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Lumped population dynamics models

Lumped population dynamics models. Fish 458; Lecture 2. Revision: Nomenclature. Which are the state variables , forcing functions and parameters in the following model: population size at the start of year t, catch during year t, growth rate, and annual recruitment.

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Lumped population dynamics models

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  1. Lumped population dynamics models Fish 458; Lecture 2

  2. Revision: Nomenclature • Which are the state variables, forcing functions and parameters in the following model: • population size at the start of year t, • catch during year t, • growth rate, and • annual recruitment

  3. The Simplest Model-I • Assumptions of the exponential model: • No emigration and immigration. • The birth and death rates are independent of each other, time, age and space. • The environment is deterministic. • is the initial population size, and • is the “intrinsic” rate of growth(=b-d). • Population size can be in any units (numbers, biomass, species, females).

  4. The Simplest Model - II • Discrete version: • The exponential model predicts that the population will eventually be infinite (for r>0) or zero (for r<0). • Use of the exponential model is unrealistic for long-term predictions but may be appropriate for populations at low population size. • The census data for many species can be adequately represented by the exponential model.

  5. Fit of the exponential model to the bowhead abundance data

  6. Extrapolating the exponential model

  7. Extending the exponential model(Extinction risk estimation) Allow for inter-annual variability in growth rate: This formulation can form the basis for estimating estimation risk: • ( - quasi-extinction level, time period, critical probability)

  8. Calculating Extinction Risk for the Exponential Model • The Monte Carlo simulation: • Set N0, r and  • Generate the normal random variates • Project the model from time 0 to time tmax and find the lowest population size over this period • Repeat steps 2 and 3 many (1000s) times. • Count the fraction of simulations in which the value computed at step 3 is less than . • This approach can be extended in all sorts of ways (e.g. temporally correlated variates).

  9. Numerical Hint(Generating a N(x,y2) random variate) • Use the NormInv function in EXCEL combined with a number drawn from the uniform distribution on [0, 1] to generate a random number from N(0,12), i.e.: • Then compute:

  10. The Logistic Model-I • No population can realistically grow without bound (food / space limitation, predation, competition). • We therefore introduce the notation of a “carrying capacity” to which a population will gravitate in the absence of harvesting. • This is modeled by multiplying the intrinsic rate of growth by the difference between the current population size and the “carrying capacity”.

  11. The Logistic Model - II where K is the carrying capacity. The differential equation can be integrated to give:

  12. Logistic vs exponential model(Bowhead whales) Which model fits the census data better? Which is more Realistic??

  13. The Logistic Model-III r=0.1; K=1000

  14. Assumptions and caveats • Stable age / size structure • Ignores spatial, ecosystem considerations / environmental variability • Has one more parameter than the exponential model. • The discrete time version of the model can exhibit oscillatory behavior. • The response of the population is instantaneous. • Referred to as the “Schaefer model” in fisheries.

  15. The Discrete Logistic Model

  16. Some common extensions to the Logistic Model • Time-lags (e.g. the lag between birth and maturity is x): • Stochastic dynamics: • Harvesting: where is the catch during year t.

  17. Surplus Production • The logistic model is an example of a “surplus production model”, i.e.: • A variety of surplus production functions exist: the Fox model the Pella-Tomlinson model Exercise: show that Fox model is the limit p->0.

  18. Variants of the Pella-Tomlinson model

  19. Some Harvesting Theory • Consider a population in dynamic equilibrium: • To find the Maximum Sustainable Yield: • For the Schaefer / logistic model:

  20. Additional Harvesting Theory • Find for the Pella-Tomlinson model

  21. Readings – Lecture 2 • Burgman: Chapters 2 and 3. • Haddon: Chapter 2

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