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Modeling Butterfly Populations
Richard Gejji
Justin Skoff
Overview
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
Introduction
Model looks at effects of weather on populations Specifically- body temperature
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
Beginning Assumptions
Ignore mating/males All butterflies are female and are always
fertilizedAll adults die at the end of a season,
leaving the eggs to hatch next seasonNo predatorsThe 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)
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
Model Derivation
A butterfly’s body temperature (BT) and PGI type (γ) affect its chances of flying
Body Temperature
Flight Probability
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:
Genotype Subscript Notation (i) Efficiency Stability
AA 1 “somewhat” efficient, γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3 “pretty” stable
Body Temperature
Flight Probability
Model Derivation
Equations that model overheating effects:
θ is the critical temperature
if BT ≥ θi, then pi(t, BT) = γi P(BT)(1 - t δ) (1) if BT < θi, then pi(t, BT) = γi P(BT)
Model Derivation
Variables:
xi(n) = number of eggs of type i fi(n) = number of flying adults of type i ri(n) = number of grounded adults of type i pi = probability of flight for type i β = flying rate α = landing rate BT = body temperature
Ni1
3ri fi Ai ri fi
ai AiN
Model Derivation
The change in x, f, and r over a time step Δt are represented by the following equations:
fi(t + Δt) = fi(t) + β pi(t, BT) ri(t) Δt – α[1 - pi(t, BT)] fi(t) Δt ri(t + Δt) = ri(t) + α [1 - pi(t, BT)] fi(t) Δt - β pi(t, BT) ri(t) fi(t) Δt xi(t + Δt) = xi(t) +
j1
3
1 p jBTfjtPjproduces it
Need to find this
Model Derivation
Use genotype ratios and Mendelian genetics to find P(j produces i)
i 1 2 3 1 a1+1/2*a3 0 a2+1/2*a3
j 2 0 a2+1/2*a3 a1+1/2*a3 3 1/2*a1+1/4*a3 1/2*a2+1/4*a3 ½
Table 2 This table will be referred to as matrix T2(j, i). E.g., P(j produces i) = T2(j, i)
Model Derivation
Skipping a bunch of steps (in the interest of time) we get the final equation for number of eggs:
/3]tδcρ/2t)δcρc(2ρ)tcρc[(ρ * α (t) x 32jji,
2j
2jji,
2jij,
2jji,
2jij,j
ji
ρ j= γ j P(bt) and cj i=Aj T2j,i
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.
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
Results
First, here is the actual probability curves we used:
Genotype
Subscript Notation (i) Efficiency Stability
AA 1 “somewhat” efficient, γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3 “pretty” 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 Efficiency Stability
AA 1 “somewhat” efficient, γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3 “pretty” stable
Results
Bt = 32
Genotype Type Efficiency Stability
AA 1“somewhat”
efficient, γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3“pretty”
stable
Results
Bt = 37
Most efficient fliers die off because they don’t want to land. So both not flying too much and flying too much is a death sentence
Genotype Type Efficiency Stability
AA 1“somewhat” efficient,
γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3 “pretty” 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 Efficiency Stability
AA 1“somewhat” efficient,
γ2 unstable
BB 2 Non-efficient, γ1 “very” stable
AB 3 “very” efficient, γ3 “pretty” 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:
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
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
IntroductionBeginning AssumptionsModel DerivationResultsCritique of Results and AssumptionsConclusion
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.