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Starting Calculus for Biologists

Shared Common Good Models

James K. Peterson

Department of Biological Sciences and Department of Mathematical SciencesClemson University

April 22, 2014

Outline

1 Opening Thoughts

2 A Shared Common Good Model

3 The Abstract Version

4 A First Whisper of Hamilton’s Rule

Abstract

This lecture begins our discussion of shared common good models.

Opening Thoughts

We are now ready to look at another interesting model fromevolutionary biology. Recall in the Chapter on Viability, wediscussed how a phenotype of type A might spread through apopulation. We didn’t say much about what the phenotype A mighthave been, but what we were really thinking about was an allelethat codes for the high level trait we might call altruistic behavior.

So type A is the allele that codes for altruism and type B the allelethat represents the lack of altruistic behavior. Note our discussionis about characteristics of human populations that are very highlevel – not at all like our previous efforts on low level things likeprotein transcription.

Our discussions here are based on some of in Chapter 3 ofMcElreath and Boyd’s book Mathematical Models of SocialEvolution: A Guide for the Perplexed from 2007.

Opening Thoughts

A major problem in biology is how altruistic behavior could evolve ina population. Key work was done by W. Hamilton in 1963 in thepaper The Evolution of Altruistic Behaviour in The AmericanNaturalist and 1964, in the paper The genetical evolution of socialbehavior in the Journal of Theoretical Biology. You should look atthose papers. Our discussion in this chapter just touches on allthese interesting things.

Before we go on, we want to take a minute and discuss thephilosophy of this course again. Since this could be your last coursein mathematics (although we hope we have encouraged you to takemore!), we have been working hard to show you that to developinsight into complicated problems requires you to think hard anddeep about the biology and behavior you see in your data and soforth and then use your training in mathematics and computationaltechniques to help you build simple models that confirm or denyyour suspicions.

This is a very iterative process and we go back and forth betweenour different versions as we hone in on a better understanding. Butthe key really is to learn how to think for yourself.

Opening Thoughts

A proper point of view for teaching mathematics and computationto biology majors involves the inculcation of digital literacy andproblem solving capability.

An interesting scenario would be to require calculus for biologist’scourse MATHSC 106-Bio to be a prerequisite for BIOL111.Students would be introduced to computational tools via MatLabin MATHSC 106-Bio and the statistics package R. BIOL110 wouldteach R to its students and MATHSC 106-Bio would teach MatLabto its students.

MATHSC 106-Bio covers the basics of calculus and modeling up toand including portions of calculus of several variables. Regression iscovered carefully in MATHSC 106-Bio also as welll as a generaldiscusion of the evolutionary biology law known as Hamilton’s Rule.

Opening Thoughts

All of this background is then made a prerequisite for the six coursesBIOSC 443 (Ecology) , BIOSC 302 Animal Diversity), BIOSC335 (Evolutionary Biology), BIOSC401 (Physiology), BIOSC461 (Cell Biology) and GEN 440 (Bioinformatics). This is donevia the inclusion of quantitative modules which use mathematicsand computational modeling to illustrate important concepts ineach class.

The instructors can assume the students have familiarity with theuse of R for statistical modeling and MatLab for mathematicalmodeling. Such an assumption frees the instructor to cover materialat a more sophisticated level enabling a better transition to modernbiological usage.

We would then see the a web of intellectual tools spread throughthe biology major with the dependences as shown in the next figure.

Food for thought, eh?

Opening Thoughts

EXST 301

M106Bio

BIOL110

BIOS302

BIOS443

BIOS335

BIOS401

BIOS461

GEN440

BIOL111

Opening Thoughts

Now in the sections that follow these remarks, we are going toexplore the idea of altruistic behavior and how it might be a winningthing to pass on the future generations in a variety of ways using anumber of different modeling paradigms. It is a great example toshow you how those of us who model keep trying to find better andbetter ways to understand.

Opening Thoughts

A Shared Common Good Model

We could define altruism as a behavior which reduces the fitness ofthe person who has this behavior but increases the fitness of theperson who benefits from this altruistic behavior.

