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A Square Brown Linoleum Tile
Imagine a couple of pine cones pushed by a storm across the beach on a new island, a tree species that never grew there before.
But it finds fertile ground, and grows, and reproduces, and succeeds. What are the trends and influences on the shape of its population graph over time in its new ecosystem?
First, there is frontier or edge growth at a time constant related to the distance in parameter one and proportional in quantity over time to the function in #7, as the first colonizers reach out from the leading edge of the population into untouched hinterlands.
For example, if the pine cones fell from an airplane in the middle of an infinite plane, then the leading edge growth would be proportional to the (circumference which is proportional to the) radius at a given time, and the radius at the given time would be about the average distance from parent to child times the number of generations since zero day. So that's just an expanding circle, and 2*pi*R has its role describing it.
On the other hand if the seed landed on the tip of a fixed-width peninsula, then the leading edge, as it travels down the peninsula increases the population about linearly at a rate proportional to the peninsula width.
Eventually, the entire island, continent, or reachable habitat has been reached by the frontier, and the edge effect hits zero. Even on a sphere such as the Earth, once the travel speed accounts for half the circumference, frontier growth goes to zero.
Second, there is in-fill growth. The first seed that rolled by and grew here maybe sent some more seeds along further, expanding the frontier, but it probably also send some seeds sideways to equally-populated zones, and backwards to previously-colonized areas that are by now probably more populated than this particular growth band. Although dispersal no doubt continues laterally within in-fill areas, cells that are similar to adjacent cells may be assumed to give and get equally, so that each can be considered as growing by itself. This can be calculated as exponential growth within cells of the landscape, assuming no limitations in density or required resources, because each growing organism has its multiplier in the succeeding generation. The formula's exponent may be adjusted based on average lifespan of the organism, since there is loss as well as gain based on each member in the population, when that one dies. But it remains exponential in the living successor count from one generation to the next. 10 new born and surviving to reproduce - 1 old parent dying, per generation, means P*9^N, with P the starting population, and N the number of generations. (As an exercise: Modify the formula for 2 parents dying per 10 surviving.)
So long as resources are unlimited, we get this linear-ish edge growth plus locally exponential in-fill growth,
But then limitations kick in. Assume there is a substratum of limited environmental resources which unlimited growth will use up, along with a substratum of renewable resources, renewing at a lower rate. For example fixed nitrogen in the soil, etc. The rate of usage is proportional to the population, each member raping the land equally. When that locally exponential growth crosses a certain line, having eaten up the limited resource, population then falls off a cliff and mass death returns the population to the sustainable density supported by the renewal rate of the renewable resource, which might be zero. If the species can mine the resource here and send it there, the locality of this effect is reduced, but on a global scale the effect remains, if the resource is unrenewable.
Infra-species density limitations also kick in at some point. For example, parents may have more children on the frontier than in a metropolis, by whatever mechanism, which may be unrelated to environmental resource constraints. Or, competition could induce shutdown, if there were a genetically shared submission algorithm whereby between two conspecifics when one detects it is losing the race it may be designed to quit competing early so that at least the other has a stronger chance of survival, thus an improved chance for the species. Trees need light, and may win when taller than the next one, and trigger a mechanism that leads to old-growth forest, giant trees comfortably spaced.
Next, disasters occur at random times with at random degrees of enormity. More or less complete extinction perhaps at the frequency of giant asteroid impacts. Disease waves wiping out substantial fractions of the population at rates of, perhaps, 30% once in 20 generations, if you take the black death, and the Spanish flu as recent examples in our species, based on 600 years, 30 years per generation. The frequency may be proportional (a) to the frontier if the disease vectors emerge on contact, or (b) to the surface covered, if the reservoir of disease vectors is geographically distributed and randomly emerging. And proportional (c) to the genetic mutation and selection rate of the potential vectors. These effects seem small relative to the resource limitation effects since they only set the population curve back briefly.
Imagining the human species population trajectory under this analysis, there would be something like an S curve. The frontier effect finished its first sweep across the entire globe during pre-history, thousands of years ago, after humans came out of Africa; mixed with Neanderthals; the species spread across eurasia; Bering Strait supported migration to the protoAmericas and Patagonia saw its first inhabitants, when the polynesian diaspora went from Australia to Hawaii. In-fill has been the process throughout history. And it has been largely exponential in-fill, given essentially unlimited resources of unfarmed fertile lands, water, etc., up to my or my grandparents' generation.
