Copyright 2006 Hal Caswell Applications of Markov chains in demography and population ecology Hal Caswell Biology Department Woods Hole Oceanographic Institution

June 2006Hal Caswell -- Markov Anniversary Meeting 2 Markov chains in demography particle = individual organism states = stages in the life cycle chains absorbing questions relate to absorbtion –when –where –timing –perturbation analysis

June 2006Hal Caswell -- Markov Anniversary Meeting 3 Patterns and rates of transition reflect hypotheses about biological processes –growth –development –maturation –reproduction –mortality

June 2006Hal Caswell -- Markov Anniversary Meeting 4 Discrete demography life cycle stages projection interval transitions reproduction

June 2006Hal Caswell -- Markov Anniversary Meeting 5 Age-classified life cycle

June 2006Hal Caswell -- Markov Anniversary Meeting 6 Size-classified life cycles 1.Dormant seeds 1 2.Dormant seeds 2 3.Small rosettes 4.Medium rosettes 5.Large rosettes 6.Flowering plants Teasel (Dipsacus sylvestris)

June 2006Hal Caswell -- Markov Anniversary Meeting 7 Wandering albatross (Diomedea exulans) extracted from Croxall et al Stage-classified life cycles

June 2006Hal Caswell -- Markov Anniversary Meeting 8 North Atlantic right whale calf post-mommommatureimmature

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June 2006Hal Caswell -- Markov Anniversary Meeting 11 Analysis within the life cycle partition entries into transitions and reproduction recognize that death is present as an absorbing state life cycle describes dynamics on transient states of an absorbing Markov chain

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June 2006Hal Caswell -- Markov Anniversary Meeting 13 transition probabilities (s x s) absorbtion probabilities (a x s) mortality probabilities (1 x s) reproduction rates (s x s)

June 2006Hal Caswell -- Markov Anniversary Meeting 14 1=calf 2=immature 3=mature 4=mom 5=post-mom 6=dead

June 2006Hal Caswell -- Markov Anniversary Meeting 15 Importance? population dynamics conservation (growth and persistence, invasion and recolonization) evolution (success of life cycles, aging, environmental response)

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June 2006Hal Caswell -- Markov Anniversary Meeting 17 Magnus and Neudecker 1988, Nel 1980

June 2006Hal Caswell -- Markov Anniversary Meeting 18 hours of fun for everybody!

June 2006Hal Caswell -- Markov Anniversary Meeting 19 The North Atlantic right whale (Eubalaena glacialis)

June 2006Hal Caswell -- Markov Anniversary Meeting 20 calving feeding Endangered, by any standard N < 300 individuals Minimal recovery since 1935 Ship strikes Entanglement with fishing gear

June 2006Hal Caswell -- Markov Anniversary Meeting : died October 1999 entanglement 1014 “Staccato” died April 1999 ship strike Mortality and serious injury due to entanglement and ship strikes

June 2006Hal Caswell -- Markov Anniversary Meeting 22 1=calf 2=immature 3=mature 4=mom 5=post-mom 6=dead

June 2006Hal Caswell -- Markov Anniversary Meeting 23 Parameter estimation photographic ID catalog multistage mark- recapture analysis maximum likelihood estimates

June 2006Hal Caswell -- Markov Anniversary Meeting 24 Results data series of statistical models evidence for declining trend in survival of moms, and of birth rate population growth rate declined to negative levels Fujiwara and Caswell 2001 Caswell and Fujiwara 2004

June 2006Hal Caswell -- Markov Anniversary Meeting 25 Time-invariant model

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June 2006Hal Caswell -- Markov Anniversary Meeting 27 E(number of reproductive events)

June 2006Hal Caswell -- Markov Anniversary Meeting 28 calfimmmaturemompost-mom Expected occupancy (yrs) Right whale calf

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June 2006Hal Caswell -- Markov Anniversary Meeting 32 calfimmmaturemompost-mom CV of stage occupancy time Right whale calf

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June 2006Hal Caswell -- Markov Anniversary Meeting 40 calfimmmaturemompost-mom Life expectancy (yrs) Right whale

June 2006Hal Caswell -- Markov Anniversary Meeting 41 calfimmmaturemompost-mom CV of life expectancy Right whale Stage

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June 2006Hal Caswell -- Markov Anniversary Meeting 51 The birth interval interval between births determines birth rate varies over time responds to environment especially important in species producing few offspring a special case of the “timing problem” calculation uses machinery for absorbtion times, but conditional times

June 2006Hal Caswell -- Markov Anniversary Meeting 52 Modified life cycle graph died before breeding reproduce before dying

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June 2006Hal Caswell -- Markov Anniversary Meeting 55 calfimmmaturemompost-mom Starting stage Mean time to reproduction (yrs) Right whale

June 2006Hal Caswell -- Markov Anniversary Meeting 56 Time variation the vital rates of the right whale changed between 1980 and 1998

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June 2006Hal Caswell -- Markov Anniversary Meeting Year Life expectancy period

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June 2006Hal Caswell -- Markov Anniversary Meeting 67 Summary stage-classified demography from Markov chains –expected stage occupancy times –variance in stage occupancy times –expected longevity –variance in longevity –net reproductive rate –generation time –birth and other intervals –period and cohort effects in variable envionments perturbation analyses connection to data things don’t look good for the right whale