabilizes close to K

  25. Nt = 2, r = 0.15; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random

  26. Nt = 2, r = 0.15; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random

  27. Nt = 2, r = 0.15; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random

  28. Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random General Trend: • Variation observed mainly at or near K

  29. Environmental Stochasiticity- Example of random K • Serengeti wildebeest data set – recovering from Rinderpest outbreak • Fluctuations around K possibly related to rainfall

  30. Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random • Mean N always less than mean K • Population rate of change differs above or below K

  31. Environmental StochasiticityC. Incorporating into Logistic Model • Constant r, K is random • More variable environment = smaller average population size

  32. Large r = track changes in K, N = close to K • Small r = slower to track changes in K • Random K, influence of r on population fluctuations R = 0.1 R = 0.8

  33. Nt = 2, r = 0.15; SD = 0.1; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Random r & K

  34. Nt = 2, r = 0.15; SD = 0.1; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Random r & K

  35. Nt = 2, r = 0.15; SD = 0.1; K = 100, SD = 20 Environmental StochasiticityC. Incorporating into Logistic Model • Random r & K

  36. Environmental StochasiticityC. Incorporating into Logistic Model • Random r & K General Trend: • Variation observed throughout population sizes

  37. Environmental StochasiticityE. Implications & Caveats • Application of principle to Population Viability Analysis (PVA) & population forecasting

  38. RAMAS EcoLab

  39. RAMAS EcoLab

  40. Environmental StochasiticityE. Implications & Caveats • Sampling variation & parameter uncertainty • All measurements have error…parameter uncertainty = variation in estimate of parameter due to accuracy & precision of sampling protocol • must account for portion of variation in estimates of vital rates determined by sampling (i.e., separate from “natural” sources of variation)

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