Safety in Numbers.

Cue

The Range, a habitat that rewards being many and a population to fill it. The Throng drawing together, the density high, each member keeping to the crowd because the crowd is the thing that works.

Action

Aggregating: The Throng, holding the throng's seat, gathers and confers on each member the benefit of its density.

Stage

The Throng, on Draw Together, makes the density high and confers the benefit on each member, the mate found, the predator diluted, the intruder mobbed, the ground foraged in company. Each member draws the gain the crowd gives, and every count taken over the range reads as a population that will go on.

Tension

The safety the crowd confers is genuine, and its being genuine is what makes the numbers the safety. There is nothing false: the density really does dilute the predator and hold the mates within reach. But the benefit is per-capita and rises with the numbers, so the plot closes with the throng dense and the safety real, the whole of it resting on a density that can be thinned and a line beneath it no count reveals.

Cue

The Refuge, and someone unwilling to trust a count that reads ample. The Keeper setting out to hold the density above the line before the attrition can carry it across.

Action

Holding: The Keeper, holding the keeper's seat, protects the range, restrains the harvest, and supplements the throng to keep the numbers over the critical point.

Stage

The Keeper, on Hold the Number, keeps the density above the critical number, holding the benefit and keeping the population off the spiral. Against the holding stand a count that reads ample and a harvest that reads sustainable, so the guarding must be spent before the numbers give any reason for it.

Tension

Holding the numbers above the line is the one move that keeps the safety, and it asks the outlay the ample count makes hard to justify. It spends on a population that looks fine, restrains a harvest that reads sustainable, and guards against a threshold no count reveals. So the plot closes one of two ways: the density held above the line and the spiral never entered ... or the guarding waived as spending on nothing, the headcount taken for safety, and the line left for the thinning to cross.

Cue

The Range, and attrition no member authored, a harvest, a drought, a road, a hard winter. The numbers beginning to fall for reasons outside anyone in the throng, the density lowering.

Action

Thinning: The Threshold, holding the threshold's seat, lowers the density by the losses any population takes and carries the numbers toward the line.

Stage

The Threshold, on Thin the Ranks, lowers the density and with it the benefit each member draws, the mate harder to find, the dilution weaker, the mob too small. The members carry on unchanged, each still fending for one, and the count still reads survivable while the critical number beneath it draws nearer.

Tension

The attrition is ordinary and no member's fault, and the same behaviours carry on because nothing about the member has changed, only the density. There is nothing careless: the losses are the losses any population takes. But the benefit falls with the numbers and the line gives no warning, so the plot closes with the throng thinned and the benefit reduced, the count reassuring at the very density where the safety is failing and the line still unseen.

Cue

The Edge, the density come close to the critical number. The throng thinned near the line, the headcount a figure that a year ago was plenty, the crossing due.

Action

Thinning past the line: The Threshold, holding the threshold's seat, carries the density across the critical number, where the per-capita growth turns negative.

Stage

The density crosses the critical number, and the per-capita growth passes from positive to negative, so the population no longer merely grows slower but shrinks, the benefit now too thin to carry it. The members still fend for one exactly as before, and the count is still a number that reads survivable, marking the crossing with nothing at all.

Tension

The line is real and fixed, and its being invisible from above is what lets the numbers cross it unremarked. There is nothing sudden the members did: each behaves as the crowd rewarded and the density passed a mark no count shows. But below the line the growth is negative and the same numbers cannot hold, so the plot closes with the line behind the population, the count still reassuring and the descent about to feed itself.

Cue

The Edge, a population below the line with its per-capita growth turned negative. Each member fending for one as it always did, the crowd the one thing that is going.

Action

Spiralling: The Member, holding the member's seat, keeps to the behaviour the throng rewarded while the density dependence runs in reverse.

Stage

Each member, on Fend for One, draws the thinning benefit from a falling density, and each loss lowers the density and the density lowers the benefit and the benefit lowers the growth. The decline feeds the decline, the spiral running faster the fewer remain, until it is the vortex, a population drawn to extinction by the very benefit that had held it up.

Tension

Every member behaves exactly as the crowd rewarded, and the faithfulness is exactly why the collapse has no author. There is no fault at the edge: each does what worked and the crowd is what fails. But the density dependence runs both ways, so the plot closes with the spiral running to extinction, no member having erred, the safety in numbers become a peril in fewness and the population drawn down by its own thinning.

Tenor

Sound, thriving, safe in the crowd. The plain good sense of a population doing what a throng rewards, each member finding its mate, diluting the predator, mobbing the intruder, foraging in company, the numbers high and the count reading as a population that will go on. Nothing false in it on its face: the safety is genuine, the crowd truly confers the benefit, and every behaviour is the one the numbers made correct.

Undertow

Under the thriving, a benefit that is per-capita and rises with density, so the safety the crowd confers is the crowd's to lose. Because a benefit that rose with the numbers falls with them, a population thinned below a critical line does not merely grow slower but turns to negative growth, and the decline feeds the decline, faster the fewer remain, drawn down by the very density dependence that had held it up. What makes a crowd safe, that fitness rises with numbers, is what makes fewness fatal once the numbers fall.

Nerve

The feeling of a population sound and secure, the throng dense and the safety real, each member doing exactly what works, because in a crowd it does work and the numbers are there, while below a line no count reveals the same behaviours cannot sustain the few, and the safety in numbers becomes a peril in fewness.

Echo

After the run: a population drawn to extinction not by any member's fault but by the loss of the numbers that were its safety, no one having over-bred and no behaviour having changed, only a per-capita benefit falling with a density thinned past the line where growth turns negative and the spiral runs, and the one move that would have kept it, holding the numbers above the line before the crossing, forgone because it means spending on a population that reads ample and restraining a harvest that reads sustainable.

Projection

Watchful. Willing to spend early. Suspicious of a healthy count. Is the conservator, the steward, the manager who guards the numbers, the one who keeps the density over the line where a headcount alone would not. No alarmism in it: the Keeper does not doubt that the count reads survivable, only doubts that a survivable count is a safe distance from the line.

Action

Protects the range, restrains the harvest, and supplements the throng, holding the density above the critical number before the attrition can carry it across. Treats an ample headcount as suspect precisely because the threshold shows only after it is passed, and spends on the numbers while they still look fine. Does what keeps the population off the spiral: it holds the density above the line, and holding it is the one thing that keeps the safety the numbers confer.

Shadow

Not a free rescue, a costly and unthanked spending, and that is the whole difficulty: the holding that keeps the population above the line is exactly the outlay a healthy count makes hard to justify. Because the numbers read ample right up to the line, the Keeper must spend on a population that looks fine, restrain a harvest that reads sustainable, and guard against a threshold no count reveals, so the density is held only against every signal that it need not be, the spiral it prevents leaving no trace and the cost of holding plain for all to see, the extinction averted looking, afterward, like one that was never going to happen. It holds the numbers to keep the safety, and holding them is read as spending on a population that was doing fine.

Tell

Says, in effect, "keep the numbers well above the line, before the count gives you reason to." Says it over a population that reads survivable, and is heard, often, as spending too early, guarding against nothing, the one who will not trust a healthy count.

Projection

Sound. Thriving. Doing what works. Is any one of the many in the throng, the bird in the flock, the fish in the shoal, the individual that must find a mate, avoid the predator, and feed, and does each of these by being among the rest. No fault in it: the Member behaves exactly as the crowd rewards, its reliance on the throng not a habit to drop but the behaviour the numbers made correct.

Action

Keeps to the many, finds its mate in the throng, is diluted among the crowd against the predator, mobs the intruder with the rest, and forages the ground in company. Does the same at low density as at high, because nothing about the member has changed, only the number around it. Does what a member must: it lives the way the crowd rewards, and the way the crowd rewards is the way that thrives while the crowd is there.

