Calculating r from a life table

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Calculating r from a life table

• Assume stable age distribution, constant lx and
  mx
• Sx0 e-rx lx mx 1
• cannot solve for r algebraically
• find r by trial and error

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End 15th lecture
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Approximating r from a life table

• r ? ln(R0) / Tc
• ln( ?x0 lxmx )
• ____________________
  ____________
• ?x0 xlxmx /
  ?x0 lxmx
• e.g., Barnacles R0 1.2852, Tc 3.1003
• r ? ln(R0) / Tc ln(1.2852)/3.1003 0.0809
• l e0.0809 1.0843

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Interpreting r l

• dN / N dt r per capita increase
• dN / dt is the slope of the Nt vs. t curve
• r 0 population is not changing (l 1.0)
• r gt 0 population is increasing (l gt 1.0)
• r lt 0 population is decreasing (l lt 1.0)

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Mouse Problem

• x lx mx lxmx
• 0 1.00 0 0
• 1 0.50 3 1.50
• 2 0.25 3 0.75
• 3 0.125 3 0.375
• 4 0 -
• R0 2.625

• Female Mice
• 50 of survive from one year to the next
• Give birth to 3 female offspring / year, starting
  in yr. 1
• After 3 breeding seasons, death by old age
• construct life table
• calculate R0

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Reproductive value

• What is the expected contribution of a female to
  future generations?
• Reproductive value at age x
• Vx Stxlt mt / lx
• Note V0 R0
• Vx is maximal at the start of reproductive age
  classes
• Natural selection acts most strongly on age
  classes with high Vx

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Reproductive value(mice)
x lx mx lxmx 0 1.00 0 0 1 0.50 3 1.5
0 2 0.25 3 0.75 3 0.125 3 0.375 4 0 - R0
2.625
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A life table problem

• x lx mx qx
• 0 1.00 0 0.90
• 1 0.10 0 0.50
• 2 0.05 10 0.80
• 3 0.01 60 1.00
• 4 0 - -

• Calculate R0 Tc
• assume stable age dist.
• Approximate r l
• Assume deaths of 3 year olds due to hunting
• PROPOSAL open hunting for 2 year olds (new q2
  1.00)
• How is answer related to Vx?

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Stable age distribution

• Cx proportion of population in age class x at
  stable age distribution
• Cx l-x lx / Si0 l-i li
• Where l er, lx survivorship to age class x
• See p. 147
• Practice do problem 10.3, p. 155

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Life history

• Set of major events or transitions in an
  organisms life
• Age at reproduction
• Number of offspring (fecundity)
• Offspring size (affects survivorship)
• Reproductive allotment (offspring size x number)
• Longevity
• Directly related to fitness can be subject to
  strong selection
• Strongly affect R0 and r

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Early vs. Late reproduction
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Concepts

• Trade-off reproductive benefits of two traits
  are negatively related
• Phenotypic plasticity expression of different
  phenotypes by the same genotype, as a function of
  different environments
• Canalization expression of the same phenotype
  regardless of the environment

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End 16th lecture
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Life history trade offs

• Offspring size vs. offspring number
• Size at maturity vs. age at maturity
• Current vs. future reproduction
• Current reproduction vs. survival

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Plasticity of reproduction
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Exponential population growth

• Life tables -- detailed information on
  individuals
• Consider instead population as a whole
• Ignore age structure
• Ignore individual (genetic) variation
• Individual differences disappear

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Population change

• Nt number of individuals at time t
• What are the things that can change population?
• B total births in the population
• D total deaths in the population
• I number of immigrants
• E number of emigrants

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Population change

• Nt1 Nt B I - D - E
• ?N Nt1 - Nt B I - D - E
• If the population is closed then E 0 and I 0
• So ?N B - D
• In this case, time is discrete and the interval t
  is long
• For many species, t 1 year is a natural time
  unit

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Continuous population growth

• Consider a long lived iteroparous organism
• Consider t becoming very small
• Instantaneous birth rate b
• B b N or b B / N
• Instantaneous death rate d
• D d N or d D / N
• Births or deaths per individual

