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logistic population growth

Filling carrying capacity. Filling carrying capacity. Filling carrying capacity. Filling carrying capacity. Logistic growth: Numerical Example (let r0 = 0.10, K = 100). Logistic growth: Numerical Example (let r = 0.10, K = 100). . Nt. dN /Ndt. K=100. K / 2=50. r0 = 0.10. 0. Logistic growth: Numerical Example (let r0 = 0.10, K = 100).

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logistic population growth

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    1. Logistic population growth dN / N dt = r0 [(K - N) / K ] Interpreting (K - N ) / K Proportion of unused carrying capacity consider carrying capacity based on space (e.g., plants, barnacles)

    2. Filling carrying capacity

    3. Filling carrying capacity

    4. Filling carrying capacity

    5. Filling carrying capacity

    6. Logistic growth: Numerical Example(let r0 = 0.10, K = 100)

    7. Logistic growth: Numerical Example(let r = 0.10, K = 100)

    8. Logistic growth: Numerical Example(let r0 = 0.10, K = 100)

    9. Logistic population decline

    10. Assumptions of logistic growth model K is constant over time does not vary year to year etc. dN / Ndt declines linearly with N alternative … nonlinear decline Effect of density N on dN / Ndt is instantaneous … no delays alternative … density now affects dN / Ndt some time in the future (time lag) Continuous overlapping generations

    11. Logistic growth: Real data

    12. Discrete Logistic: non-overlapping generations e.g., Seasonal reproduction Difference equation model DN = Nt+1 - Nt = r0 Nt [ (K - Nt ) / K ] Note: this notation is different from that in Krebs (pp. 158-159) See Gotelli Ch. 2, pp. 37-39 Unlike continuous logistic, many odd and complex dynamics result Note: not necessarily semelparous

    13. Discrete Logistic Outcomes r0 < 2.0 … Damped Oscillations

    14. Discrete Logistic Outcomes2.00 < r0 < 2.45 … Stable 2-point cycle

    15. End 19th lecture

    16. Discrete Logistic Outcomes 2.45 < r0 < 2.50 …more complex cycles

    17. Discrete Logistic Outcomes 2.50 < r0 < 2.57 …more complex cycles

    18. Discrete Logistic Outcomes r0 > 2.57 … Chaos

    19. Cycles & chaos in discrete logistic low r0 (<2.0) … very simple dynamics as r0 increases … 2, 4, 8, 16, 32 , etc. point cycles r0 > 2.57 … deterministic chaos non-repeated oscillations slightly different starting conditions yield completely different dynamics

    20. Discrete Logistic Outcomes Chaos … dynamics depend on initial conditions

    21. Importance of cycles and chaos Annual, discrete generations common Insects (and others) frequently go through cycles and complex fluctuations in nature Traditional interpretation … annual random variation in conditions Cycles and chaos may be products of the deterministic dynamics, not variation

    22. aphid -- 2 point cycle aphid -- chaos moth -- damped moth -- damped moth -- damped yellow jacket -- damped

    23. Denisty dependent population growth Logistic growth implies density dependence Population is Regulated Density dependence is what stops population increase Density independent effects also implact logistic populations Population Limitation

    24. Density independent effects Logistic population growth Density dependent b Density independent d e.g. due to weather If density independent d differs, K will differ But b is what regulates population

    25. Harvesting We harvest natural populations How much can be harvested in any year? Is the population to remain stable? “Stock” = harvestable part of population Stock: decreases due to mortality and harvest increases due to growth and recruitment “recruitment” = fish attain catchable size

    26. Harvesting S2 = S1 + R + G – M – F Where: S1 = Stock at start of year S2 = Stock at end of year R = Recruits G = Growth M = Natural mortality F = Fishing yield

    27. Harvesting

    28. Exam #2

    29. Continuous vs. Discrete generations

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