Modeling Butterfly Populations

1
Modeling Butterfly Populations

• Richard Gejji
• Justin Skoff

2
Overview

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

3

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

4
Introduction

• Model looks at effects of weather on populations
• Specifically- body temperature

5

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

6
Beginning Assumptions

• Ignore mating/males
• All butterflies are female and are always
  fertilized
• All adults die at the end of a season, leaving
  the eggs to hatch next season
• No predators
• The change in number of flying and grounded
  adults with respect to time is zero

7
Beginning Assumptions, Continued

• Reproduction is reliant upon flight
• The probability of egg laying while flying is
  100
• The probability of flight is based on
• Body Temperature
• Genotype
• Time (sometimes)

8

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

9
Model Derivation

• A butterflys body temperature (BT) and PGI type
  (?) affect its chances of flying

10
Model Derivation

• There are 3 PGI genotypes
• Stability is defined how flight probabilities
  react to temperatures
• Lower stability - probabilities affected more by
  overheating
• Due to PGI denaturing
• Efficiency

11
Model Derivation

• Equations that model overheating effects
• ? is the critical temperature

12
Model Derivation

• Variables

13
Model Derivation

• The change in x, f, and r over a time step ?t are
  represented by the following equations

Need to find this
14
Model Derivation

• Use genotype ratios and Mendelian genetics to
  find P(j produces i)

15
Model Derivation

• Skipping a bunch of steps (in the interest of
  time) we get the final equation for number of
  eggs

16
Model derivation

• Dealing with seasons
• Use x(t length of season) to find the number of
  eggs laid
• Assume 5 survive to make it to next season
• Use the number of these new adults as the
  population parameters for this second season.
  I.e, the P(j - i) table is reclaculated.

17

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

18
Results

• First, here is the actual probability curves we
  used

19
Results

• We use a ß 3 chosen arbitrarily, and ?1 38,
  ?2 45, and ?3 41 which were chosen to fulfill
  the table definition of stability. We start with
  an even initial population of 30 of each type.
• For bt 31

20
Results

• Bt 32

21
Results

• Bt 37
• Most efficient fliers die off because they dont
  want to land. So both not flying too much and
  flying too much is a death sentence

22
Results

• Bt 39
• Overheating does not seem to have too much effect
  because for these body temperature ranges, the
  flight probability is still large

23
Results

• BT 40
• As the body temperature increases, the flight
  probability decreases and efficient types can
  once again be efficient. We also see overheating
  take effect and show the near extinction of type
  1, while type 2 and 3, which are more stable
  thrive.

24
Results

• At the right body temperature type 3 alone can
  support the species

25

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

26
Critique

• Many of our assumptions have little or zero
  experimental evidence
• Linear changes of flying and landing adults are
  proportional to the probabilities of flight and
  non flight
• Fast flying mechanics
• Flying rate coefficient is equal to the landing
  rate coefficient
• Assumption of constant body temperature incorrect

27
Critique

• The result that too much flying will cause a type
  to die off is flawed
• In real life weather fluctuations would change
  that

28

• Introduction
• Beginning Assumptions
• Model Derivation
• Results
• Critique of Results and Assumptions
• Conclusion

29
Conclusion

• The original impetus of this experiment was to
  investigate whether or not it is possible for a
  fit species to die out due to the decrease of the
  unstable types from higher body temperatures.
• According to this model, we can predict that the
  size of the type 1 and type 2 populations are
  enough to control whether or not type 3
  increases or decreases, however, if the weather
  is favorable, it is possible for type 3 to not
  only survive, but to generate the existence of
  the other types.

30
Conclusion

• Investigation needs to be done on how reasonable
  the flight/landing assumptions are. If they are
  accurate, investigate if it is possible that
  butterflies can die out due to a high flight
  probability
• According to the model, fluctuations in the size
  of type 1 and type 2 can determine growth or
  decline of type 3. Also, it is possible for a
  collection of heterogeneous genotypes to sustain
  the population.
• As far as global warming goes, the equation
  predicts for a small range, the unstable genotype
  will almost die out while the stable types
  survive and sustain the dying genotype. However,
  if we exceed this range, all the butterflies die.