Framing Sociobiology

 

Section II : Negative Returns on Marginal Fertility

 

In this section we intend to show that marginal return of fertility is not always positive.  In other words, there exist a collection of cases in which the assumption that ‘having more offspring is superior to the alternative’ is not always correct.  There are, in fact environmental conditions under which having an incrementally larger number of offspring has profoundly negative effects. 

 

The Optimal Population Level

 

Before we start our work we must ask a simple question:  In which real environment is monotonically increasing population possible?  Imagine a real population. Its members’ eat, act, and reproduce.  They gather energy and they expend it.  Now imagine that the fertility level, the rate of reproduction, is held constant in time.  Individuals reproduce at some finite rate in the absence of any restraints being imposed by the physical world.  In other words, there is not starvation, predation, or disease.  What happens to the population?  In the absence of disease, starvation, predation and other losses, the population grows exponentially.

 

     (1)

 

The number of individuals in time period t is equal to the number of individuals at some earlier starting point multiplied by a factor, 1+r once for each elapsed time period.  In most problems involving long-lived animals t will be one year. 

 

So how long is it until humans completely overrun the planet?

 

1)    Assume the average person weighs 50kg and has a density equal to that of water.  Then his volume must equal 50/1000 =.050 cubic meters. 

2)    The surface of the earth is approximately 5x10⁸ km²  = 5x10¹⁴m² .  A 20 meter shell around the earth’s surface would possess a volume of 1x10¹⁶m²

3)    If one were to fill this volumetric shell with human bodies, it would require 2x10¹⁷ bodies.

4)    If we start with current population n₀ of 6 billion human bodies, and we cause them to reproduce at a modest increase, r, of  3% per year, how many years would it take to reach 2E17 bodies, enough to fill up this volume ? t= log(2E17/6E9)/(log1.03) = 586 years.

 

Few species reproduce so slowly as man, yet even at man’s own low reproduction rate, less than six centuries would suffice for man to cover whole surface of the earth to more than 60 feet deep in human bodies, stacked like cord wood.  But nature has never produced this many instances of any life form.  Why?  It is not for lack of reproductive prowess.  This we have just proven.  Rather it is because of other material constraints.  Principally, it is because there is a finite amount of energy committed to promoting life on the planet and all species compete for their share of that energy. While high reproductive rates might sometimes be advantageous in filling a frontier, once population levels begin to approach equilibrium levels, it is frequently the case that survival is linked to reproduction in a rather distant way.  What really matters in the real world, most of the time, is the next meal – either one’s own or someone else’s.  Sometimes too much incremental fertility makes the next meal impossible.

 

Imagine a population with relatively static qualities living in a relatively static environment. The population has fixed eating habits and is well adapted to eating a particular set of materials produced by nature at a strictly or relatively regular, steady rate. What is the population count that represents the highest level of ‘fitness’ for the population?  There is a sense in which a greater population is indicative of greater fitness.  A larger number of individuals expresses more genotypic and phenotypic variety and therefore would be more adaptable to any hypothetical changes.  If the materials that the species in question use for food are materials used by other species in neighboring areas, then a failure to fully exploit the niche by consuming all these materials invites increased competition.  So by almost any measure, the population is best suited to its environment, is most fit, is most likely to survive when it fully exploits its niche and it does so in a completely sustainable way. 

 

Assume that any real population of n individuals consumes an amount of energy :

 

  (2)

 

where e is the average rate at which an individual consumes energy.  This formulation assumes no variation in energy consumption among individuals. Assume also that the area inhabited by the population is fixed and that it produces a steady state flow of energy, E_p:

 

  (3)

 

Finally, assume that the optimal population level occurs when the population consumes precisely the amount of available food energy.  It cannot consume more.  If, hypothetically there were a way for a population to sustain a higher level of consumption indefinitely, then E_p would be equal to a new, larger constant.  But if the population consumes more in a way that draws down production stocks, then the population is headed for a crash, just like the population of whale hunters. Thus, the population level is sustainable so long as the food energy usage of the population does not exceed the food energy production of the closed area:

 

    (4)

 

Because we assume that a larger population is better up until the point that increasing population might cause some problems, the optimal population level is defined in the case of equality.   We will show below that populations that exceed this level produce an array of undesirable consequences.

 

This derivation assumes that energy is the sole constraining resource, or at least the limiting one.  In places where there exists another, more seriously constrained resource, obviously that resource would constrain population levels. In the case of plants the constraint might be suitable soil, or sunlight or water. In the case of animals it might be suitable habitat and sufficient food.  

 

Jared Diamond’s Collapse suggests that fertile soil along with adequate supplies of water are two requisite resources for reliable supplies of food energy, and that a number of flourishing civilizations have collapsed because of a collapse in soil fertility and, therefore, agricultural productivity. In every case, a population ceases to be sustainable when it consumes more than the freely available amount of any constraining resource. When agriculture collapses, the populations’ energy supply disappears.  so too does the society it supports.  

 

 

The Limiting Cases

 

Assuming then, that there exists some hypothetical constraining resource, how does this affect Williams’ argument?  If we can find a single case in which Williams’ thought experiment predicts the wrong outcome, then that thought experiment fails as an article of faith.  It would gain the status of as questionable hypothesis or a conditional conjecture.  It ought to be demoted from the status of religious orthodoxy to being appropriate matter of serious scientific treatment.  If we can find multiple cases in which the thought experiment fails then we might be interested in admitting scientific enquiries that are based on alternative hypotheses. If for a wide range of populations we can find a way in which it must always fail, we may wish to view existing bodies of research in another light, and consider admitting alternative points of view in research efforts.

 

There are two limiting cases we will examine. One limiting case is where periodic and absolute food shortages occur regularly.  Such a case might occur in a temperate zone where food is available seasonally.  Alternatively it might occur in a more torrid zone where there exist seasonal rain patterns – a rainy season followed by a dry season.  The second limiting case occurs at the opposite end of the spectrum. In this case there exists a completely steady supply of food.  The population rises to the equilibrium level, and we explore what happens thereafter.  We have constructed these limiting cases to be idealizations. We intend to demonstrate that these cases come much closer to describing real physical phenomena than does Williams’ thought experiment.

