Exponential Population Change

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Exponential Population Change

• Next Time Logistic Population Change

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Kudzu Introduced
Whitetails Range Expansion Colonization
Lesser Snow Geese Anthropogenic
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Population Dynamics

• Why do we care?
• Can past dynamics inform us about current and
  future dynamics
• Nt1 Nt B I - D - E
• Simple, but
• Models Simplifications of Reality
• Simple enough to be interpretable and complex
  enough to be realistic
• Managing in the face of uncertainty, with the
  goal of reducing it
• Adaptive Management

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Exponential Population Models

• Ingredients
• Lambda (?) finite rate of change (increase or
  decrease)
• The proportional change in abundance over an
  interval of time

?
?
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How do we interpret Lambda?

• ? 1.00, abundance is constant
• ? gt 1.00, abundance is increasing
• Rate (? - 1.0)100
• ? lt 1.00, abundance is decreasing
• Rate (? - 1.0)100

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How big, or small, is Lambda?
?
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Assumptions

• No immigration or emigration
• Relax later
• Constant birth and death rate
• No sex, time, or space specificity
• Rate of change is independent of abundance
• Growth occurs in discrete steps

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Exponential Change

• Change by constant factor each step
• Multiply by constant (geometric)
• Not Arithmetic change change by constant
  difference (add a constant value)
• Nt1 Nt ? (e.g, 2004 to 2005)
• Nt2 Nt ? ?
• Nt2 Nt ?2
• and
• Nt2/ Nt ?2 . ? sqrt (?2 ) or tth root
• not ?/2

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Exponential Change in Cows

• 1.44
• N1 10

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Other Characteristics of ?

• Can not changed by scaling
• ?/2 NO
• Can not be added together
• Geometric
• Sometimes expressed as R
• R (? - 1)

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Instantaneous Rate of Change

• r Malthusian parameter
• Continuous not Discrete
• Differential Equation
• Very small time interval

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Characteristics of r

• r 0.0, abundance stable
• r gt 0.0, abundance increasing
• r lt 0.0, abundance decreasing

• er
• r ln(?)

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Other Characteristics of r

• Can be summed and changed by scaling
• E.g., if r 0.101 day, what is r per hour?
• (0.101/24 0.0042)
• What is ? per hour?

?
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Can you Determine Doubling Rate or Halving Rate
for a Population?
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Graphic Representation of Exponential Change for
No 10 and ? 1.2
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(No Transcript)
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Kudzu Southeastern U.S.

• Introduced
• Japan 1978
• Now covers 7 million acres
• Other introduced plants
• Purple Loosestrife
• Russian Knapweed

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George Reserve Deer
In 1928, six white-tailed deer (2 males, 4
females) were introduced to the Edwin S. George
Reserve, a two-square-mile fenced area in
southeastern Michigan. Six years later, in 1933,
the first drive census showed 160 deer. This
remarkable growth of deer population became a
landmark in the young field of wildlife
management.
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Lesser Snow Geese - Anthropogenic
Mid-continent
La Perouse Bay