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# Age-structured models (Individual-based versions)

Age-structured models (Individual-based versions). Fish 458, Lecture 6. Individual-based models (IBMs)-I. When population size is very small (&lt;50): The age-structured models described previously begin to be invalid (what does 0.5 of a bowhead mean?).

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## Age-structured models (Individual-based versions)

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1. Age-structured models(Individual-based versions) Fish 458, Lecture 6

2. Individual-based models (IBMs)-I • When population size is very small (<50): • The age-structured models described previously begin to be invalid (what does 0.5 of a bowhead mean?). • Demographic stochasticity (individual births and deaths) need to be modeled explicitly. • IBMs therefore provide a more defensible basis for estimation of extinction risk.

3. Individual-based models-II • Other uses for individual-based models: • Detailed process modeling: • Predation, movement, births, mating strategy • Questions outside the realm of age-structured models. One can keep track of far more information (who were an animal’s parents, the genetic structure of the population).

4. A first IBM (demographic stochacity + females only) • Set up an initial population – assign each animal an age – assume that they are all females. • For each animal, generate a random number, Z, from U[0,1]. If Z < half the probability of giving birth, 0.5 b, “it gives birth to a female” so add a new animal. • For each animal, generate a random number, Z, from U[0,1]. If Z < the probability of death, d, “kill” it. • Increment the age of each animal. • Repeat steps 2-4 for each year of the simulation period.

5. Estimating Extinction Risk • Run the algorithm a large number of times and count the number of extinctions (or quasi-extinctions). • The probability of extinction depends on: • The difference between d and b – larger implies a greater probability. • The magnitude of (b+d)/2 – larger implies a greater probability. • The initial population size – smaller implies a greater probability • The length of the simulation – the longer the simulation higher the probability of extinction.

6. The Simplest Extensions • Allow d to be age-dependent • (e.g. higher for juveniles and old animals). • Allow b to be age-dependent • (e.g. explicitly allow animals to “mature” / include senility)

7. Some Key Extensions • Environmental variability: • Allow b and d to vary between years (e.g. d=d+ where  is normally distributed). • Allowing environmental variability in the death rate can increase the extinction risk noticeably. • Catastrophic events: • Allow a catastrophic event (e.g. 20% of all animals die) to occur with a certain probability.

8. Other Extensions - I (mainly mammals / birds) • Time between births. • Prevent females from giving birth in successive years. • This involves keeping track of when each animal last gave birth. • Allow females to give birth more quickly than expected if their offspring die.

9. Other Extensions – II(mainly mammals / birds) • Harvesting: • Each animal would have a (time-dependent) probability of death due to harvesting. • The may require modelling males! • What happens if a mother with a young offspring is harvested? • Density-dependence: • In principle any / all of the parameters of an IBM may be density-dependent (the most common would be the birth / death rates). • Movement.

10. Basic message – Individual-based models are as complex or as simple as you want them to be.

11. Should I used an IBM?? • Advantages: • “Realistic” and very flexible. • Disadvantages: • Computationally very intensive for “large” population sizes. • Very data intensive in principle. • Hard to know when to stop! • The same results can often be obtained using a model that lumps all animals of the same age. • Hard to find “general” results as each IBM is highly case-specific.

12. Estimating Bowhead Extinction Risk • The bowhead population is estimated to have been very small near the turn of the 20th century (some models suggest a total (both sexes) population size of only ~200). What was the probability of extinction given: • Demographic stochasticity only. • Demographic stochastcity and environmental variability in the death rate.

13. Estimating Bowhead Extinction Risk-II • Basic specifications: • s=(1-d)=0.98; b=0.325 • No calves if the population size exceeds 1000. • CV of 0.6 of log(M) ; M=-log(s).

14. Demographic and environmental stochascity But these projections assume s0=s1+

15. As before but s0=0.3 Rather than s0=0.99

16. What if there were only 20 females??

17. Readings • Burgeman et al. (1994); Chapters 1 and 2.

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