 

 

 

 

 

Seasonal Absolute Famine

 

Imagine a population living where there exist regular, periodic seasonal shortages of food. Such conditions exist throughout most of the temperate parts of the world.  And there are many locations in the subtropics where seasonal wet and dry seasons cause wide fluctuations in food availability for certain populations.  The populations in question are ones in which the normal, full lifetime of a member will exceed two or three full calendar years.  Species such as aphids or mosquitoes that tend to repopulate during favorable conditions but go dormant during unfavorable ones will not be well represented by this model.

 

The energy consumed by a population of n individuals during some specified period t₂ is:

 

  (5)

 

Each individual consumes energy e in a series of tasks.  We will assume, for the sake of simplicity that each individual in the population of n consumes approximately the same quantity of energy, and that this amount is not time variant. Typically, we would expect t₂ to occur roughly one year after t=0, although one might imagine other conditions. These assumptions are for modeling convenience because variations that other accounting methods might represent do not strike us as being germane to this model.

 

The total amount of usable food energy produced by the environment and available to the population over the year is:

 

  (6)

 

 

If we start the system with the population being considerably less than the equilibrium or optimum population, the population will increase with each successive year.  The optimum population occurs when there is a rough equilibrium between the energy consumed over the year and the energy available.  If we specify a condition in which no members of the population die from hunger then,

 

   (7)

 

So long as the energy required to sustain the population through the whole year is less than the available food energy, the population continues to thrive unaffected by food supply.  Namely, lack of food is not a cause of mortality in the population.  Again, we are assuming individuals of identical fitness with respect to the gathering, consumption, and storage of food.

 

Substituting equations five and six into seven gives:

 

  (8)

 

We start the clock at t=0.  This is the first calendar day of the year in which more food energy becomes available to the population than is consumed by it.  We assume that the food availability function on the right begins as a monotonically increasing function for some time, and then it decreases monotonically to zero.  Actually, it does not matter whether the energy availability function, itself, behaves this way; it only matters that the time integral of the function behaves this way, and that the food energy that is produced is stored up and available for later use.

 

For the duration of t1, production outstrips consumption and depletion.  After that, no food energy is produced in the environment; and any food energy that is used by the population comes from stores of some sort.  Either those stores exist in-situ in the environment as plant or animal matter, or the stores are harvested materials  such as nuts stored by squirrels or honey stored by bees, or the stores are internal to the organisms, stored as fat.  At least one of several means of storing food energy is available to all members of the population.

 

During the period of shortage, the population draws down on the energy stores.  Each day the level of stored energy decreases.  Meanwhile, the calendar begins to approach the start of the cycle.  If the population has stored enough food energy, it will survive the period of shortage without losing any members to starvation or to weaknesses caused by insufficient energy levels. 

 

 As the period of dearth wears on, individuals lose weight.  They grow weak.  They approach death.  But before that process reaches an irreversible state, the new season of plenty starts, and the whole population is revived. What is the criterion that predicts survival?  So long as the population remains less than or equal to the optimum population, all is well.

 

What happens, however, when the population gets too large?  We start the clock at the beginning of the period of plenty.  Imagine that one individual in a population reproduces in a way that violates the energy inequality above.  Then the period of zero production hits. It is tempting to argue that, even if it is a relatively small population, the excess reproduction will cause no harm because other members will simply “cut back” on consumption.  But remember that the population has already expanded to the very edge of sustainability. Remember that before this last individual was produced, the whole population was on the brink of collapse the day before food became available.   Because of Williams’ hypothesis, the entire population is just seconds from expiring when the excess food shows up in the new season.  So any increment to the population must necessarily place sufficient strain to cause it to collapse completely.  Every member dies.

 

The cost of Williams-cheating in a periodic famine model is that the entire population dies of hunger partway through the period of dearth. If cheating is rampant and the population is much bigger than the sustainable level, the whole population will go extinct somewhere in the middle of the famine.

 

We must infer that any population that admits cheaters and fails to isolate their effects through some cultural or physical method is, inevitably extinct. This, we would argue, is a rather powerful selective effect.   It is hard to understand how one might construe this as being a compelling means by which cheaters gain any incremental advantage over non-cheaters: they end up being every bit as dead as the members who did not cheat.  It is also interesting to note that if one fails to take into account cultural effects that might isolate Williams-cheaters, the effect of over-population is borne by the whole population.

 

It might be helpful to cast this model into a kind of game that biologists are more used to thinking about.   Imagine two new neighboring populations A & B going through these seasonal cycles year after year. In the absence of this model there is a temptation to believe that the population that reproduces fastest is unconditionally more fit. Williams’ thought experiment implicitly assumes that increased fertility rate always generates an advantage. 

 

 

 

Two Villages

 

Imagine that population B consistently reproduces faster than population A by an incrementally small amount.  It will approach the equilibrium point much faster than A.  Now, suppose that it continues to reproduce at exactly the same rate.  In an environment where there exist periodic complete shortages, population B will perish, and be extinct.  Once B is extinct, little effort will be required for members of population A to colonize the area.  A has proven more biologically fit by reproducing at a slower rate.  

 

Curiously, this example does not even require that A possesses a mechanism to control either its population or its rate of reproduction.  It simply requires that A&B start at the same point in time, and continue under the same uniform conditions.  Then, thanks to higher fertility, population B outstrips its resources and goes extinct before the same thing happens to A.  So, even if we were to buy Williams’ conclusion that altruism is impossible, it is impossible to buy his assumption that the marginal benefits of higher fertility must always be positive. There are environmental conditions under which lower fertility confers a powerful incremental advantage to a population.

 

In fact, if we imagine that Williams is right, and the population does from time to time produce cheaters, and if the population survives, then one plausible explanation is that the population actually has evolved ways of isolating the negative effects of cheating. And if one believed this, one would have to look at something in the behavior of organisms that lies outside the realm of copulation. If the religious orthodoxy of evolutionary biology prohibits this, then it would be impossible for evolutionary biologists to discover the problem.

 

In certain northern European and Asian agricultural societies with seasonal food supplies, organization has tended to be along the lines of  nuclear families.  By contrast,  agricultural societies in locales with more steady supplies of food throughout the year tended to organize in more extended families and in larger groups along feudal lines. It might be possible toable to explain the difference by using this model. A nuclear family isolates the effects of seasonal shortages to the location of cause quite effectively.  It is a profoundly effective way of punishing Williams-cheaters. Whether this is a defensible example, we do not know. We pose it only as a speculation to stimulate further work.