Infra-species competition normally led the disaffected, to migrate increasingly locally, since the spaces between began so vast and have become so small. But where confined inescapably, perhaps in some urban hellhole or a bounded or perhaps socially controlled environment, the competition submission effect has some role, inducing social hierarchy to the benefit of the subservient who may at least then survive or alternatively simply the removal (death) of the non-dominant. Suicide of the romantically rejected may be an example of a submission algorithm as described above. Migration or escape is obviously preferred if possible, and has been available for most of human history till now, but plants grow where they root, before they may know their competition, and humans may have similar rootings and competitive outcomes. Not just density but also other issues may induce dispersal. It is worth considering, how human evaluations of their fellows, so often critical or repelled, might be more positive if they were the last very few on earth, or in a given river basin or distant homestead. If so, then density is indeed at the essence of the need to disperse.
Random disasters have certainly applied to the species, removing fractions during an otherwise exponential growth phase. A wave of travel and transmission may, certainly has already, spread the disaster frontier perhaps many generations after one subpopulation was decimated to suddenly decimate another when the subpopulations come into contact. So such catastrophes are local and have a spreading, perhaps sometimes halting, trajectory. But even a 90% death wave under a generational doubling only sets a robustly-doubling population back three or four generations, say with humans as a whole 100 years. Subpopulations that do not robustly double, or that robustly double at rates less than other populations, may see varying trajectories or relative trajectories, as that of Jews after Hitler, or populations of native America after the combined scythings of the Gallows of Columbus' and the voracious smallpox.
The main limitation across say 10^3 or 10^6 generations is resource limitation, whether environmental resources or infra-species density limitations.
The huge bump (drop) in the population trajectory comes when the limited environmental resources hit their exhaustion point. At that time the entire population must be reduced to the level sustainable by the renewal rate of the resource.
If our lives depended on helium, for example, and it gets mined out and dumped to the atmosphere as it presently is being done, and the helium leaving the ground also leaves the atmosphere to outer space, then the renewal rate is zero and the sustainable population would be zero. A resource-capture-and-re-use program would be the only way to milk a population's survival time further, and any losses would be proportional to long-term species population reduction.
Technology could be developed and deployed to fix nitrogen, if you were a species of smart pine trees, for example, so as to raise the sustainable population level by some amount. Or new resources could be found, or alternatives made useable perhaps by mutation of either the resource provider (as in foodstuff hybridization) or the resource user (as in the indigenous northern European human adaptation for milk consumption.)
One could imagine an energy-bounded, solar-powered sci-fi world where Earth's surface is covered by a maximum sustainable number of floors filled with minimal-energy-consumption cells with solar power captured on the entire surface powering a maximum density of population. Population adjustments could be enabled by: conservation as in lower energy farming methods, expansion of supply as in, say, off-planet energy capture, or alternative supply, as in nuclear energy.
However, the list of resources whose limitations limit our species is a list with unknown membership. Perhaps that bee species is a keystone in the life cycle of a certain essential foodstuff. Perhaps that ocean gyre causing European weather and farmability depends on a certain artic icepack condition to drive a vortex to continue spiralling north atlantic surface waters inward and downward, and your Suburban just pushed enough C02 into the atmosphere to melt it over an unknown (or ignored) threshold. Poof.
The concerned husbandman who seeks a gentle trajectory for a species might hold a thoughtfully heavy hand on population growth especially before drop time, that moment when exponential growth is about to cross into exhaustion of non-renewables. Since knowledge at such a time has the value of a large fraction of the lives of the population, he -- or more likely she -- might spend a fat fraction of resources on a scientific efflorescence to understand the dependencies in the ecosystem, on precise monitoring of usage and a careful measurement of the differences between limited and renewable, and on early practice attempts at adequate preparations for widespread implementation of renewable resource expansion programs or capture/reuse programs.
Otherwise later generations, looking back at those who squandered those limited resource reservoirs, will curse them, their numerous, greedy, and stupid, forefathers and mothers, as they wither and die.