Shadow

Not over-breeding, not erring, a member behaving faithfully as it always did, and that is the whole difficulty: the behaviour is sound, and its soundness depends on a density the member cannot hold up alone. Because the safety was in the numbers and not in the member, when the numbers fall the same reliance draws a thinner benefit, the mate harder to find, the dilution weaker, the mob too small, so the member below the line is drawn down not by any change in itself but by the loss of the crowd, doing to the last exactly what worked while the crowd was there. It behaves as the numbers rewarded, and the reward was the numbers, so behaving well is no defence once the numbers go.

Tell

Does, in effect, "what I have always done, because it works." Does it while the density falls around it, and cannot behave its way out, because there is no better behaviour than the one the crowd rewarded, and the one thing failing is the crowd.

Projection

A line, unmarked. A force, downward. Is the critical density itself, the point in the numbers where growth crosses zero, and the standing pressure of attrition, mortality, harvest, hard seasons, that lowers the density toward and past that point. No malice in it: the Threshold is a property of the population's own fitness curve and the ordinary losses any population takes, neither of which any member authored.

Action

Lowers the numbers, and holds the line beneath them: attrition thins the ranks toward the critical density, and at the density the per-capita growth rate crosses from positive to negative. Gives no sign of itself from above, and no way back from below, the same numbers that read as plenty above the line unable to sustain the population beneath it. Does what a threshold does: it marks where the benefit can no longer carry the growth, and where it is passed the population falls.

Shadow

Not a cliff anyone can see, a line without a marker and a pressure without a face, and that is the whole difficulty: the density where growth turns negative is real and fixed, and nothing in the headcount shows where it lies. Because the Threshold is invisible from above, a population reads survivable at a number just over the line as at a number far above it, so the attrition that lowers the density crosses the line unremarked, and below it the population is not slowed but reversed, drawn toward extinction by a growth rate gone negative, the line that gave no warning giving no way back. It is a line no count reveals, and being unrevealed is exactly what lets the numbers cross it before anyone acts.

Tell

Is, in effect, "the number below which the same numbers no longer hold." Is it silently, marked by nothing the headcount shows, so the population passes it reading survivable and learns of the line only from the far side, where growth has already turned.

Projection

Dense. Many. The safety itself. Is the crowd as a body, the flock, the shoal, the herd, the colony, the aggregate whose size is the thing that protects and provides for each within it. No illusion in it: the Throng genuinely confers the benefit, the dilution and the mobbing and the mate-finding real gains that scale with how many it holds.

Action

Gathers, and by gathering makes each member safer and better fed than it would be alone, the predator diluted across many, the intruder driven off by the mob, the ground foraged in company. Confers a benefit that rises with its own density, so the more of it there is, the more each within it draws. Does what a crowd does: it is the numbers, and the numbers are the safety, so the Throng is the benefit made of bodies.

Shadow

Not a permanent shelter, a safety made entirely of its own size, and that is the whole difficulty: the protection is real and it is the density's, so it can only be as large as the numbers are. Because the benefit it confers is per-capita and rises with density, a thinning of the Throng lowers what each member draws, and there is no floor of safety it keeps once thinned, the dilution weaker, the mob smaller, the mates sparser, so the crowd that was the safety becomes, below the line, too small to confer it, the very thing that held the population up now too thin to hold it. It is the safety while it is dense, and being the safety is exactly why losing its density loses the safety.

Tell

Is, in effect, "the many that keep each one." Is it only while the many are there, and cannot keep any one once thinned below the line, because the keeping was the number, and the number is what has gone.

Shown

The place the population comes to near the line, a thinned range, a remnant, a sparse patch, where the throng is smaller and the benefit each member draws has fallen with it. The Members still fending for one, the density near the critical number, and the count a figure that a year ago was plenty, giving no sign the line is close.

Holds

The crossing, and the spiral that waits below it. Here the density is near the critical number, so the Edge holds the point where the safety fails: a per-capita growth about to turn negative and a benefit too thin to carry it, so it is where the same behaviours that thrived can no longer sustain the few, and where, once the line is behind, each loss lowers the density and the density lowers the benefit and the decline feeds itself.

Offers

A population that still reads survivable, and a line about to be crossed. Because the Edge is where the numbers are near the threshold but not yet marked by it, it offers a headcount that still looks like plenty and a growth about to turn, so it hands out a last stretch of apparent safety, and hands it out over a line no count reveals, the crossing due and the count not showing it.

Withheld

Any warning that the line is here. The Edge holds the population at the brink but cannot mark the brink, because the threshold is invisible from above and the count reads survivable to the last figure before the turn, so it withholds the crossing until it is past, letting the number stand for safety at the very density where safety fails, and, all apparent plenty and no marker, it withholds the spiral until the growth has already gone negative and the fewer remain the faster they fall.

Shown

The place the population lives and the density is felt, a habitat, a patch, a breeding ground, where the throng draws together and each member finds its mate, is diluted against the predator, and forages in company. The Throng dense across it, the Members drawing the benefit, and every count taken over it reading survivable.

Holds

The density, and the safety it confers. Here the numbers are high and the benefit real, so the Range holds the point where the crowd works: a per-capita fitness that rises with density and a member that thrives by keeping to the many, so it is where the safety in numbers is genuine and felt, and where nothing in the abundance shows the line beneath it.

Offers

A thriving population and a safety that works. Because the Range is where the density is high and the benefit conferred, it offers a sound population and a real protection, the mates found and the predator diluted and the ground foraged in company, so it hands out the safety the numbers make, and hands it out on the density and not on any floor, the whole of it resting on numbers that can be thinned.

Withheld

Any sight of the line the density sits above. The Range shows the population thriving but cannot show the critical number beneath it, because the threshold carries no marker and the count reads survivable at any density above it, so it withholds the distance to the line, letting the abundance stand for safety, and, all high numbers and no visible floor, it withholds the fact that the safety is the density's until the attrition thins it toward the edge.

Shown

The place the density is held above the line before it can fall to it, a reserve, a protected patch, a supplemented population, where the harvest is restrained and the numbers kept well over the threshold. The Keeper holding the density up, the throng kept dense enough to confer its benefit, and the population kept off the edge where the growth would turn.

Holds

The holding, and the safety it keeps. Here the numbers are kept above the critical line before the thinning can reach it, so the Refuge holds the point where the safety is preserved, not restored: the density held over the threshold and the benefit held with it, so it is where the population is kept off the spiral, the one arrangement in which the safety in numbers is guarded against the fewness that would end it.

Offers

A way to keep the density above the line, and the safety kept rather than lost. Because the Refuge holds the numbers over the threshold before the crossing, it offers the one thing the Range could not once thinned, the safety preserved while it can still be preserved, so it hands out a population kept off the spiral, and hands it out at the price of spending on numbers that read ample and restraining a harvest that reads sustainable.

Withheld

Any ease, or any proof that the holding was needed. The Refuge can keep the density above the line but cannot make the spending look justified or make its worth show, because the spiral it prevents leaves no trace and the cost of holding is felt at once, so it withholds the standing of one who spends only when the count demands it, exacting the look of guarding against nothing, and, all preserved safety and no visible payoff, it withholds the thanks for a spiral that now never runs, the averted extinction seeming, after, like one that was never going to happen.

Place

Conferred across the range, where the density is high and the gain each member draws is full. Reduced toward the edge, where the thinning lowers the density and the benefit with it.

Load bearing

Without it density is just a number and there is nothing the crowd gives, and no safety to rise and fall. With it the trap turns: the per-capita gain each member draws from the density, the mate-finding, the predator dilution, the cooperative defence and foraging, each a component effect that improves with how many are present (Courchamp, Berec and Gascoigne, on the mechanisms of the Allee effect; Allee, on the survival value of aggregation). The Benefit is load bearing because it is what makes the safety the numbers' and not the member's: a gain that rises with density and falls with it, so the piece carries the return to crowding that is real at high numbers and thin at low, the same reliance paying less as the throng thins though the member has not changed.

Apparent

A trait of the member, a fitness it seems to carry in itself. Seen at the surface, the Benefit looks like the member's own soundness, a bird that finds mates and evades hawks because it is a capable bird, and the truth that the gain is per-capita and density-dependent, that it is drawn from the crowd and not held by the member, that it falls when the numbers fall for no change in the member at all, stays hidden behind how much a thriving member looks like the author of its own thriving.

Yearbook

Conferred by the throng through aggregating, drawn by the member through fending for one, the gain the numbers give. It is the turn of the trap: the Benefit is drawn from the density and rises with it, falls as thinning lowers the numbers past the critical number, and its loss below the line drives the spiral within the vortex.

Place

Taken across the range and the edge, where the figure reads survivable at any density above the line. Distrusted only at the refuge, where the keeper treats an ample count as suspect.

Load bearing

Without it the danger would show in the numbers and the crossing could be seen coming, and someone would act before the line. With it the trap is deceptive: the headcount that reads as a viable population and carries no mark of the threshold, so a number that was plenty yesterday reads plenty today and nothing in it warns that the line is near (the deceptive cue of a headcount that stays reassuring while the density approaches the critical number). The Count is load bearing because it is what hides the line: the figure that looks safe at the very density where safety fails, so the piece carries the reassuring number that reads survivable to the last figure before the turn, the cue that misleads not by error but by carrying no sign of the threshold at all.

Apparent

A reliable measure, a healthy figure that means a healthy population. Seen at the surface, the Count looks like exactly the signal it should be, a number that tracks the population's health, and the truth that it reads the same just above the line as far above it, that it gives no distance to the threshold, that a survivable count is not a safe count, stays hidden behind how much a healthy figure simply feels like proof of safety, right up until the crossing.

Yearbook

Taken over the density as it falls, distrusted only by the keeper through holding, the figure that reads survivable to the last. It is the deceptive cue of the trap: the Count reads safe over the benefit already thinning and the critical number already near, showing nothing of the spiral until the vortex is running.

Place

Approached at the edge, where the thinning carries the density toward it. Held off at the refuge, where the numbers are kept well above it before the crossing.

Load bearing

Without it the benefit only slows the growth and there is no line to cross, and a thinned population merely grows more slowly. With it the trap has its threshold: the density beneath which per-capita growth falls below zero, so a population thinned past it shrinks rather than slows, the mark that makes the effect strong rather than weak (Stephens, Sutherland and Freckleton, on the definition of the Allee effect and the critical density; Dennis, on the critical population size and the threshold below which growth cannot sustain the population). The Critical Number is load bearing because it is the line that turns decline into collapse: the point at which the safety fails outright, so the piece carries the threshold no headcount marks, above which the numbers hold and below which they cannot.

Apparent

Nothing, on the surface: a number like any other, no different from the counts above it. Seen at the surface, the Critical Number is invisible, because a density just over it reads as a density far above it and the count gives no sign of the line, so the population reads survivable to the last figure before the turn, and the truth that there is a real threshold, that it is fixed and unmarked, that the same numbers cannot hold below it, stays hidden behind how ordinary the number looks right up to the crossing.

Yearbook

Approached as thinning lowers the density, held off only by the keeper through holding, the line the safety sits above. It is the threshold of the trap: the Critical Number is where the benefit can no longer carry the growth, unseen beneath the count, and once crossed it opens the spiral that is the vortex.

Place

Gathered across the range, where the throng draws together and the numbers are high. Lowered toward the edge, where the thinning carries the density toward the line.