 

 

Maximum Sustainable Rate of Reproduction

 

There is a trend in evolutionary theory to think in terms only of reproduction, but not in terms of survival between reproductive acts.  So, we ask ourselves, based on a model of seasonal variation of food supply whether evolution always favors an incrementally greater reproductive rate,  r.  If it does, then strategies that trade some other measure of fitness for reproductive fitness will be inferior to reproductive fitness.  If, on the other hand, increasing r can be shown to cause a decrease in fitness, then there exist circumstances in which decreases in reproductive quantity r, provide competitive advantage.  Increases in reproductive fitness as expressed by incrementally positive fertility rates, we are suggesting, may sometimes lead to lower fitness of the population.  The marginal fertility rate in such a case yields negative returns.

 

If population survival is not a monotonically increasing function of fertility, then there must be an optimal point.  If the rate of reproduction gets bigger than this number, then things get worse for the population rather than better.

 

We start by assuming that the population at the beginning of the new period of plenty is less than the optimal population.  In a real population it is normally true that periods of dearth will cause some mortality, so this is not a far-fetched idea.  We assume also that there is exactly one breeding season in a year and that during that time the population increases from n to (1+r)n where r is not a very big number.  Probably r is never as big as 1 for most species of any sort that live two or more years. For very large and long-lived mammals and trees, r is likely to be only a tiny bit larger than null.  In fact, for a species in perfect equilibrium with its environment, we will argue that r will evaluate to zero, on average.

 

To simplify the math we will make these assumptions.

1)    Energy consumption during the period of plenty can be characterized by ne₁t₁

2)    Energy consumption during the period of dearth can be characterized by (1+r)ne₂(t₂-t₁).

3)    Total energy production over the plentiful period can be characterized as a constant E_p.

 

This formulation implicitly assumes that the size of the population does not fluctuate widely, and that it changes in relatively predictable ways.  In other words, we can choose n to represent the number of individuals at the beginning of the time period or the end of the time period, and even if these are different, the product net2 can be used to accurately characterize the energy consumption of the population.

 

Now, let us imagine a world in which the energy produced E_p is fixed from year to year.  Imagine the first day of the year was the first day of the period of plenty. We ask the question, what condition is required for the population to survive during the period of dearth, t₂-t₁?

 

   (9)

 

 

Rearrange and solve for r:

 

  (10)

 

This formula suggests that r has some maximum value, and that maximum value depends on how long the period of dearth is compared to the period of plenty, upon how plentiful the period of plenty is, and upon the population level and energy consumption levels during the two periods in question.  If the dearth does not exist, the equation converges on

 

   (11)

 

 

At the optimum population, the energy consumed is equal to the energy produced and we can substitute for E_p  in equation 10,

 

   

 

  (12)

 

 

Where the population is far from the equilibrium value, r can be relatively high.  But when the population reaches a kind of seasonal equilibrium, r is relatively small. At the optimum point, r evaluates to zero.  This keeps the number of members in the population level.  It magically causes new members to be produced at precisely the same rate they die or exit the population by other means.

 

What happens when a population reproduces at a level beyond r?  If all the individuals of the population start the time period of dearth in exactly the same condition and they all survive on exactly the same level of energy, then they will survive identical amounts of time.  If such a population is in perfect equilibrium with the environment, then it will reproduce and sustain a number of members that causes the energy supply to be completely exhausted on the first day in which there is food energy available in a new time period.  A population with fewer members has failed to maximize its place in its environmental niche. 

 

If, on the other hand, this ideal population has just one member too many, then all sources of energy will be depleted some critical amount of time before the new season of plenty arrives.  And since we assumed that all members of the population are identical, then necessarily, the whole population dies.

 

Snide Comment

 

If evolutionary success is to be defined by the assumption of ever-positive returns of marginal fertility, then we are forced by logic to take the inevitable natural result associated with a population living far beyond its means – namely the total extinction of the population - as an example of evolutionary success. 

 

 

Level Equilibrium Cases

 

We have examined periodic shortage; let us look at the opposite extreme. The advantage of this approach is that one might imagine that all real world situations might be thought of as some linear combination of the two extremes.  Thus, if both extremes show the same behavior it is not impossible to imagine that all linear combinations of these conditions would display the same behavior.

 

Imagine a population that is at perfect equilibrium with a static environment.  Imagine further that this population magically produces exactly the right number of offspring. Imagine that as soon as a member of the population dies, exactly one new one takes its place instantaneously.  This population completely fills its niche, leaving no room for competition.  At the same time, it manages not to deplete resources.  It is the optimum population level. 

 

Such a condition might describe a large population of herd animals living on a plain so large that they consume precisely the amount of food it produces in the year. The  weather  might admit of some cyclic qualities, but at no time would the herd be required to store any energy.  And therefore this theoretical animal is incapable of storing any energy. It must graze continuously.

 

We do not know how the feat is accomplished, but we see that the population is in perfect balance with the environment.   Now, imagine that a Williams’- cheater  enters the herd and produces individuals over and above the optimal level. We will look at the case of the Williams cheater under a number of scenarios, because the effects on the population depend upon how members of the population interact.

 

Herd Animals

Imagine a population of herd animals. These are animals that live in a large, cohesive group. Bats, elephants, and zebra come to mind as fair approximations. Because there is no individual territory, there is no way for one group of animals to enforce low fertility constraints on the rest of the animals.  If Williams’ argument holds, and it is always to the advantage of any individual to bear more offspring, then the system moves toward one extreme, reproduction continues, the population keeps growing.

 

Energy resources are divided perfectly evenly; therefore, the effects of shortages that have been caused by overpopulation are shared equally among all individuals.  The food shortage gets worse with each new birth, and individuals get weaker and weaker. Then, at the last moment a new individual is born.  The attempt to feed this new individual draws the energy level of every member of the population below the critical level and every member of the population dies. 

 

What we are saying here is that there is a kind of inevitable equivalence between Williams-cheaters and extinction in populations that have no social organization at all.  As a thought experiment this proves that all undifferentiated herds living in a realm of constant energy supply and containing Williams-cheaters are extinct. Again, we view extinction as being a rather high cost to pay for evolutionary success as Williams has defined it.