However, another notion of altruism is that it is a behavior whichreduces the fitness of the person who engages in it relative to thepeople who benefit from it.

So there are two types of altruism: absolute and relative. Let’sdesign a simple model to help us think about this more clearly. Let’salso call the person or individual in the population that uses somebehavior an actor.

We have some animals that live in a pool: following McElreath,these are salamanders. Each salamander can choose to poop in thepool or walk outside of the pool to do it.

So the act of walking outside the pool is a cost to the salamanderthat does it. But every creature in the pond shares the commonbenefit of having cleaner water.

Consider the following matrix for interaction of two salamanders,Salamander 1 and Salamander 2. This table records the fitnessconsequences of each of four possible interactions.

In the interaction of two salamanders, each has a 50% chance ofeither cooperating by going out of the pool to poop or staying inthe pool to poop. There are four types of interactions.

they both go outside the pool to poop is calledV(walk out, walk out).the salamander 1 is nice and walks outside the pool to poopbut salamander 2 poops in the pool. Call thisV(walk out, stay in).salamander one poops in the pool and salamander two walksout of the pool to poop. Call this V(stay in, walk out).salamander one and salamander two both poop in the pool.Call this V(stay in, stay in).

Since going outside of the pool to poop is an action that helps thegroup, we can call this cooperation.

If a salamander stays in the pool to poop, the salamander goesagainst the group benefit; we say the salamander defects. Thegeneral form of our matrix is

Salamander TwoSalamander One Go outside to poop Stay in to poopGo outside to poop V(walk out, walk out) V(walk out, stay in)Stay in to poop V(stay in, walk out) V(stay in, stay in)

Let B be the total benefit for the population; we call this the groupbenefit. We let c be the private cost to a salamander to leaving thepool to poop.

What should these four fitness consequences be?

V (walk out, walk out): since this option is chosen 50% of the timethe the average group benefit seen is B/2. However, Salamanderone uses this option at the same time salamander two uses it. Sothere is an additional B/2 that comes from that choice. The cost ofgoing outside to poop is always c so the total fitness for player oneis upgraded by B − c .

V (stay in, walk out): since Salamander one stays in the pool thereis no private cost incurred. Thus, the upgrade in fitness forSalamander one is B/2.

V (walk out, stay in): since Salamander one goes out of the pool topoop, the fitness increase Salamander one sees is B/2 − c . There isno effect from Salamander two’s choice on Salamander One’sfitness.

V (stay in, stay in) If both salamanders poop in the pool, this doesnot change the group benefit, so the fitness increase of salamanderone should be 0.

We can summarize what we have said into a table as shown below.The table shows the fitness consequences to Salamander one. Thisis called the Payoff Matrix for Salamander One.

Salamander TwoSalamander One Go outside to poop Stay in to poopGo outside to poop B − c B/2 − cStay in to poop B/2 0

When does the action to be altruistic and help the group spreadthroughout the population? This happens when the best choice foreach salamander to make is the one that increases the overall good.In our example above, if it was true that B/2 − c > 0, then we havethe other terms are all larger than B/2 − c as

B − c = B/2 + B/2 − c > B/2 − c and B/2 > B/2 − c

In this case, the best choice to make is always that bothsalamander’s cooperate for the group benefit and go outside thepoop to poop. So this action spreads throughout the populationand becomes dominant.

B − c = B/2 + B/2 − c > B/2 − c and B/2 > B/2 − c

Example

Let B = 3 and c = 1 in our salamander model. Write down the payoffmatrix and determine if the action to go outside the pool, i.e. bealtruistic, spreads throughout the population.

Solution

For these values, the payoff matrix is

Salamander Two

Salamander One Go outside to poop Stay in to poopGo outside to poop 2 1/2Stay in to poop 3/2 0

and since B/2 − c = 1/2 > 0, this action will spread throughout thepopulation.

Example

Let B = 2.0 and c = 0.8 in our salamander model. Write down thepayoff matrix and determine if the action to go outside the pool, i.e. bealtruistic, spreads throughout the population.

Solution

Salamander Two

Salamander One Go outside to poop Stay in to poopGo outside to poop 1.2 0.2Stay in to poop 1.0 0

and since B/2 − c = 0.2 > 0, this action will spread throughout thepopulation.

Example

Let B = 2 and c = 1.3 in our salamander model. Write down the payoffmatrix and determine if the action to go outside the pool, i.e. bealtruistic, spreads throughout the population.

Solution

Salamander Two

Salamander One Go outside to poop Stay in to poopGo outside to poop 0.7 −0.3Stay in to poop 1.0 0

and since B/2− c = −0.3 < 0, this action will not spread throughout thepopulation.

Homework 76

76.1 Let B = 4 and c = 3 in our salamander model. Write down thepayoff matrix and determine if the action to go outside the pool, i.e.be altruistic, spreads throughout the population.

The Abstract Version

Now let’s make this a bit more abstract. Label the cooperativeaction where the salamander goes out of the pool to poop as C andthe action where the salamander goes against the group benefit andpoops in the pool by D. Then our matrix can be written moresuccinctly as

Salamander TwoSalamander One C DC B − c B/2 − cD B/2 0

Note that our simple (and silly ) example could have been aboutsomething else. It works as long as there is a shared common goodand a private cost for helping the group.

Now let’s make this a bit more abstract. Label the cooperativeaction where the salamander goes out of the pool to poop as C andthe action where the salamander goes against the group benefit andpoops in the pool by D. Then our matrix can be written moresuccinctly as

Salamander TwoSalamander One C DC B − c B/2 − cD B/2 0

Note that our simple (and silly ) example could have been aboutsomething else. It works as long as there is a shared common goodand a private cost for helping the group.

For example, it is well known that if a squirrel is on the ground andsees a predator, often the squirrel will stand on its feet and utterloud piercing cries to alerts its mates of the danger. Of course, thishas a great personal cost: the squirrel that calls out the alarmincreases it own risk greatly!

So the squirrel example fits into our common good model nicelytoo. So you don’t have to think about poop but we figured wecould grab your attention by starting with it!

In general, instead of salamanders in our example, we would callthem agents and we would rewrite our table as

Agent TwoAgent One C DC B − c B/2 − cD B/2 0

Finally, we define the following four things:

V(C, C): payoff or fitness consequence to agent one’s choiceof C given that agent two chooses C.V(C, D): payoff or fitness consequence to agent one’s choiceof C given that agent two chooses D.V(D, C): payoff or fitness consequence to agent one’s choiceof D given that agent two chooses C.V(D, D): payoff or fitness consequence to agent one’s choiceof D given that agent two chooses D.

which are just like the V’s we defined earlier.

which are just like the V’s we defined earlier. Then the generalfitness matrix or payoff matrix is

Agent TwoAgent One C DC V(C, C) V(C, D)D V(D, C) V(D, D)

Finally, we define the following four things:

V(C, C): payoff or fitness consequence to agent one’s choiceof C given that agent two chooses C.V(C, D): payoff or fitness consequence to agent one’s choiceof C given that agent two chooses D.V(D, C): payoff or fitness consequence to agent one’s choiceof D given that agent two chooses C.V(D, D): payoff or fitness consequence to agent one’s choiceof D given that agent two chooses D.

which are just like the V’s we defined earlier.

which are just like the V’s we defined earlier. Then the generalfitness matrix or payoff matrix is

Agent TwoAgent One C DC V(C, C) V(C, D)D V(D, C) V(D, D)

The total fitness consequence to a choice of D is then called W(D)and is given by

W(D) = W0 + p V(D, C) + (1 − p) V(D, D).

where in general p is the probability the choice of C is made and1 − p, the probability the other choice of D is made.

Also the term W0 is a baseline fitness level the population hasbefore any agent choices have been made.

Similarly, the fitness consequence to a choice of C is then calledW(C) and is given by

W(C) = W0 + p V(C, C) + (1 − p) V(C, D).

W(D) = W0 + p V(D, C) + (1 − p) V(D, D).

W(C) = W0 + p V(C, C) + (1 − p) V(C, D).

W(D) = W0 + p V(D, C) + (1 − p) V(D, D).

W(C) = W0 + p V(C, C) + (1 − p) V(C, D).

We can simplify this a bit more (well, we think it simplifies but thenwe like abstraction!) using a little matrix and vector multiplication.The definitions above can be written more succinctly as[

W(C)W(D)

[V(C, C) V(C, D)V(D, C) V(D, D)

1 − p

So for our example, we have[W(C)W(D)

[B − c B/2 − cB/2 0

] [0.50.5

[W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W0 + (0.5)(B/2) + (0.5)(0)

]We conclude

W(C) = W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W(D) = W0 + (0.5)(B/2).

W(C)W(D)

[V(C, C) V(C, D)V(D, C) V(D, D)

1 − p

]So for our example, we have[

W(C)W(D)

[B − c B/2 − cB/2 0

] [0.50.5

[W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W0 + (0.5)(B/2) + (0.5)(0)

We conclude

W(C) = W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W(D) = W0 + (0.5)(B/2).

W(C)W(D)

[V(C, C) V(C, D)V(D, C) V(D, D)

1 − p

]So for our example, we have[

W(C)W(D)

[B − c B/2 − cB/2 0

] [0.50.5

[W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W0 + (0.5)(B/2) + (0.5)(0)

]We conclude

W(C) = W0 + (0.5)(B − c) + (0.5)(B/2 − c)

W(D) = W0 + (0.5)(B/2).

Now for cooperation to spread through the population, the changein difference W(C) − W(D) should be positive. Note this change isjust the amount above the baseline W0 and so

W(C) − W(D) = (0.5)(B − c) + (0.5)(B/2 − c) − (0.5)(B/2)

= B/4 + 2 (B/2 − c) − B/4 = 2 (B/2 − c).

This difference is positive if we assume B/2 − c > 0. Hence, wehave our first inkling that this relationship, B/2 > c is an importantone that is linked to the spread of altruistic behavior throughout thepopulation. Indeed, it implies that the benefit to cooperation isalways increasing!

Now for cooperation to spread through the population, the changein difference W(C) − W(D) should be positive. Note this change isjust the amount above the baseline W0 and so

W(C) − W(D) = (0.5)(B − c) + (0.5)(B/2 − c) − (0.5)(B/2)

= B/4 + 2 (B/2 − c) − B/4 = 2 (B/2 − c).

This difference is positive if we assume B/2 − c > 0. Hence, wehave our first inkling that this relationship, B/2 > c is an importantone that is linked to the spread of altruistic behavior throughout thepopulation. Indeed, it implies that the benefit to cooperation isalways increasing!

Example

Determine if action C will spread throughout the population forV(C, C) = 3, V(C, D) = 2, V(D, C) = 1 and V(D, D) = 0.5 when theprobability of action C is p = 0.5.

Solution

The payoff matrix isAgent Two

Agent One C DC 3 2D 1 0.5

The payoff for action C is then

W(C) = W0 + p V(C, C) + (1 − p) V(C, D)

= W0 + (0.5) (3) + (0.5) (2) = W0 + 2.5.

Solution

and the payoff for action D is

W(D) = W0 + p V(D, C) + (1 − p) V(D, D)

= W0 + (0.5) (1) + (0.5)(0) = W0 + 0.5.

The difference W(C) − W(D) is 2.0 and so action C will spread.

Example

Determine if action C will spread throughout the population forV(C, C) = 2, V(C, D) = −2, V(D, C) = 1 and V(D, D) = 0 when theprobability of action C is p = 0.5.

Solution

The payoff matrix isAgent Two

Agent One C DC 2 −2D 1 0

The payoff for action C is then

W(C) = W0 + p V(C, C) + (1 − p) V(C, D)

= W0 + (0.5) (2) + (0.5) (−2) = W0 + 0.0.

Solution

and the payoff for action D is

W(D) = W0 + p V(D, C) + (1 − p) V(D, D)

= W0 + (0.5) (1) + (0.5)(0) = W0 + 0.5.

The difference W(C) − W(D) is −0.5 and so action C will not spread.

Homework 77

77.1 Determine if action C will spread throughout the populationfor V(C, C) = 1, V(C, D) = 2, V(D, C) = 1 andV(D, D) = 0 when the probability of action C is p = 0.5.

77.1 Determine if action C will spread throughout the populationfor V(C, C) = 4, V(C, D) = −3, V(D, C) = 2 andV(D, D) = 0 when the probability of action C is p = 0.5.

77.3 Determine if action C will spread throughout the populationfor V(C, C) = 3, V(C, D) = 2, V(D, C) = 1 andV(D, D) = 0 when the probability of action C is p = 0.4.

77.4 Determine if action C will spread throughout the populationfor V(C, C) = 2, V(C, D) = −1, V(D, C) = 1 andV(D, D) = 0 when the probability of action C is p = 0.6.

A First Whisper of Hamilton’s Rule

Now let’s talk about what is called Hamilton’s Rule. Hamilton had

the insight that kinship could lead to altruism evolving. As

McElreath says ,

He reasoned that an allele that codes for altruismcould selectively help other copies of itself if altruisticbehavior was preferentially directed toward kin.

In our cooperate/ defection model interaction in the last section, weassumed random sampling of cooperative/ defecting individualsfrom the population. So the chance of an agent being cooperativeor defecting was the same. This is what determined our fitnesscalculations.

Now let’s talk about what is called Hamilton’s Rule. Hamilton had

the insight that kinship could lead to altruism evolving. As

McElreath says ,

He reasoned that an allele that codes for altruismcould selectively help other copies of itself if altruisticbehavior was preferentially directed toward kin.

In our cooperate/ defection model interaction in the last section, weassumed random sampling of cooperative/ defecting individualsfrom the population. So the chance of an agent being cooperativeor defecting was the same. This is what determined our fitnesscalculations.

It turns out non random interactions are the key to the formation ofaltruism. Let

P(A|A) be the probability an altruist A is paired with anotheraltruist A.P(A|N) be the probability an altruist A is paired with a nonaltruist N.P(N|A) be the probability a non altruist N is paired with analtruist A.P(N|N) be the probability a non altruist N is paired withanother non altruist N.

We also need the fitness values for these interactions.

V(A|A) is the fitness change for the A and A pairing.V(A|N) is the fitness change for the A and N pairing.V(N|A) is the fitness change for the N and A pairing.V(N|N) is the fitness change for the N and N pairing.

It turns out non random interactions are the key to the formation ofaltruism. Let

P(A|A) be the probability an altruist A is paired with anotheraltruist A.P(A|N) be the probability an altruist A is paired with a nonaltruist N.P(N|A) be the probability a non altruist N is paired with analtruist A.P(N|N) be the probability a non altruist N is paired withanother non altruist N.

We also need the fitness values for these interactions.

V(A|A) is the fitness change for the A and A pairing.V(A|N) is the fitness change for the A and N pairing.V(N|A) is the fitness change for the N and A pairing.V(N|N) is the fitness change for the N and N pairing.

We can then calculate fitness like usual with these values. Thefitness W(A) and W(N) are given by

W(A) = w0 + P(A|A) V(A|A) + P(A|N) V(A|N)

W(N) = w0 + P(N|A) V(N|A) + P(N|N) V(N|N)

Now let’s go back and think about our previous cooperate/defecting strategy analysis. If the agents are not being chosenrandomly with equal probability, we have to rethink how wecalculated our payoff matrix. Let b be the benefit gain to thepopulation for an altruistic act; we call this the group benefit. Welet c be the private cost to a altruist. We have to decide what thesefour fitness consequences should be. And that is our next lecture!

Also notice how each step in our analysis, we add a little morecomplexity and build on how we worked things out before. Thisincremental approach is a nice way to develop models!

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