Effects of plant diversity on nutrient cycling in a California serpentine grassland

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Fishing and Population Growth

• I. Fishing the context how many fish can be
  sustainably harvested?
• A. Natives
• B. Increased fishing in the Pacific Northwest
• C. International issues
• D. Current catches
• II. Fundamentals of population growth
• Geometric and exponential growth
• Logistic growth
• Density dependence/density independence
• III. Maximum sustainable yield?
• A.     The approach
• B.      Problems with the MSY approach
• C. Alternatives to MSY

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II. Population Biology Basics

• Geometric and exponential growth
• Logistic growth
• Density dependence/independence

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A. Geometric and exponential growth

• How quickly do populations grow when they arent
  restricted (by resource limitation, predators,
  disease, whatever)?
• Relates to a populations capability to recover
  following collapse or upon colonizing a new
  habitat.

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1. The Simple Case Geometric Growth

• Constant reproduction rate
• Discrete breeding seasons (like birds, trees,
  bears, insects, and salmon)

• Suppose the initial population size is 1
  individual.
• The indiv. reproduces once dies, leaving 2
  offspring.
• How many if this continues for 5 years?

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Equations for Geometric Growth

• Growth from one season to the next Nt1 Nt?,
  where
• Nt is the number of individuals at time t
• Nt1 is the number of individuals at time t1
• ? is the rate of geometric growth

• If ? gt 1, the population will increase
• If ? lt 1, the population will decrease
• If ? 1, the population will stay unchanged

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Equations for Geometric Growth

• From our previous example, ? 2
• If Nt 4, how many the next breeding cycle?

• Nt1 Nt ? (4)(2) 8

• How many the following breeding cycle?
• Nt1 (4)(2)(2) 16

• In general, with knowledge of the initial N and
  ?,
• one can estimate N at any time in the future by
  
• Nt N0 ?t

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Using the Equations for Geometric Growth

• If N0 2, ? 2, how many after 5 breeding
  cycles?
• Nt N0 ?t (2)(2)5 (2)(32) 64
• If N0 1000, ? 2, how many after 5 breeding
  cycles?
• Nt N0 ?t (1000)(2)5 (1000)(32) 32,000

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2. Exponential Growth Continuous breeding,
overlapping generations
dN/dt rN, where dN/dt is the instantaneous
rate of change in number of individuals in the
population r is the intrinsic rate of increase
ind./ind/unit time
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Exponential Growth - Continuous Breeding
r explained r b - d, where b is the birth
rate, and d is the death rate Both are expressed
in units of indivs/indiv/unit time When bgtd,
rgt0, and dN/dt (rN) is positive When bltd, rlt0,
and dN/dt is negative When bd, r0, and dN/dt 0
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Equations for Exponential Growth

• If N 100 indiv., and r 0.1 indivs/indiv/day,
  how much growth in one day?
• dN/dt rN (0.1)(100) 10 individuals

• To predict N at any time in the future, one
  needs to integrate that equation
• Nt N0ert

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Exponential Growth in Rats

• In Norway rats that invade a new warehouse with
  ideal conditions, r 0.0147 indivs/indiv/day
• If N0 10 rats, how many at the end of 100
  days?
• Nt N0ert, so N100 10e(0.0147)(100) 43.5
  rats

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Comparing Exponential and Geometric Equations

• Geometric Nt N0?t
• Exponential Nt N0ert
• Thus, a reasonable way to compare growth
  parameters is er ?, or r ln(?)

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Assumptions of the Equations

• All individuals reproduce equally well.
• All individuals survive equally well.
• Conditions do not change through time.

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B. Logistic (sigmoidal) growth

• Exponential growth cant go on forever!

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Right?
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11.8
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Logistic growth

• Effects of lower birth rates and/or higher
  mortality at higher population sizes
• The curve
• The equation
• dN/dt rmaxN(1-N/K)

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Two questions relevant to sustainable salmon
harvest

• At what point in the curve is r (intrinsic rate
  of increase) at a maximum?

Graph it, based on a discussion of the logistic
curve
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11.13
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Two questions relevant to sustainable salmon
harvest

• 2) At what point in the curve is dN/dt (the
  number of new individuals added per unit time) at
  a maximum?

Graph it, based on a discussion of the logistic
curve
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C. Density dependence/density independence

• The shape of the logistic curve depends on higher
  d or lower b as population approaches carrying
  capacity. Why might this be?
• Density dependent process rates depend on the
  number of individuals relative to resource
  availability.
• Density independence b, d rates independent of
  number of individuals

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Summary
With unrestricted resources, populations
typically grow geometrically/exponentially -
Discrete breeding seasons (especially with
non-overlapping generations) geometric -
Continuous breeding exponential With
restricted resources or increased predation
pressure logistic growth. number of
individuals relative to K determines b and
d. May not get exact curve (overshoots,
timelags, etc.)
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