 

The model exists as a thought experiment, but the consequence in the real world may not always be profoundly different.   Herbivorous herd animals, if they drop below a certain high level of fitness simply provide more frequent nutritious meals to predators, causing the predators to gain population.  Usually this would soon reduce the herbivore population to normal bounds.  But if the overpopulation were too extreme and predation failed to do bring it back in bounds quickly, predators may be produced in great abundance, and not long afterward the herd could get hunted to the edge of extinction. Or beyond.  In populations that have no predators, weakness and high population density will frequently lead to high mortality from disease.

 

If we are not mistaken, almost every existing population whose members survive two or more years has a social hierarchy.  We imagine some biologists have studied this hierarchy by studying mating cultures in herd animals, but we insist they would have been prepared to measure, assess, and judge social interactions, acts that might not be directly related to copulation.

 

It is interesting to note that bats, elephants, zebra and quite a large number of other long-lived animals that live in large social groups without fixed individual or family territories tend to bear one offspring at a time.  It is as if the whole population is programmed by nature to avoid the problems associated with severe overpopulation.

 

 

 

Snide Comment

Evidently these species have not read Alcock and have not learned that higher fertility rates are unconditionally good, which is ironic because these are reputed to be among the more intelligent members of to the animal kingdom.

 

Lack of Mechanism

One might object that self-regulation is impossible because there is no apparent mechanism.  This is an attractive argument, for we all want to know how and why things happen.  It may be true that extinction explains that regulation occurs, but it is not a compelling explanation of how or why. Still, we caution that not knowing how or why a phenomenon works is necessarily different from denying its existence.  We do not know precisely why gravity works, yet we are willing to believe in it enough not to jump wantonly from tall buildings.

 

Speculation might help us look for better explanations.  There is quite a lot of evidence to support the theory that various sorts of stress lower fertility.  So, too, does malnourishment.  We also know that shortages of food and other needful things induce various sorts of stress.   It is reasonable to assume that stress is a means by which a number of mammals differentially regulate reproductive tendencies. We recall [ref tbd.] a population model built to explain fluctuations in fertility rates in Sweden over several centuries.  The model led one to conclude that relatively high fertility correlated very well to the plentitude of food.  In other words, mammals experience stress when they perceive a shortage of resources.  When they experience stress, they reproduce less plentifully.  Our analysis explains the necessity for the regulative mechanism.  The Swedish model suggests how it works.

 

Snide Comment

One might argue that the analysis is wrong because it resorts to population theories rather than theories about individual behavior: evolutionary biology is not about populations but about individuals. Perhaps it is heretical to say so, but in evolutionary terms, if a population jumps over a cliff, why would it matter much whether all its members die individually or the population vanishes collectively?  In one case all the individuals are extinct; in the other the population is.  For most practical or theoretical purposes the two effects are equivalent.

 

Territorial Animals

Because sociobiology tends to studiously avoid social interaction there is a temptation to treat all members of a population as being undifferentiated. Yet at any given point in time no two members of a population are in precisely the same space, performing precisely the same function. So there is reason to believe that any two individuals of a population, by their unique actions, are affecting the population in different ways.  If we suspend disbelief for a moment and assume that some social interactions occur in animals and that these social interactions have an effect on natural selection, we can examine the case of shortages in a slightly different way.

 

 Many species of birds are territorial.  Many predatory mammals are territorial.  Imagine such a population occupying a given area.  Imagine that the area produces a steady, level supply of food.  Assume that this level supply is invariant with time.   Assume that the animals in question are incapable of drawing down the supply of food in an unsustainable way, even under conditions of overpopulation. Thus, when the population level exceeds the optimal level, one of two things must happen, either territories get smaller and smaller, or the territories stay the same size and the portion of the population that occupies a territory changes. 

 

If the territory size scales perfectly with population, then the reaction of the system is indistinguishable from that of herd animals: there is no differentiation.    We  will assume instead, that territories are allotted in such a way that one family unit – however that might be defined – occupies a territory in which at least one attribute does not change with population level. For the sake of convenience we will assume that this attribute is energy production.  But it could be territorial size or some other quality that relates to partitioning the supply of a critical, limited resource to a population. Assume further that members of the population without territories are left without the means to reproduce.   

 

We assume for this model that territories are allotted in some time-invariant manner and that they have some time-invariant, minimal size. Assume territory has been allotted to these animals by some means and the territories are fixed. The food supply in every territory supports  up to n individuals at an optimal level of fitness. 

 

Now, imagine that a territory produces Williams-cheaters.   The number of animals increases to n+m, then the energy available to the population is n/n+m.  This puts the animals in this territory at an incremental disadvantage compared to animals in neighboring territories that have not reproduced past their means; for if these animals were to mount an attack on the territory, the greater number of offspring would be of no incremental benefit in defending the territory, but they would prove a liability in terms of how energetically the territory might be defended. Remember that the defenders are suffering the ill effects of undernourishment; but the attackers are not.

 

This means that the Williams-cheaters are also more likely to perish from other causes such as predation or illness. This comparable disadvantage might be enough to put an end to Williams-cheaters in the first generation. But if, as Williams suggests, cheating is an inherited trait, and if the territory is likewise inherited then it will not be many generations before the population in that territory collapses completely for lack of food.  At this point, individuals from families of lower fertility would colonize the territory. This proves that in territorial species that has been for an arbitrarily large number of generation living near the optimum population level for an environment,  Williams-cheaters are extinct.

 

Once again, the price of Williams’-cheating in a well-developed population is extinction.

 

Imagine instead that these territorial animals were migratory birds.  It would be reasonable to imagine that the culture among some territorial migratory birds is to give up territorial claims when the season ends and the birds migrate.  At the start of the next nesting season negotiations for territory might start anew. Territorial domains would be re-taken each year with no prejudice from history. More fecund families would gain more territory in successive years. Perhaps this happens in all migratory bird species.  Or perhaps it happens in some.  Or in none. To the extent that it did it would allow larger families that, in a previous season had over-reproduced to pick up new territory provisioned with new and larger seasonal allotments.  This would give Williams’-cheaters a short-term advantage.

 

But there is a downside risk.  A parent raising n+1 offspring might be materially more challenged to bring even n of these offspring to maturity with the level of fitness that would be required for a long migration. Remember that we have already specified that the population exceeded the optimum level.   Benefits of higher fertility would necessarily be offset by losses in individual fitness. Were that to be expressed as individual losses during migration, Williams’ cheating might lead to a complete loss of the brood, causing n+1 to evaluate to zero in evolutionary calculus.  We know that sometimes this must happen.  And, once again, we assert that complete extinction seems a high price to pay for competitive success.

 

There is a point beyond which increases in fertility must produce negative marginal returns in terms of total fitness in territorial animals. If territorial bounds are fixed once in history, then families that over-reproduce will be pressed hardest by resource shortages.  The Williams cheaters will be the ones most likely to perish from lack of nutrition, illness, or predation. The net effect will be to drive members of the species to longer life spans and lower fertility rates. The effect one would observe will be approximately what one has observed of mammals since they first evolved.  The effect one would observe will be approximately what could be observed of reptiles as they evolved.

 

In other words: we are forced to assert that there are environmental conditions under which fitness is increased by decreases in fertility.  Or, there are environmental conditions under which fitness does not increase monotonically with fertility, but rather, for which there exists some optimal level of fertility. Or, the incremental return on reproduction is not invariably positive.  In the case where populations exceed sustainable equilibrium levels established by the available resources, the incremental return on reproduction is invariably negative.  The assumption underlying Williams’ profound thought experiment – namely that the return on marginal fertility is invariably positive -  is correct only for populations on frontiers or ones recovering from some catastrophe.  In such populations, growth is of paramount importance.  Elsewhere, populations generally exist  near the optimal population level.  Populations near the optimal value, attach a marginal value of zero to fertility; ones far beyond attach a negative value to fertility.

 

 

The Case for Control

 

The consequence of this argument is that all populations successful in geological time have some means at hand to control population. The exact nature of these natural equilibrating forces we are not prepared to assert.  We agree with Williams that it is not useful to think of this in terms of ‘altruism, ’ but the fact that altruism cannot reasonably be treated as the motivating factor does not make the phenomenon of regulation less real. Nor does it require that all factors that create the force are exogenous to the population.  

 

Regulation  may appear to be absent in species that are unleashed on in vast, new, and almost unlimited environments, as man, Japanese beetles, and kudzu in the New World, or as tomatoes, potatoes, maize, and peppers in the Old World. In the absence of significant levels of control on the part of diseases or predation, they will quickly gain unusual levels of success. But at some point environmental forces will begin to limit this success.  It may happen that as such populations experience shortages of resources they begin to express  some internal control.  Or it may be that their plentitude is controlled entirely by external elements: predation, disease, or starvation. The evolutionary forces driving predation and disease grow exponentially with population density, so there is a sense in which “success tempts fate.”   In an evolutionary sense, a population is best served when these factors come to its rescue before it has exhausted the resource base on which it depends, and collapses completely.

 

And in populations where the forces of predation, disease, and age fail to control populations, violent acts within the species will often serve to control populations long before starvation does.  

 

 

 

Reproduction Strategies: Quality vs. Quantity

 

Again, implicit to Williams’ argument is that increased fertility outweighs other measures of fitness.  In other words, when an organism is faced with a tradeoff between two choices:

1)increase fertility and a decrease in other qualities or decrease in fertility or

2)increase in other qualities at the expense of fertility.

Williams argument implicitly assumes former is always better than the latter.  But does evolutionary history bear this out?

 

Let us frame this argument against the general arc of evolution.  We see, when we examine fossil records, the evolution of ever more complex and long-lived creatures.   We see, when we examine fossil records, that in periods of favorable, stable environmental conditions, animals of a certain type tended to evolve into ever larger sorts of animals.  The largest of the great lizards evolved from the smallest ones.  The largest of the mammals evolved from smaller ones. The largest of trees evolved from smaller plants. 

 

Large animals and large plants reproduce more slowly than small ones.  So the only way to explain the evolutionary forces  at work is to imagine that in times of environmental stasis large animals must have evolutionary advantages that far outweigh the significant disadvantages of slow reproduction.  Single-cell organisms reproduce faster than any others.  But multi-cellular organisms evolved, survive, and exploit environmental niches single celled organisms cannot.   Thus, as important as reproduction is to survival, there are a number of other factors that need to be considered as part of fitness.  And over the vast horizon of ecological time, reproduction capacity, reproduction rate, alone actually constitutes an ever- shrinking portion of total fitness.  To the extent that evolutionary biology can be concerned about multi-cellular creatures it is time to look beyond copulation as the sole evolutionary act.

 

Quality vs. Quantity

In terms of reproduction, one can find two general strategies in nature.  One strategy is more consistent with the Williams idea of ‘more is better.’ This strategy is expressed more often in areas where conditions are only suitable for survival for short periods of time during the year, making the whole environment a kind of seasonal frontier.  

 

The other strategy is more an appeal to quality.  Because the natural world has so many different niches it is natural that one might find plants and animals adopting slightly different strategies.  The first strategy was appeal to quantity, but it would appear that – at least in most stable environments - the niches that remain to be exploited by appeals to quantity are more rare than those that appeal to quality.  In a macroscopic sense, at least, more of natures’ most recent creations have appealed to some version of quality over some version of quantity either in the reproductive area itself or in some other area such as durability or longevity or size.

 

 

The Plant Kingdom

 In the plant kingdom annual grasses and clovers, poppies, and a number of desert and steppe flowers are examples of small plants that grow and reproduce quickly. Chief among virtues is the ability reproduce quickly.  They produce a huge number of seeds, and leave the rest to chance. That’s a good strategy for colonizing a frontier. But it is not always the best strategy when the world is already full of competitors. So why do they do it?  They do it because they live at the fringe of sustainability.  Punishing conditions might actually preclude life for much of the year.  Thus, each season when the weather permits, the desert is a frontier, a tabula rasa in terms of living plants.  Fecundity is completely vital to survival.

 

There is an alternative strategy that appeals to quality.  This approach tends to be expressed where conditions are more conducive to life year-around. Consider fruiting plants; apples, watermelons, walnuts, and avocados.  Plants in these species expend vast amounts of energy on at least one of two appeals to quality.  One appeal is a large seed, a seed that stores a huge amount of food energy.  This makes a seed much more likely to be successful when challenged by competitors.  The second is a fruiting body.  This tempts large animals to eat the fruit and spread the seeds.  In order to implement these reproductive strategies a plant must create a vastly smaller number of seeds than it otherwise might.