Load bearing

Without it there is no crowd and no benefit, and no numbers to thin below the line. With it the trap has its substance: the numbers per area on which the whole safety rests, so that a member's fitness is not fixed but rises with the density around it, positive density dependence and not a constant (Allee, on animal aggregations, the finding that survival and growth improve with numbers and that under-crowding harms as over-crowding does). The Density is load bearing because it is the variable the safety hangs on: the quantity that confers the benefit when high and withdraws it when low, so the piece carries the numbers that are the safety itself, real while they are many and the safety's to lose the moment they thin.

Apparent

A count, a population size, a figure that is high and reads as secure. Seen at the surface, the Density looks like a stock that is either present or not, a healthy number that means a healthy population, and the truth that the safety is a rising function of it and not a constant, that a benefit high at high density is low at low density, that the numbers are the protection and not merely its measure, stays hidden behind how much a large number simply looks like enough.

Yearbook

Gathered by the throng through aggregating, lowered by the threshold through thinning, the numbers the safety is made of. It is the substance of the trap: the Density confers the benefit while it is high, is read as the count that looks survivable, falls toward the critical number, and below it drives the spiral at the heart of the vortex.

Place

Runs at the edge, where the population past the line is drawn down by its own thinning. Kept from starting, if ever, at the refuge, where the density is held above the line.

Load bearing

Without it a population below the line simply declines and the loss is a subtraction, not an acceleration. With it the trap has its reckoning: below the critical number each loss lowers the density, lower density lowers the per-capita benefit, and lower benefit lowers the growth further, so the decline compounds and the population falls faster the fewer remain, the density dependence running in reverse (Dennis, on the accelerating extinction risk below the critical density; the demographic Allee effect turning a thinned population toward collapse). The Spiral is load bearing because it is what the loss of the numbers finally costs: not a slower population but a self-feeding descent, so the piece carries the acceleration that no member started and none can halt by behaving well, the safety in numbers become a peril in fewness.

Apparent

A sudden crash, a collapse that seems to come from nowhere. Seen at the surface, the Spiral looks like a shock from outside, a population fine one season and failing the next, and the truth that the descent is the population's own density dependence in reverse, that each member is behaving exactly as the crowd rewarded, that the acceleration was latent in the benefit all along, stays hidden behind how much a compounding decline looks like a blow that struck from without.

Yearbook

Runs from the benefit lost below the critical number, kept from starting only by holding, the descent each member's faithful behaviour feeds. It is the reckoning of the trap: the Spiral is the density falling faster the fewer remain, the true cost of the safety that was the numbers', the engine of the vortex that the count gave no warning of.

Place

Turns across the edge, where the population past the line is pulled down by its own thinning. Averted, if ever, at the refuge, where the numbers are held above the line and the vortex never opens.

Load bearing

Without it the decline is a loss and no more, and nothing draws the population all the way down. With it the trap is complete and authorless: a population below the critical number pulled toward extinction by mutually reinforcing feedbacks, each small step lowering the density and each lower density deepening the descent, the benefit that had made the crowd safe now the engine of its loss (Gilpin and Soulé, on the extinction vortex, small populations drawn to extinction by the reinforcing loops their own smallness sets off; Courchamp, Berec and Gascoigne, on the Allee effect and extinction). The Vortex is load bearing because it is the phenomenon whole: not one bad season but a self-deepening spiral no member started and none can halt by behaving well, so the piece carries the extinction the loss of the numbers finally becomes, the safety that was real drawn down by its own dependence on the density.

Apparent

A tragic collapse, a population that failed against the odds. Seen at the surface, the Vortex looks like bad luck or a hostile world, a species that simply could not make it, and the truth that the descent is the density dependence in reverse, that every member behaved as the crowd rewarded, that the safety was genuine and lost only with the numbers that held it, stays hidden behind how much an extinction looks like a fate rather than a benefit running backward.

Yearbook

Turns from the spiral below the critical number, averted only by the keeper through holding. It is the phenomenon whole: the Vortex is the density drawn down by the loss of the benefit it conferred, the extinction the count gave no warning of, the safety in numbers at the centre of the play become a peril in fewness.

