458 Lumped population dynamics models Fish 458; Lecture 2

458 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

458 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).

458 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.

458 Fit of the exponential model to the bowhead abundance data

458 Extrapolating the exponential model

458 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)

458 Calculating Extinction Risk for the Exponential Model The Monte Carlo simulation: 1. Set N 0, r and  2. Generate the normal random variates 3. Project the model from time 0 to time t max and find the lowest population size over this period 4. Repeat steps 2 and 3 many (1000s) times. 5. 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).

458 Numerical Hint (Generating a N(x,y 2 ) 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(0,1 2 ), i.e.: Then compute:

458 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”.

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

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

458 The Logistic Model-III r=0.1; K=1000

458 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.

458 The Discrete Logistic Model

458 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.

458 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.

458 Variants of the Pella-Tomlinson model

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

458 Additional Harvesting Theory Find for the Pella-Tomlinson model

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