 

The oak or walnut tree might have had a thousand times more reproductive chances to establish a new tree had it behaved like the carrot in making seeds.  So why does it behave this way?  It must be because it gains an advantage from doing so.   At the far end of the extreme is the avocado.  The pit of this fruit is the size of a squash ball.  The amount of energy expended to produce a single seed is monstrous once one takes into consideration how much energy goes into the fruit as well as the seed.  Perhaps one could make a thousand or ten thousand tiny seeds for the same expenditure of energy. So is the avocado unusually maladapted?  Were there evolutionary forces at work when it evolved this strategy that caused the strategy to exploit some previously unexploited niche? Did dinosaurs cultivate avocados?  Or was the avocado designed by God?

 

 

The Animal Kingdom

In the animal kingdom fish, amphibians, and reptiles tend to produce a large number of eggs that hatch into a large number of new individuals. Were all of these to make it to adulthood, a reptile could reproduce twenty fold, fifty fold or a hundred fold or more in one year. But this does not happen.  The mortality rate of very young reptiles, fish, and amphibians is generally quite high.  They are targets of predation.  Sometimes by close relatives.

 

Imagine a species that has a habit of nurturing its young.  Ants, bees, mammals, and avian species come to mind.   William’s argument implicitly places no value on this nurturing process because it implicitly argues that there is no optimal point of reproduction: more is invariably better.  So the question we wish to ask is this:  Are there situations in which the advantages of producing fewer offspring can be proven to outweigh the advantages of producing more?  If we can conceive of a single situation, then William’s thought experiment fails.  We would argue that in every case mentioned the species have traded fecundity for some other quality, namely survivability.  If we were to imagine that reproduction is costly, then such a trade would make sense.  But even if we could not imagine reproduction to be costly, the simple fact that animals that nurture young evolved from animals that did not nurture young suggests that there is some advantage to the strategy.

 

A trade-off that both plants and animals make in reproduction is “quality vs. quantity.” Some successful plants produce hundreds or thousands of miniscule seeds.  These seeds will grow in tiny spots left temporarily uncolonized  by other plants. Such plants we frequently refer to as weeds. As any gardener will know, weeds are quite successful.  But some plants have specialized in a different way.  A number of large trees produce nuts.  A nut contains a huge reservoir of energy and nutrients that can set a plant well on its way to a long and happy life.  An acorn or a walnut might be thousands of times as large as a carrot seed. 

 

 

Again, the trend of plants to grow larger, to live longer, and to produce more viable offspring parallels the trend among animals to do the same.  The long-term trend in evolution suggests a kind of ‘flight to quality.” 

 

Once, all organisms were single-celled. Then multicellular organisms evolved.  These organisms succeeded in places where single celled ones did not.  And they did it not just despite the fact that most of the cells in question had to cease reproductive function, but because they did.  Multicellular organisms evolved to be able to express different tissue types and create different organs.  Ever larger animals evolved.  These larger animals all reproduced slower than smaller ones.  But despite this factor, smaller animals tended to evolve into larger ones.  Then, about a hundred million years ago mammals evolved.  The novel quality of mammals was that they nurtured their young.  Nurturing the young is many times more costly than simply laying eggs and going away. This meant that instead of being able to reproduce a hundred fold like a turtle or a frog or a crocodile, the mammal could only reproduce one at a time, or two at a time, or six at a time.  Yet mammals are certainly no less successful than frogs or crocodiles.  Crocodiles are found in just a few of the world’s major rivers and swamps.  Mammals live in every corner of the world, even inhabiting Antarctica. 

 

In the world of arthropods, ants are arguably the most successful group of species. Few related groups of animals represent so much living mass as ants.  Only humans come close.  One might argue that ants reproduce as fast as they can.  But to do so would be to make a categorical mistake.  Most ant colonies have but a single queen who does all the reproduction.  The remainder of the population does the farming and the defense of the anthill. It may be that the queen produces offspring as fast as is possible for her to do.  But a typical ant colony may have ten thousand female workers but only one queen.   Those female workers all have the genetic capacity to be queens. That capacity is suppressed at some point in the nurturing process.  If the marginal return to fertility were unconditionally positive, every ant colony would be better off with 10,000 queens and one worker, rather than the other way around.

 

Viewed against Williams’ metric, the ant has one of the lowest fertility rates in the animal kingdom; it ought to be the least successful of all beasts. But, in fact, the ant is arguably the best adapted of creatures on the whole planet judged in terms of territory inhabited and total impact on planetary life.  If one were to refute Williams’ assumption that more fertility is always better than less, ants provide quite a sound example. 

 

There is another sense in which ants refute Williams’s ideas.  The consequence of Williams’s thinking is that males are the exponents of evolution.  In the ant colony the male has absolutely nothing to do with its success. He is nothing but a vector to pass along some genes from one successful ant colony to another. The male is a proxy of success, but he has nothing whatsoever to do with actively causing success.  This may be important. If one wishes to learn anything about why an ant colony succeeds, by what measures it is fit, studying the male or the particulars of the mating process will provide no useful information.

 

Williams’ ground might be defended by arguing that we are confusing ‘apples and oranges.’ We are making an argument that compares species to each other rather than groups or individuals within a species.  But this is probably a faux distinction if one imagines that speciation occurs in the way Darwin proposed.

 

Imagine that at every point in evolutionary history speciation occurred as a selection between two populations.  One group was always marginally more fecund than the other.  This group would have the quality of always trading away some other quality for fertility, by some incremental amount.  Imagine, on the other hand, a population that reproduces in a way that keeps it close to the equilibrium point.  It, rather, trades longevity and individual success for fecundity. 

 

Now if, for whatever reason, Williams’ populations proved more successful we would see the path of evolution trading away other measures of fitness for fertility.  But in both the plant kingdom and the animal kingdom we see larger, longer-lived, less fertile species evolving from smaller, shorter-lived, more fertile species.

 

The actual path of evolution appears to be leading away from fertility as the primary element in the fitness equation. In fact, it is easy to argue that this step was certainly well under way when the first successful multi-cellular species became established. At least since that point in evolutionary history nature has been trading fertility for other qualities. The reason is simple.  No animal can beat the multi-cellular organism for reproductive punch.  That ecological niche is full.  If nature is to advance by filling new niches, it must proceed from highly fecund single-cell organisms to highly advanced multiple cellular ones that are comparatively less fecund.

 

 

Williams Species

 

It would be a mistake to argue that there are no species that might possibly benefit from extremely high reproductive capacities. We will call such a species a Williams species. There are several possible ways of describing a Williams species. It is a species for whom fitness regularly evaluates to reproductive capacity.  And for this to be true it is a species that, in some sense always exists on a frontier. Among examples of species for possible inclusion would be animals or plants that grow from eggs, spores, or seeds. They would do so seasonally when conditions permit, and they would compete with similar plants or animals to colonize in areas of new opportunity. Annual grasses might qualify. Or various fungi.  Or aphids. Or mosquitoes.  Mice might qualify.

 

We know that some life on earth must exist as Williams’ species.  The infamous ‘Yeast in a bottle’ experiment might be a perfect example.  The experimenter puts sugar, water, and yeast in a jar.  Then, at regular intervals he measures the yeast population.  The population grows exponentially.  Then it grows linearly. Then it grows sub-linearly.  Finally, the population sags or crashes. 

 

The observed phenomenon is explained in terms of the yeast’s relationship to its environment.  It eats the sugar, turning the available energy to reproduction, and producing as waste, alcohol and carbon dioxide.  Early on, there exists much sugar and hardly any waste. The yeast cells reproduce quickly and the population grows exponentially.  Then, as the yeast cells have to compete for sugar, and the waste products begin to reach concentrations that slow them down, they reproduce more slowly.  Finally, when the available energy is exhausted and/or the wastes have reached high levels, the yeast die. 

 

Now, we observe with yeast that they reproduce at an accelerating pace.  Then they slow down. Then they hardly reproduce at all. Depending on the initial conditions and the yeast and so on it is possible that all the yeast will finally die.  But those who are familiar with the yeast population’s fate will see how well it parallels the whale-hunters’ fate. In both cases, the system itself provides some negative feedback as the resources of the environment become changed by the presence of the population living in it.  Self-regulation need not be considered the sole regulatory mechanism.  Nor is it necessarily the primary regulatory mechanism.  It is certainly not the ultimate regulatory mechanism. 

 

Yeast did not evolve in a bottle and its tiny size means that it could happily ride the wind currents to find a new home far away. In the case of yeast it would clearly be a mistake to imagine that fertility is not important.

 

In the cases of very tiny species that can be assumed to be food for some very common and widely available predator in the immediate environment, it is possible to imagine no means of self-regulation.  Predation, shortage, and disease might suffice to control the species. 

 

In the case of aphids, a female is capable of reproducing asexually, and this allows for much faster reproduction.  Aphids are tiny, defenseless creatures that survive by sucking juices out of the soft tissues of plants.  In the absence of ministrations by protective ants, aphids tend to be consumed by predatory insects.  The ladybug is an example.   In a suburban garden, aphid populations in the absence of predators can soar quickly.  In the presence of predators, aphids quickly become an undetectable presence.   

 

It would be possible to imagine that aphids are a species that benefit from extremely high fertility. Yet even in the case of the aphid whose very existence is assured by its extreme fertility, one might reasonably expect that a modicum of resistance to predation might make the aphid more successful yet. This explains its relationship with ants.  Aphids produce a sugary fluid that some ant species use as food.  Some of some ant species will protect an aphid colony that is producing this fluid.  In my own garden the single case of this resulted in the plant in question dying quickly of dehydration. So, once again, we run into a population reproducing beyond the available means of support.  Other plants, evidently, were not judged tasty enough.  Or were unprotected by ants.  The aphid infestation disappeared when the source of food did.

 

It would be interesting to find out whether aphid populations that have had a long-standing and dependable relationship with ants tend to be Williams’ species to the same extent as aphids that lack this protection.  If one were to search for a group of closely related organisms that live under two different sets of environmental conditions, any differences between protected aphids and unprotected aphids might be worth studying.  Protected aphids would run the risk of destroying all the plants that fail to completely resist their attacks if they always reproduced as Williams’ species.  So it is not out of the question that one could find some population regulation behavior arising among protected aphids – at least if one ran the experiment in geological time.

 

In the case where an organism is at equilibrium with its environment for a reasonable amount of ecological time, it would seem that there really is an optimal level of fertility beyond which a species endangers its very own existence. Evolutionary success has preferentially attached to species that trade away fertility advantages to gain advantages in other areas.  As correct as Williams’ statement might be “all other things being equal,” the truth is that they never are equal.  There simply must be occasions, especially when a population exists near it evolutionary equilibrium point, where lower fertility actually provides an incremental advantage to higher fertility. 

 

There are clearly a number of cases of frontier species for which Williams’ argument holds.  If one imagines Homo sapiens var. angloamericanus to be a unique species, there are reasons to believe it is probably one such Williams’ species.  There are, of course, other identifiable populations in the traditionally defined species that have a habit of  expanding unconditionally.  But there are pockets where this tendency has been overcome. In any case, one wonders whether Williams’ argument is simply a case of the rather normal human tendency of projection.

 

There are a lot of frontiers and a lot of unique species that occupy them. At  frontiers marginal fertility might be considered to turn negative at far higher population levels than it does in more steady environments, but it must necessarily turn negative at some point. Even for frontier species the marginal utility of fecundity must have some theoretical limit.  Under more normal conditions, marginal fertility could only be unconditionally positive where the size and energy consumption of the species scales inversely with its population.  We know of few populations where this is true.  It can also occur where the population is extinct.

 

Arguably, Williams’ thesis is strictly wrong for every multi-cellular organism of sufficient complexity that exists with continuity through fluctuating conditions.  Arguably, Williams thesis is strictly wrong for multi-cellular organisms that exist in persistent colonies with populations very close to the optimal level.  At least before the success of man, a huge portion of populations on earth might have been categorized as belonging to one of these two categories. The question looming before us is whether man, is any better at anticipating and adapting to resource shortages than is yeast.

 

More Realistic Assumptions & Other Considerations

 

Unequal Status

One particular assumption in these models potentially overstates the consequences of exceeding the optimal population level.  Probably the most problematic is the assumption of strict equality. If one introduces inequality into the society, the catastrophic consequences associated with tiny population excursions beyond the optimal point might sometimes be avoided.  Under all but the worst cases of over-population, recovery would be possible, likely, or assured. 

