1. Evaluating Persistence Times in Populations Subject to Catastrophes Ben Cairns and Phil Pollett Department of Mathematics

2. Persistence Most populations are certain not to persist forever: they will eventually become extinct. When is extinction likely to occur?When is extinction likely to occur? We consider such measures as the expected time to extinction (or ‘persistence time’). We have developed methods for finding accurate measures of persistence for a class of stochastic population models.

3. Overview Modelling populations subject to stochastic effects: –Birth, death and catastrophe processes. Calculating measures of persistence: –Bounded models. –Unbounded models: Analytic approaches. Accurate numerical approaches.

4. Population Processes Populations are subject to a variety of sources of randomness, including: There are other sources of randomness (e.g. environmental) but we focus on the above. Any model must account for the uncertainty introduced by this stochasticity. –Survival and reproduction –Survival and reproduction (uncertainty in the survival and reproduction of individuals) –Catastrophes –Catastrophes (events that may result in large, sudden declines in the population by mass death or emigration, often with external causes)

5. Modelling Populations Birth, death and catastrophe processesBirth, death and catastrophe processes, a class of continuous-time Markov chains, are stochastic models for populations. As their name suggests, BDCPs incorporate both demographic stochasticity and catastrophic events. They are both powerful and simple, allowing arbitrary relationships between population size and dynamics. They model discrete-valued populations that are time-homogeneous.

6. A General BDCP BDCPs are defined by rates: birth rate –B(i) is the birth rate; i – 2i – 1ii + 1 D(i)D(i) B(i)B(i) C(i)F(1|i) C(i)F(2|i) C(i)F(k|i)C(i)F(k|i) death rate –D(i) is the death rate; catastrophe rate –C(i) is the catastrophe rate, with catastrophe size distribution –F( i – j | i ), the catastrophe size distribution.

7. A General BDCP BDCPs are defined by rates: birth rate –B(i) is the birth rate; death rate –D(i) is the death rate; catastrophe rate –C(i) is the catastrophe rate, with catastrophe size distribution –F( i – j | i ), the catastrophe size distribution.

8. Important Features Jumps up limited in size to 1 individual (only births or single immigration). most general modelThis is the most general model of its type: it allows any form of dependence of the rates on the current population. boundedThe population is bounded if it has a ceiling N (then B(i) > 0 for x e < i < N, and B(N) = 0). unboundedIf there is no such N, and B(i) > 0 for all i > x e, the population is ‘unbounded’. extinction levelA population is quasi-extinct (or functionally extinct) at or below the extinction level, x e.

9. Simulation Example

10. Persistence: Bounded Pop’ns Suppose the population is bounded with ceiling N. Extinction is certainExtinction is certain in finite time; persistence times are the solution, T, to M = [ q ij ], for x e < i, j  N. 1 is the unit vector. T = [ T i ], T i = persistence time from size i. –If N is not ‘too large’, we can easily find numerical solutions (e.g. see Mangel and Tier, 1993).

11. Unbounded Populations Unbounded models (ones without hard limits) can still be reasonable population models. explode never go extinctHowever, such models could explode to infinite size in finite time, or never go extinct. –We would generally want to rule out this kind of behaviour for biological populations. If extinction is certain, the persistence times are the minimal, non-negative solution to:

12. An Unbounded Model Suppose there is an overall jump rate, f i, depending only on the current population, i. jump size distributionLet the jump size distribution, given that a jump occurs, be the same for all i > 0. Then: (Let x e = 0, and deaths be ‘catastrophes’ of size one.)

13. An Unbounded Model Models like this may be useful when: a)Individuals trigger catastrophes a)Individuals trigger catastrophes (e.g. epidemics) at rates with forms similar to their birth rates (e.g. by interaction,  i(i – 1).) b)Catastrophes are localised b)Catastrophes are localised but the population maintains a fairly constant density, so that the catastrophe size distribution is fixed. Another advantage: they are quite general and are amenable to mathematical analysis. (We find analytic solutions for persistence times and probabilities for this model.)

14. Unbounded Model: Example Suppose the overall jump rate is f i =    i –1 and, given a jump occurs, –it is a birth with probability a; –catastrophe size has geometric distribution: d k = (1-a)(1-p)p k–1, 0  p < 1. We show that the persistence times are if  = 1, or if   1, whenever p + b/a  1, where b = 1-a, and  = 1/ .

15. Unbounded Model: Example

16. Approximating Persistence Suppose our preferred model is either unbounded or has a very large ceiling. What if we cannot find complete solutions? truncatingWe can still make progress by truncating our model: we approximate the population, introducing some form of boundary. We must, however, show that the chosen truncation is appropriate, or else our approximate persistence times may be nothing like the true values!

17. Approximating Persistence Interesting properties of extinction times (expectations, etc.) are all solutions to Solutions have the form (Anderson, 1991) where k * = sup i > x e [ b i / a i ] ensures this is the minimal, non-negative solution.

18. Approximating Persistence See Anderson (1991) for details of the (fairly simple) derivation of the sequences {a i } and {b i }. These sequences are unique. In our work, we do not use Anderson’s results to calculate persistence times directly, but rather obtain from them a quantitative indicator of the accuracy of a truncation. However, could in theory find {a i } and {b i } for x e < i  N+1, then let k *  k N+1 = b N+1 / a N+1 …

19. Accurate Approximations Then This will be an accurate approximation provided k N+1 is close to k *, which we can judge in two ways: –Plot k N+1 versus N+1 and look for convergence. –If a plot of  k N+2 = k N+2 – k N+1 is linear on log-linear axes, k N+1 appears to converge geometrically (fast) to k *.

20. Accurate Approximations  k N+2 = k N+2 – k N+1 approaches 0 geometrically. k N+1 appears to converge…

21. Absorption vs. Reflection absorbing boundaryDirect use of Anderson’s approach is not satisfying in many cases: it allows the population to become ‘extinct’ by going above N! Then, N+1 is an absorbing boundary. reflecting boundaryIn our approach, we take a truncation with a reflecting boundary, so that the N is a true ceiling and only states  x e correspond to quasi-extinction. We calculate persistence times etc. as for bounded populations, and … we use the convergence of k i as an indication of the accuracy of the truncation.

22. Example: Numerical Solutions

23. Conclusions We can calculate various measures of persistence for general birth, death and catastrophe processes, a useful class of population models: –Bounded processes with a low ceiling: Solutions to questions of persistence are very easy to obtain. –Unbounded processes: In some cases we can find analytic solutions. –Processes that are unbounded or have a high ceiling: We can get approximate solutions and obtain quantitative indicators of their accuracy.

24. Acknowledgements Dr Phil PollettDr Phil Pollett (supervisor and co- author) Prof. Hugh PossinghamProf. Hugh Possingham (associate supervisor) … and the organisers of MODSIM 2003: