# Plotting Recruitment Curves

I was asked how I plotted the recruitment curve from my last post. It's pretty easy to do in R.

R:
1. Dn <- expression(r * (1 - N/K) * N)
2. r <- 0.004
3. K <- 10^10
4. N <- seq(0,10^10,by=10^7)
5. png(file="recruitment.png") ## sends output to .png file
6. plot(N,eval(Dn),type="l", xlab="Human Population", ylab="Recruitment")
7. dev.off()

Now we can generalize to the generalized $theta$ logistic model where

This model changes the shape of the density dependence when $\theta<1$ the density dependence is felt only at higher densities, whereas when $\theta>1$ the density dependence kicks in at low population size.

Consider our problem of what the optimal harvest size of humans by Kanamits consumers would be if human population size was actually regulated according to the theta-logistic model. First, we assume that $\theta<1$.

R:
1. Dn.gen <- expression((r * (1 - N/K))^theta * N)
2. theta <- 0.1
3. png(file="recruitment-thless1.png")
4. plot(N,eval(Dn.gen),type="l", xlab="Human Population", ylab="Recruitment")
5. title("theta=0.1")
6. dev.off()

Now, we assume that $\theta>1$.

R:
1. theta <- 10
2. png(file="recruitment-thgr1.png")
3. plot(N,eval(Dn.gen),type="l", xlab="Human Population", ylab="Recruitment")
4. title("theta=10")
5. dev.off()

I sent the output to a portable network graphics (png) device because I wanted to include the figures with this post. Normally, in interactive work, you would send this to the default on-screen device (e.g., x11 or Quartz). When making a figure for publication or to include with a $LaTeX$ document, one would send the output to a pdf device.

From this example, we can see that the shape of the density dependence can pretty drastically change the MSY, at least in the case where the discount rate is zero. When $\theta=10$, we can see that the Kanamits would only be able to sustainably harvest on the order of a billion people, whereas if $\theta=0.1$, they could take nearly 9 billion and still have a viable population. I'm guessing that the former is closer to the truth than the latter, but let's hope that the answer remains in the realm of absurdist speculation where it belongs.

# To Serve Man

So there I was. 23 hundred hours, New Year's Eve 2008 at my in-laws' in Corvallis, Oregon. Kids asleep in the other room. Spouse headed in the same direction. Insomniac from lack of activity and feelings of general physical displacement. Not much prospect for a party or other New Year's antics. Sounds like the perfect set-up for a Twilight Zone marathon, no? I think that I managed to make it through about 8 episodes in the wee hours of the first day of 2009. I wonder what that augers for the new year?

One of the episodes I managed to watch was "To Serve Man" (viewable freely in its entirety here). It's your standard aliens land, aliens altruistically transfer advanced technology, aliens solve all of humanity's problems, aliens turn out to have ulterior motives, aliens plan to eat all of humanity kind of plots. You see, the Kanamits have come to Earth to bring humanity back to their home world to eat them. The book that the Kanamits spokesman leaves in the United Nations (the big tease...) turns out to be a cookbook, a fact that the crack cryptographic team headed by protagonist Michael Chambers discovers too late. Chambers is already boarding the ship bound for his vacation getaway to the Kanamits' home world when the real brains of his outfit, the beautiful Pat, comes running to inform him (and, maddeningly for humanity, only him) of the horrible news. He's shipping off to get eaten. Get it? "Serve" as in provide assistance vs. "serve" as in provide a meal?

Toward the end of the episode, Michael Chambers, says in soliloquy that the recollections he has just shared represent the life cycle of Man "from dust to dessert." He comments that every person will ultimately share his fate (i.e., being served): "Sooner or later, we'll all of us be on the menu." This is what got me. Surely not. This doesn't seem like very good natural resource management on the part of the Kanamits.

The classical theory of natural resource management suggests that there should be a maximally sustainable yield for farmed humans. We will assume that human population growth can be written in terms of its current size and some function of size $\dot{N}= N f(N)$, where $\dot{N}$ indicates the derivative of population size with respect to time $t$. For simplicity's sake, let's assume that the effect of density on the human growth rate is linear such that $f(N) = r(1 - N/K)$, where $r$ is the intrinsic rate of natural increase $K$ is the carrying capacity (i.e., the maximum number of people the Earth could potentially support).

Now, I should note that this is a purely hypothetical exercise meant simply to illustrate basic concepts. In any real Soylent Green like future where we are faced with fattening Peter to save Paul, we would need a much more sophisticated population model. Among other things, the idea of a carrying capacity for humanity is a maddeningly elusive one. The eminent demographer Joel Cohen has written a whole book on the matter and comes up with no particularly satisfying answer to the seemingly basic question of How Many People Can The Earth Support?

For argument's sake, let's just say that the maximum intrinsic rate of increase is $r=0.04$ (4% annual growth, which is high, but remember, humanity has just received this remarkable technology transfer) and the carrying capacity is $K=10^{10}$ (i.e., 10 billion). Population growth changes with population size. We can plot the incremental increase (recruitment) as a function of population size. The maximum point of this so-called recruitment curve, where $d\dot{N}/dN=0$ is the MSY.

For the simple linear model of density-dependence (i.e., the logistic growth equation) and with no time preferences on the part of the Kanamits, the maximum sustainable yield (MSY) occurs at $MSY=K/2$ or 5 billion souls. So, you can now understand my dismay at 0200 (or whatever it was) on the first day of 2009. Why would everyone necessarily meet the fate of Michael Chambers? Under optimal harvest management, only half should be harvested. I can think of at least three explanations for this conundrum:

1. Michael Chambers is either misinformed (he was a cryptographer, after all, and not a wildlife manager) or was exaggerating for dramatic effect.
2. The Kanamits, while advanced in the physical sciences, had not developed an adequate theory of natural resource management or population dynamics more generally. Perhaps this is why they needed to come all the way to Earth to acquire their dessert?
3. Perhaps the Kanamits actually had a very sophisticated understanding of population biology and realized that the human intrinsic rate of increase was less than the prevailing discount rate. Under these conditions, the optimal resource-management strategy is liquidation of the stock.

I'm leaning toward explanation #3. That was an awful long way to come, after all.

Sobering thoughts with which to commence the new year.

I am currently teaching a class entitled "Demography and Life History Theory."  For this class, we read the classic paper by Madhav Gadgil and Bill Bossert, "Life Historical Consequences of Natural Selection." In preparing for class, I re-read this paper for about the twelfth time.  Something happened this time.  It really dawned on me what a spectacularly important paper this is.  Just about every important theme in life history theory is addressed in this paper and the analyses remain remarkably relevant.

One of the fundamental ideas this paper brought to life history theory is thinking about the convexity of the functions that describe both the fitness benefits and costs associated with the degree of reproductive effort.  In particular, Gadgil & Bossert show that iteroparity (i.e., repeated breeding) can only evolve if the function relating benefit to effort is concave or the function relating cost to effort is convex.  The figure shows a concave profit function and a linear cost.  Clearly, the maximal value of the difference between the profit and cost happens at some intermediate level of effort $\theta_j$.

Maddeningly, Gadgil & Bossert invert the terms "convex" and "concave."  I'm sure there is a good historical explanation for this, but contemporary usage indicates that a continuous, twice differentiable function $f(\theta)$ is convex if $f''(\theta)>0$.  That is a convex function shows increasing marginal returns to effort, whereas a concave function shows diminishing marginal returns to effort.

Their analysis focuses on a discrete-time form of Lotka's characterisitc equation:

where $m$ is the Maulthusian parameter (the intrinsic rate of increase, $r$, in a density-independent population), $l_x$ is the fraction surviving to age $x$, $b_x$ is the birth rate (in daughters) to females age $x$.

What makes this approach so interesting and important is that the vital rates are functions of reproductive effort $\theta_x$ at each age.  In addition to $l_x$ and $b_x$ being  functions of effort, they are also functions of "satisfaction" $\psi_x$, a measure of environmental quality.  Fertility at age $x$ increases with effort and satisfaction.  Survivorship and growth decrease with effort and increase with satisfaction.  Fertility also increases with body size.

Gadgil and Bossert then maximize $m$ subject to the biological constraints on intrinsic mortality, growth rate, and initial size, and the environmental constraints of satisfaction and mortality due to predation. They used an automatic computer to numerically solve their optimization problem.  They derive a number of quite general results for age-structured populations.

1. Only when the profit function is concave or the cost function is convex can repeated breeding be optimal.
2. Reproductive effort increases with age in repeated breeders.  George Williams arrived at a similar conclusion in 1966, but failed to consider the possibility that the fitness profit could decline with age.  Gadgil & Bossert arrive at this result with a much more general approach than that used by Williams.
3. When mortality increases following some age $j$, reproductive effort increases for ages less than $j$.
4. When reproductive potential increases slowly with size, reproductive effort will be lower at maturity, rise with age, and growth will continue beyond maturity.
5. A uniform change in mortality -- affecting all ages equally -- will have no direct impact on reproductive effort.
6. If the population is resource-limited, such a uniform change in mortality will increase satisfaction $\psi_j$ with its consequent effects, including a lowering of age at first breeding and increase in reproductive effort.

I would have to say that this is the desert island paper for life history theory.  If you only ever read one paper (conditional on having the mathematics background to make it worthwhile), this is the one to read, even after 38 years.

# Value of Human Life Declining Dramatically

An AP story suggests that, according to the EPA, the value of a human life has declined from $8.7 to$6.9 million since 2004 (in 2008 dollars).  That is an annual decline of 5.5%.  This means that if this decline continues at a constant rate, a human life will be worth one dollar in just under 272 years!

# Declining Life Expectancy in the Richest Country in the World

A new paper (discussed in this New York Times article) shows what many demographers have suspected for a while.  Life expectancy in the nation's poorest counties has actually declined since 1983, while life expectancy continues to rise for the more affluent counties.  This finding stands in stark contrast with the trends recorded from 1961-1983.  During this period, life expectancy increased in nearly every county in the United States.  The basic idea is that if you were well off when Ronald Reagan entered the White House, you have done well in terms of mortality since.  However, if you had the misfortune of being poor, then you have typically fared less well.  The result? Diverging health fortunes among the haves and the have-nots in American society.

This finding relates to what my colleagues Shripad Tuljapurkar and Ryan Edwards refer to as "the ultimate inequality."  The manifestation of inequality does not get much more stark than through systematic differences in the number of years lived by people of different socioeconomic or ethnic backgrounds.

In the 1983-1999 period, mortality from the largest cause of death, cardiovascular disease, declined in both rich and poor counties.  So what has caused the divergence in life expectancy between rich and poor counties?  It appears to be driven largely by increased mortality among the poor from diabetes and other non-communicable diseases on the one hand, and lung cancer and chronic obstructive pulmonary disease (COPD; of which emphysema is an example) on the other.   Among men in the poor counties, the probabilities of death from homicide or HIV/AIDS have also increased.  While the probabilities of death due to either homicide or AIDS increased only a little, homicide and AIDS can have a big impact on life expectancy for a population.   This is because victims of homicide and AIDS are usually young, so that every homicide or AIDS death leads to a relatively large number of lost years for the sub-population. So much for that goal of  "accelerat[ing] CDC’s health impact in the U.S population and to eliminate health disparities for vulnerable populations as defined by race/ethnicity, socio-economic status, geography, gender, age, disability status, risk status related to sex and gender, and among other populations identified to be at-risk for health disparities."  We can do better...

# The Requirements for an Ecological Anthropology Curriculum

A question was posted today on the ecological anthropology listserv: What are the basic requirements for an ecological anthropology graduate program? I don't claim to be qualified to say what these are for the field as a whole, but I am qualified to say what we have decided on in setting up our new ecological and environmental anthropology Ph.D. program at Stanford. Here I include an edited version of the reply I sent to the thread.

At the risk of essentializing, there are, broadly speaking, two general classes of ecological anthropologists: (1) those who use human relationships with the environment as a lens through which to study problems in cultural anthropology (e.g., agency, social structure, the construction of meaning, etc.), and (2) ecologists who study humans as their primary organism. The majority of practitioners currently falling under the latter category are probably human behavioral ecologists, though I can think of some notable exceptions to this. This is the approach our program emphasizes.

In addition to departmental requirements, EE students are required to take the following:

• Evolutionary Theory
• Research Methods in Ecological Anthropology
• Data Analysis in the Anthropological Sciences

All students need to know how to integrate theory, method, and application, but the specific nature of the courses in which they learn that doesn't matter that much. Therefore, we require three courses from a list of theory-driven graduate classes, including (but not limited to):