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
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) gt 0 for xe lt i lt N, and B(N) 0).
• If there is no such N, and B(i) gt 0 for all i gt
  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 Popns

• 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 lt 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 gt 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 r bi 1
  and, given a jump occurs,
• it is a birth with probability a
• catastrophe size has geometric distribution
• dk (1-a)(1-p)pk1, 0 p lt 1.
• We show that the persistence times are
• if b 1, or
• if b ¹ 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 gt 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 Andersons 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 lt i N1, then let k kN1 bN1 / aN1

19
Accurate Approximations

• Then
• This will be an accurate approximation provided
  kN1 is close to k, which we can judge in two
  ways
• Plot kN1 versus N1 and look for convergence.
• If a plot of DkN2 kN2 kN1 is linear on
  log-linear axes, kN1 appears to converge
  geometrically (fast) to k.

20
Accurate Approximations
kN1 appears to converge
DkN2 kN2 kN1 approaches 0 geometrically.
21
Absorption vs. Reflection

• Direct use of Andersons approach is not
  satisfying in many cases it allows the
  population to become extinct by going above N!
  Then, N1 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