Has

A doubt about a healthy count and the means to hold the numbers up: the will to keep the density above the line and the levers to protect the range, restrain the harvest, and supplement the throng. A real corrective: holding the density above the critical number is the one thing that keeps the safety the numbers confer. No cheap way to do it, and no welcome for the one who spends on a population that looks fine.

Orders

Hold the number. Keep the density above the critical line, protect the range, restrain the attrition, and supplement the throng before it thins to the threshold, and treat an ample headcount as suspect for showing the danger only once it is past. Spend early, because the line gives no warning.

Loses

The standing of one who spends only when the count demands it. Because the numbers read survivable right up to the line, holding them above it means spending on a population that looks fine, restraining a harvest that reads sustainable, and guarding against a threshold no count reveals, so the density is held only against every signal that it need not be, and the spiral it prevents leaves no trace while the cost of holding is plain, the averted extinction looking, after, like one that was never going to happen. It gains the numbers held and the spiral never entered, and loses the ease and the standing that spending late would have kept.

Drives

Toward the density held above the line and the safety kept, at the continuous cost of spending on a population that reads ample, so the seat that holds the number keeps the population off the spiral, and is read as guarding against nothing.

Has

A behaviour that works and a crowd that makes it work: the reliance on the throng to find the mate and dilute the predator, and the numbers that make that reliance pay. A real fitness: the behaviour is sound and rewarded, so keeping to it is not an error. No safety of its own and no way to hold the density up alone, because the safety was in the numbers and not in the member.

Orders

Fend for one. Live the way the crowd rewards, keep to the many, trust the throng to find the mate and dilute the hawk and raise the alarm, because that is what works while the numbers are there. Do the same at low density as at high, since nothing about the member has changed.

Loses

The safety, exactly by its being the numbers' and not its own. Because the benefit is per-capita and rises with density, when the throng thins the same reliance draws a thinner benefit, so the member below the line is not slowed by any change in itself but drawn down by the loss of the crowd, doing to the last what worked while the crowd was there. It keeps a sound behaviour and a faithful life, and loses the density that was the only thing making them safe.

Drives

Toward the behaviour the crowd rewarded, held to whatever the numbers do, its fitness rising with the density and falling with it, so the seat that fends for one behaves faithfully still as the crowd that made it safe goes, and cannot behave its way out of a fewness the behaviour did not cause.

Has

A fixed line in the numbers and the ordinary force that lowers them to it: the critical density where growth crosses zero, and the attrition, harvest, drought, and hard seasons that thin the ranks toward it. A real edge: the density where the benefit can no longer carry the growth is genuine and fixed, not a mood or a slope. No marker above it and no return below it, because the line gives no sign of itself until it is passed.

Orders

Thin the ranks. Lower the density by the losses any population takes, carry the numbers toward the critical line, and past it turn the per-capita growth negative, because that is what attrition and a fitness curve do, neither authored by any member. Give no warning, since the line is invisible from above.

Loses

Nothing of its own, and takes the safety with it: the Threshold is a property and a pressure, not a seat that can be harmed, and what it removes is the population's margin above the line. Because the attrition thins the density unremarked and the critical line carries no marker, the numbers cross it reading survivable, and below it the population is not slowed but reversed, drawn toward extinction by a growth rate gone negative, the line that gave no warning giving no way back. It gains no benefit and bears no malice, and what is lost across it is the safety the numbers had conferred.

Drives

Toward the density lowered and the line crossed, a fixed critical number and an attrition that reaches it unremarked, so the seat that thins the ranks carries the numbers past a line no count reveals, and below it the same numbers cannot hold.

Has

A size and the benefit that size confers: the many bodies and the dilution, the mobbing, and the mate-finding that rise with how many it holds. A real protection: the safety is genuine and the density's, so gathering is not a comfort but the thing that works. No safety apart from its numbers, and no floor it keeps once thinned, because the benefit is made of size and falls with it.

