Evaluating Persistence Times in Populations Subject to Catastrophes

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? 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: • Survival and reproduction (uncertainty in the survival and reproduction of individuals) • Catastrophes (events that may result in large, sudden declines in the population by mass death or emigration, often with external causes) • 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.

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

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

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

9. Simulation Example

10. Persistence: Bounded Pop’ns • Suppose the population is bounded with ceiling N. • Extinction is certain in finite time; persistence times are the solution, T, to • M = [ qij], for xe < i, j N. 1 is the unit vector. T = [ Ti ], Ti = 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. • However, 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, fi, depending only on the current population, i. • Let the jump size distribution, given that a jump occurs, be the same for all i > 0. Then: • (Let xe = 0, and deaths be ‘catastrophes’ of size one.)

13. An Unbounded Model • Models like this may be useful when: • Individuals trigger catastrophes (e.g. epidemics) at rates with forms similar to their birth rates (e.g. by interaction, i(i – 1).) • 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 fi = rbi –1and, given a jump occurs, • it is a birth with probability a; • catastrophe size has geometric distribution: dk = (1-a)(1-p)pk–1, 0 £ p < 1. • We show that the persistence times are if b = 1, or if b¹ 1, whenever p + b/a 1, where b = 1-a, and g = 1/b.

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? • We 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* = supi > xe [ bi / ai ] ensures this is the minimal, non-negative solution.

18. Approximating Persistence • See Anderson (1991) for details of the (fairly simple) derivation of the sequences {ai} and {bi}. 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 {ai} and {bi} for xe < i£N+1, then let k*»kN+1 = bN+1 / aN+1…

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

20. Accurate Approximations kN+1 appears to converge… DkN+2 = kN+2 – kN+1 approaches 0 geometrically.

21. Absorption vs. Reflection • Direct 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. • In our approach, we take a truncation with a reflecting boundary, so that the N is a true ceiling and only states xe correspond to quasi-extinction. We calculate persistence times etc. as for bounded populations, and … • we use the convergence of ki 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 Pollett (supervisor and co-author) • Prof. Hugh Possingham (associate supervisor) … and the organisers of MODSIM 2003: