Modeling Butterfly Populations Richard Gejji Justin Skoff

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

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

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

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

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

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)

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

Model Derivation A butterflys body temperature (BT) and PGI type ( γ ) affect its chances of flying

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: GenotypeSubscript Notation (i)EfficiencyStability AA1somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3pretty stable

Model Derivation Equations that model overheating effects: θ is the critical temperature

Model Derivation Variables:

Model Derivation The change in x, f, and r over a time step Δt are represented by the following equations: Need to find this

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

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

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.

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

Results First, here is the actual probability curves we used: Genotype Subscript Notati on (i)EfficiencyStability AA1somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3pretty stable

Results We use α = β = 3 chosen arbitrarily, and θ1 = 38, θ2 = 45, and θ3 =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 Genotype Type EfficiencyStability AA1somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3pretty stable

Results Bt = 32 Genotype Type EfficiencyStability AA1 somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3 pretty stable

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 Genotype Type EfficiencyStability AA1 somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3pretty stable

Results Bt = 39 Overheating does not seem to have too much effect because for these body temperature ranges, the flight probability is still large Genotype Type EfficiencyStability AA1 somewhat efficient, γ2unstable BB2Non-efficient, γ1very stable AB3very efficient, γ3pretty stable

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.

Results At the right body temperature type 3 alone can support the species:

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

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

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

Introduction Beginning Assumptions Model Derivation Results Critique of Results and Assumptions 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.

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.