Brief review of previous lecture

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Brief review of previous lecture

• Timing of sampling estimating fecundity for
  matrix
• pre-breeding vs. post-breeding census

Fx Sxmx
Fx mxS0

• Stage-structured models

• Stage-transition matrices and loop diagrams
• Stage retention, skipping, and reversal

• Uses of sensitivity analysis
• Planning future research
• Evaluating management options

• Sensitivity and elasticity values

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Lecture Outline Sensitivity Analysis, Life
Tables, and Life Histories

• Additional examples of sensitivity analysis
• Determinants of growth rate in red-billed choughs
• Invasive species control bullfrogs

• Life tables

• Types cohort/dynamic vs. static
• Survivorship and fertilities schedules
• Net reproductive rate, generation time, rate of
  increase

• Life-history patterns

• Tradeoffs
• Constraints

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Example Population Growth of Red-billed Choughs1

• Species of conservation concern in UK

• Cavity nester one clutch per year (3-6 eggs)

• Twenty-year study in Scotland

1Reid et al. 2004. Identifying the demographic
determinants of population growth rate a case
study of red-billed choughs Pyrrhococorax
pyrrhococorax. J. Animal Ecology 73777-788.
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Breeding success
Survival
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• Used an age-structured matrix model (females
  only)

• Birth-pulse dynamics

• Conducted sensitivity analysis using elasticities
  of vital rates

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• Results were robust to demographic stochasticity

Survival
Breeding success
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But results differed when DEMOGRAPHIC COVARATION
included
Survival
Breeding success
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Invasive species control bullfrogs on Vancouver
Island

• American bullfrogs are used in farming for
  gourmet frog legs and have escaped and
  established feral populations worldwide.
• Introduced bullfrogs have negative effects on
  native fauna.
• Past control efforts focused on removing tadpoles
  and breeding adults.

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Stage-structured model for complex life cycle
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Invasive species control

• Sensitivity analysis indicated that lambda for
  bull frogs was most influenced by proportion of
  tadpoles metamorphosing early (tadpole
  development rate), and by early postmetamorphic
  survival rates.
• Concluded that modeling suggested culling of
  metamorphs in fall is most effective control
  strategy.

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Reminder of important point

• Sensitivity analyses identifies the demographic
  rates to which population growth rate is
  theoretically most sensitive.

• However, lambda varies as a function of the
  degree that each rate varies in nature and its
  sensitivity to that variation. Lambda will
  remain constant if demographic rates with high
  sensitivities do not vary.

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Hal Caswell

• Woods Hole Oceanographic Institution

• Key figure in demographic analysis using matrix
  models.
• Author of most widely used book for topic,
  Matrix Population Models.
• Coauthor on the right whale paper that is an
  assigned reading for the course.

(Fujiwara, M., and H. Caswell. 2001. Demography
of the endangered North Atlantic right whale.
Nature 414537-541)
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The key ingredients of a Life Table

• X is age (years), Nx is number of individuals
  alive at beginning of age x, and Bx is number of
  offspring produced at given age.

• All other life table calculations are derived
  from these three columns.

• Common to use only females.

• These data can be gathered in two main ways.

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2. Static life table

• Counts of individuals of different age classes at
  one time step.
• Single snap-shot of population
• Easier to get data (if age of individuals can be
  determined)
• Not as reliable more assumptions

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Survivorship Curves
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• Fertility (maternity, mx) is the average number
  of offspring produced by individuals in each age
  class

(or average number of daughters per female in
each age class)
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Several useful life-history values can be
calculated from the survivorship schedule (lx)
and fertility schedule (mx)
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2. Generation time (Tg) is a measure of the
average age of reproduction.
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3. Instantaneous rate of increase (r)

• A good approximation is given by

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Life-History Patterns in a Nutshell
Why do we see certain life-history patterns?
Why do species tend to have certain combinations
of life-history traits?
What keeps species from having optimal life
histories in which survivorship and fertility are
at a maximum for all age classes?
Tradeoffs and Constraints
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Key Life-History Traits

• Size at birth
• Growth pattern
• Age and size at maturity
• Number, size, and sex ratio of offspring
• Age- and size-specific reproductive investments
• Age- and size-specific mortality schedules
• Length of life

(from Stearns 1992)
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Life-History Tradeoffs

• Tradeoffs are linkages among traits that
  constrain the simultaneous evolution of two or
  more traits.

• Key tradeoffs include
• Current reproduction vs. survival
• Current reproduction vs. future reproduction
• Reproduction vs. condition/growth
• Number vs. quality of offspring

These often represent physiological tradeoffs
within an individual due to allocation of limited
energy.
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Example Current reproduction vs. survival in red
deer
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Example Reproduction vs. growth in blue-headed
wrasses
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Evolutionary Constraints

• Some life-history traits are fixed at high
  taxonomic levels and do not vary within
  populations.
• All species of tubenose seabirds
  (Procellariformes) have one egg.
• All species of Pacific salmon are semelparous.

• Other traits vary among species and higher
  groupings in ways that suggest macroevolutionary
  tradeoffs and constraints.

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Example Offspring size vs. age at maturity in
primates
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Example Life expectancy vs. age of reproduction
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Example Number vs. size of offspring
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Life Histories and Elasticity Patterns Fast and
Slow Mammals
Age at first maturity 1 yr
Age at first maturity 2 yr
Elasticity
Age at first maturity gt2 yr
Elasticity