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

458 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.

458 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).

458 A first IBM (demographic stochacity + females only) 1. Set up an initial population – assign each animal an age – assume that they are all females. 2. 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. 3. For each animal, generate a random number, Z, from U[0,1]. If Z < the probability of death, d, “kill” it. 4. Increment the age of each animal. 5. Repeat steps 2-4 for each year of the simulation period.

458 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.

458 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)

458 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.

458 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.

458 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.

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

458 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.

458 Estimating Bowhead Extinction Risk The bowhead population is estimated to have been very small near the turn of the 20 th 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.

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

458 Demographic stochastcity only

458 Demographic and environmental stochascity But these projections assume s 0 =s 1+

458 As before but s 0 =0.3 Rather than s 0 =0.99

458 What if there were only 20 females??

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