 

If, for example, one assumed two levels of fitness in the constant-conditions model, then the group with the lower level of fitness would perish first. If this happened early enough,    and if food were stored in some exogenous form, then such deaths would free up more for the higher ranking individual.  In a sufficiently hierarchical society, the hardship at the bottom would be invisible from the top; hence, “let them eat cake.”  What appears to be the case is that social hierarchy in long-lived non-territorial species serves to protect the population from extinction in the case of severe shortages.

 

Herd animals appear to have quite a high level of equality.  The only apparent physical differences depend upon how close individuals are to the periphery of the herd.   Herds will frequently place high ranking members near the center and low ranking members on the periphery. The assumption is that this is because it is at the periphery that most predation occurs. In the case of predation, the peripheral members have a negative differential survival rate.  Being on the fringes can be dangerous. If, however,  one were thinking in terms of communicable disease in herd animals, one might argue that being at the periphery constitutes an advantage more than it did a disadvantage.

 

It is reasonable to ask to what extent the assumption of strict equality imperils the model.  And the answer is that it depends. If a population exceeds the optimum level by a tiny amount, then the death of a tiny number of its weakest members will suffice to bring it back into equilibrium.  In such a scenario, one might argue that periodic, slight over-reproduction might slightly accelerate all adaptive mechanisms in a society that has any heterogeneity.   This mechanism can only work when population exceeds the optimum level by a tiny incremental amount, because very large departures will materially weaken large portions of the population.

 

The question of departures from equilibrium is important.  There are reasons to argue that large departures would cause over-corrections that would necessarily take the population well below the optimal level.  Of course, the system response will depend on the exact mechanism by which a population is reduced to a sustainable, optimal level.   As we have argued for disease and predation, there exist examples in history where substantial corrections have occurred.  The black plague in Europe, or the near-extinction of whales.

 

Theorem In a population existing in a static environment there must exist be a level of over-population n_e that must be followed in time by depopulation to a level n_u such that n_u is not greater than the optimal population, n_o =E_p/e:   -1<n_u<n_o.

 

Proof : Assume that the level of overpopulation continues or increases.  In order for this to occur, a new and greater supply of energy will be required.  If this new and greater supply of energy is sustainable, then the population in question is not over-population. Thus, overpopulation is, by definition unsustainable.

 

Therefore, the optimal population is defined in terms of sustainability.  It only exists in a sustainable sense.The long-term response to the overpopulation can be bounded by two cases:

 

Case A: Undifferentiated – Equipartition of energy implies that all individuals perish. In this case, n_u=0.

Case B: Differentiated - Assume differentiation divides the population into two classes. One class is made up of a population of exactly the size that could be sustained on the new level of sustainable energy.  The other class is the balance of the population. In a society divided by these two classes, the first class will survive.  The second class will perish,  And the population will quickly arrive at n_u=n_o . A highly stratified society in which each member has a particular rank will be able to naturally adjust to any dearth with the minimum of disruption. ( We do not argue that this is a desirable sort of organization from any point of view outside the science.)

 

These two conditions establish the extremes of the reaction.  All real system reactions must lie somewhere in between these two cases.  In most cases is impossible to precisely divide a population as is suggested in case B: therefore, there must be some overcorrection.  Dividing an area into heritable territories does, however, address this problem to some degree. It may explain the Latin tradition of fixing property inheritance to a single male heir.  The feudal system, instead of treating territory as divisible, treated it as being a fixed societal asset.  And it arbitrarily assigned to that asset an administrator whose role was make it productive.  This put implicit pressure on all other members of the community not to reproduce, by threatening them with starvation.  It is a brutal system of apportioning food rights. It does have two positive attributes: it preserves the population near an equilibrium level, and it preserves some social order which, in turn, sustains a modicum labor in food production.

 

This strategy of apportioning territory is suggestive of a conjecture, namely, that the size and scale of the over-correction will depend upon the size and scale of the over-population.  If the over-population is extreme, and the burden on the environment such that the environment itself is materially degraded by that overpopulation, then the correction is likely to be extreme. By confining over-population effects to the smallest possible part of the population, large and painful correction effects can be localized and mitigated.

 

If the population reaches levels that far exceed the optimum level, many potential problems occur.  Almost all of its members will suffer at least some weakening from lack of energy.  This might translate into higher susceptibility to disease.  Or it might make the individuals easier targets as prey. And as we described earlier, if they present uncommonly attractive targets as prey, predators may multiply to a higher level.  If the predator level rises above a certain level, then especially for very large animals, the probability of extinction quickly approaches 1.0.  So, extreme levels of success can be said to “tempt fate.”

 

In agricultural societies, overpopulation tends to consume the very productive resources upon which it is build and thus to undermine the very energy– producing system.  When things start to break down, the population is likely to be well beyond the equilibrium population value.  Frequently, by this time the very fertility of the soil is degraded by high stresses.  And fuel supplies have been used up.  Food supplies dwindle, and as time wears on the cooperative behaviors that once led to high resource production begin to break down. At this point society can begin to come unzipped.  Believing in an unconditionally positive marginal return to fertility leads to exactly this kind of collapse scenario, if we are to believe the dozen or so examples in Diamond’s Collapse. 

 

The system behavior is derived from solid physical constraints on growth, namely energy intake.  Evolution may allow for tiny incremental advantages in energy efficiency, but these accrue slowly.  Thus, energy consumption is a very solid physical hard-stop against which population growth must end.  The only question to be resolved is how energy is to be partitioned among members of the population.  We have shown that the details of the portioning mechanism lead to differing outcomes in terms of recovery from situations of over-population. 

 

 

Fluctuation and Pseudo-Equilibrium

It is interesting to note that there exist dynamic population systems that fluctuate quite far from an equilibrium point.  Braun (Differential Equations and their Applications) presents an integrated model for the population fluctuations among mice that is informative.  In that model mice colonize an area.  They reproduce rather quickly, and the population density soars. Then at some point at least one of two things happens. The mice contract disease, or territorial predators move in.  And before long, the mouse population dips to a rather tiny fraction of its peak level.  At the low population density the mouse becomes unattractive prey and the predators move to new territory.  The low population density no longer supports disease organisms. And the environment produces lots of food. The mice reproduce quickly, and the cycle repeats itself. 

 

One particularly interesting thing about the model is that predatory behavior fails to wipe out the mice.  Until the population is extremely high, they are not attractive prey, being more trouble to catch than they are worth. Perhaps having an underground hiding pl