Alan Hastings
Dept of Environmental Science and
Policy, Univ of Calif, Davis
two.ucdavis.edu
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, I’ll look at questions that involve only algebra
• Quantitative methods in Population Biology
– First year students in graduate school
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
Important practical issues
– Epidemics
• Hoof and mouth in UK
• Childhood diseases
• AIDS
– Insect pests
– Fisheries
Overall concepts of interaction
between math and biology
Mathematical formulation
Biological question
Mathematical Mathematical Mathematical Mathematical
step
step
step
step
Biological
conclusion
Overall concepts of interaction
between math and biology
Mathematical formulation
Mathematical Mathematical Mathematical Mathematical
step
step
step
step
Should not be a linear process- feedback
Biological question
Biological
conclusion
Overall concepts of interaction
between math and biology
Mathematical formulation
Biological question
Mathematical Mathematical Mathematical Mathematical
step
step
Check step
with
biological
reasoning
step
step
Biological
conclusion
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
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
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(t+1)=R N(t)
• What is R?
• Exact example – univoltine insects
• From this equation we can predict all
future population sizes
Geometric growth continued
• N(t+1)=R N(t)
• N(t+2)=R N(t+1)
• N(t+2)=R (N(t+1))
Geometric growth continued
•
•
•
•
•
•
N(t+1)=R N(t)
N(t+2)=R N(t+1)
N(t+2)=R (RN(t))
N(t+2)=R2N(t)
N(t)=RtN(0)
This last formula gives an exact prediction
of future population size
Geometric growth continued
N(t)=RtN(0)
• This formula gives an exact prediction of
future population size
• But more interesting:
• R> 1 grows
• R < 1 declines
• R = 1 only way to get equilibrium
• Illustrate with Excel
Mud turtle
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
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
Other examples
• Epidemics and the threshold theorem
– BN/g
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
Anaphes flavipes
(Hymenoptera: Mymaridae)
A. flavipes female on host
egg.
PHOTO: 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 late pupal stage
within host. Note the
darkened body.
PHOTO: USDA, APHIS,
PPQ,
Niles Plant Protection
Center
• Pleolophus basizonus is an important
ectoparasitoid of diprionids. During
outbreaks this species can cause high
mortality rates.
• Egg of P. basizonus. The eonymph had been
paralyzed by the female parasitoid prior to
oviposition.
Pteromalid Wasp Parasitiod of Stable
Fly
and House Fly Puparia
Circular hole left by wasp parasitoid
emerging from (left) an armored scale
and (right) a soft scale.
At left, larval and (right), pupal
stages of a parasitoid.
Left: Adult female
Encarsia inaron.
Right: E. inaron exit
holes (arrow) from
Ash whitefly
nymphs. M.Rose
(both)
Braconid Larvae Emerging
from Mature Red Admiral
Caterpillar – I