Modeling Butterfly Populations Richard Gejji Justin Skoff.

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.