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Instantaneous growth rate

• As t becomes very small (differential equation)
• dN / dt bN - dN (b - d) N
• b - d r per capita rate of increase, so
• dN / dt r N
• same parameter we derived from life table
• so, Nt N0 ert
• exponential growth
• l er for a single time unit

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Exponential growth rates
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Reading

• Gotelli Chapter 1 (pp. 2 - 26)
• Milner Reserve
• Problems pp. 23 - 26
• Do problems 1, 2, 4
• Answers on pp. 24-25

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Some typical rates of increase
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Exponential growth

• No limit
• Population increases to infinity
• e.g., if t 1000, r 0.1 (flour beetles), and
  N0 100
• N1000 N0 e1000 (0.1) 100e100 2.69 x 1045
• Continuous growth, overlapping generations

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Semelparous organisms

• Spider e.g., Argiope
• Eggs hatch in spring
• spiders grow
• lay eggs in fall
• female dies
• Discrete or non-overlapping generations (parents
  never encounter offspring)

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Discrete exponential growth

• Discrete difference equation
• Nt1 l Nt
• where l finite rate of increase
• Nt2 l Nt1 l (l Nt) l 2 Nt
• Nt l t N0

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Exponential growth is unrealistic

• Growth without limits is physically impossible
• For short periods populations may follow
  exponential growth
• laboratory populations (E. coli in test tube)
• Introduced species (while spreading)
• Human populations at various times

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End 17th lecture
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Populations in nature

• Do not explode routinely. Why?
• Hypotheses
• dN / Ndt varies over time or space, averages 0
• Density independent population growth
• dN / Ndt declines as N increases
• Density dependent population growth

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Reading

• Gotelli Ch. 2 (pp. 28 - 34, 37-39, 43-49)
• Milner reserve
• Problems p. 50
• Do problems 1, 2, 3
• Answers on pp. 51-52

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Density dependent population growth

• As N goes up, dN / N dt goes down
• Stabilizes population
• Why might dN / N dt decrease with N ?
• resource depletion
• direct interference
• predation, parasitism
• emigration

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Resource depletion

• Resource -- something necessary and capable of
  being depleted (e.g., food, water, space)
• As N increases resource per capita decreases
• May reduce b
• May increase d

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Direct interference

• Interference - direct harm of one individual by
  another
• As N increases encounters and aggression increase
• May reduce b harm, waste of time
• May increase d harm, cannibalism

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Predation parasitism

• Predators learn to attack more common prey
• As N increases attacks or pathogen transmission
  may increase
• May increase d (obvious)
• May reduce b (time spent hiding)

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Emigration

• Leave when it gets crowded
• As N increases proportion leaving increases

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Density independent population growth
(exponential)
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Density dependent population growth (logistic)
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Logistic population growth

• dN / N dt r0 - (r0 / K )( N )
• linear decline of dN / N dt with increasing N
• When N K , dN / N dt 0, no population growth
• K carrying capacity
• density at which population growth ceases
• r0 intrinsic rate of increase
• rate of increase as N ? 0
• NOTE r in text
• logistic r0 is not equivalent to exponential r

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Logistic population growth

• dN / N dt b - d (for a closed population)
• density dependence in the logistic model implies
  that either
• b decreases with N or
• d increases with N
• Linear relationships of b or d to N

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Density dependent effects on b and d (logistic)
Nt
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Alternative ways of expressing logistic growth

• dN / N dt r0 (K - N ) / K
• per capita population growth
• dN / dt r0N (K - N ) / K
• total population growth
• Nt K / 1 (K-N0 ) / N0 e-r0 t
• population size

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Logistic population growth
at any point slope of curve dN / dt
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Logistic population growth
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Logistic population growth

• dN / N dt r0 (K - N) / K
• Interpreting (K - N ) / K
• Proportion of unused carrying capacity
• consider carrying capacity based on space (e.g.,
  plants, barnacles)

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End 18th lecture