Alan Hastings

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Alan Hastings

• Dept of Environmental Science and Policy, Univ of
  Calif, Davis
• two.ucdavis.edu

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Lessons from several courses

• Population Biology
• Taught from a perspective that emphasizes
  quantitative approaches and models
• Audience is typically college juniors and seniors
• Have taken some biology, and at least one quarter
  of calculus
• Today, Ill look at questions that involve only
  algebra
• Quantitative methods in Population Biology
• First year students in graduate school

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Population biology

• Quantitative reasoning essential
• We count the number of individuals
• We count the frequencies of different types in
  populations
• Find that biology students have often not been
  comfortable with mathematics
• Artificial examples
• New textbooks may help

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Important practical issues

• Epidemics
• Hoof and mouth in UK
• Childhood diseases
• AIDS
• Insect pests
• Fisheries

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Overall concepts of interaction between math and
biology
Mathematical formulation
Biological conclusion
Biological question
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Overall concepts of interaction between math and
biology
Mathematical formulation
Should not be a linear process- feedback
Biological conclusion
Biological question
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Overall concepts of interaction between math and
biology
Mathematical formulation
Biological conclusion
Check step with biological reasoning
Biological question
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Combining computer and mathematical approaches

• Problems in biology are nonlinear
• Involve quadratic and more complex functions
• So numerical (computer solutions are essential)
• What numerical platform
• Easy for students to pick up
• Easy for me to put a relatively nice interface on
• Spreadsheet, with students just having to enter
  numbers
• Other solutions possible

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Then why analytic mathematical reasoning at all?

• Need to think about qualitative behavior
• Will population grow (at all) may be much more
  important question than how fast
• Understand and get general results
• Get deeper understanding of why
• Emphasize interplay among biology, math and
  numerical/computer results

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Geometric or exponential growth

• Question how do populations grow if resources
  are unlimited?
• (Models are often most useful if their
  predictions are not upheld)
• N(t1)R N(t)
• What is R?
• Exact example univoltine insects
• From this equation we can predict all future
  population sizes

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Geometric growth continued

• N(t1)R N(t)
• N(t2)R N(t1)
• N(t2)R (N(t1))

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Geometric growth continued

• N(t1)R N(t)
• N(t2)R N(t1)
• N(t2)R (RN(t))
• N(t2)R2N(t)
• N(t)RtN(0)
• This last formula gives an exact prediction of
  future population size

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Geometric growth continued

• N(t)RtN(0)
• This formula gives an exact prediction of future
  population size
• But more interesting
• Rgt 1 grows
• R lt 1 declines
• R 1 only way to get equilibrium
• Illustrate with Excel

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Mud turtle
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Growth with two age classes

• Build on ideas of previous example
• First develop from basic principles
• Then use matrices
• Stable age structure
• Develop analytic solutions
• Then numerical ones

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Other topics

• Population genetics
• Frequency changes in one locus two allele model
• Biston betularia example nice
• Numerical solutions
• Drift
• Discuss the example of one individual
• Numerical examples
• More numerical examples

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Other examples

• Epidemics and the threshold theorem
• BN/g

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Take home messages

• Emphasize tight interplay between biology and
  quantitative reasoning
• Use the simplest analytic models possible
• Numerical approaches allow investigation
• Develop both mathematical and biological themes
  but always focus on the biological question
• Emphasize the importance of failures of models
• two.ucdavis.edu

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Anaphes flavipes(Hymenoptera Mymaridae)
A. flavipes late pupal stage within host. Note
the darkened body.PHOTO USDA, APHIS, PPQ,
Niles Plant Protection Center
A. flavipes early pupal stage within host. Red
compound eyes are the first visible
feature.PHOTO USDA, APHIS, PPQ, Niles Plant
Protection Center
A. flavipes female on host egg.PHOTO PHOTO
USDA, APHIS, PPQ, Niles Plant Protection Center
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• Pleolophus basizonus is an important
  ectoparasitoid of diprionids. During outbreaks
  this species can cause high mortality rates.

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• Egg of P. basizonus. The eonymph had been
  paralyzed by the female parasitoid prior to
  oviposition.

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Pteromalid Wasp Parasitiod of Stable Flyand
House Fly Puparia
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Circular hole left by wasp parasitoid emerging
from (left) an armored scale and (right) a soft
scale.
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At left, larval and (right), pupal stages of a
parasitoid.
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Left Adult female Encarsia inaron.Right E.
inaron exit holes (arrow) from Ash whitefly
nymphs.  M.Rose (both)
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Braconid Larvae Emergingfrom Mature Red Admiral
Caterpillar I