Orders

Draw together. Gather, keep the density high, and confer on each member the dilution and the mobbing and the mates that only numbers give, because the more of the crowd there is, the more each within it draws. Be the numbers, since the numbers are the safety.

Loses

The safety it is, exactly by being made of a density it cannot keep. Because the benefit it confers rises with its own density, a thinning lowers what each member draws and there is no protection it holds once thinned below the line, so the crowd that was the safety becomes too small to confer it, the very thing that held the population up now too thin to hold it. It gains a real safety while it is dense, and loses it precisely by losing the numbers that were it.

Drives

Toward the density gathered and the benefit conferred, the safety that is the numbers and only the numbers, so the seat that draws together is the safety while it is dense, and cannot keep any one once the density falls below the line.

Initiated by

A population and a habitat that rewards being many. At the range, the throng gathering, each member keeping to the crowd because the crowd finds the mate, dilutes the predator, and mobs the intruder.

Direction

Toward the density high and the benefit conferred. Because a crowd dilutes the predator, mobs the intruder, and holds the mates within reach, aggregating gathers the numbers and confers on each member a per-capita gain that rises with them, so the more of the throng there is the more each within it draws, the safety made of bodies and genuinely the density's.

Lever

The rise of the benefit with the numbers. Because each component gain, the mate found, the dilution, the mob, the forage, improves with how many are present, aggregating runs on a real positive return to density and not on any comfort, so the safety is assembled from no illusion, only from a crowd whose size is the thing that protects and provides for each within it.

Echo

A density gathered, and a safety made of it. The throng has drawn together and conferred its benefit, so what reads as a settled protection is, in truth, a safety that is the numbers and only the numbers, real while the crowd is dense and the density's to lose the moment the crowd thins.

Initiated by

A count that reads ample and someone unwilling to trust it. At the refuge, the keeper protecting the range, restraining the harvest, and supplementing the throng before the attrition can carry the density to the line.

Direction

Toward the density held above the line and the safety kept. Because the safety fails below the critical number and the line gives no warning, holding keeps the density well over the threshold before the thinning can reach it, so the benefit is held and the growth kept positive, the population kept off the edge where it would turn, the safety preserved while it can still be preserved rather than restored once lost.

Lever

The willingness to spend against an ample count. Because holding the numbers means spending on a population that reads survivable and restraining a harvest that reads sustainable, it runs only on someone choosing to guard against a line no count reveals, so the corrective is exacted continuously and thanked rarely, the spiral it prevents leaving no trace and the cost of holding plain for all to see.

Echo

A density held, and a spiral never entered. The keeper has kept the numbers above the line, so what reads as spending on a population that was doing fine is, in truth, the one act that keeps the safety the numbers confer, the extinction it averts leaving nothing to thank it for and the outlay it took felt at once.

Initiated by

A population below the critical line, its per-capita growth turned negative. At the edge, past the crossing, each member still keeping to the behaviour the throng rewarded while the numbers can no longer sustain it.

Direction

Toward the decline feeding the decline. Because each loss lowers the density and lower density lowers the benefit and lower benefit lowers the growth, spiralling turns each reduction into a further one, so the population below the line does not merely shrink but shrinks faster the fewer of it remain, the engine that lifted it now lowering it, no member having changed and the crowd the thing that goes.

Lever

The density dependence running both ways. Because the same positive return that made a dense crowd safe makes a thin one perilous, spiralling runs on no new force but the benefit falling with the numbers, so the acceleration is assembled from no error and no member's fault, only from each fending for one below a line where the behaviour that thrived can no longer pay.

Echo

A decline that gathers speed, and no member to blame for it. Each has done exactly what the crowd rewarded, so what reads as a sudden collapse is, in truth, the density dependence in reverse, the safety in numbers become a peril in fewness, the population drawn down by the very benefit that had held it up.

Initiated by

Losses any population takes and no member can prevent. At the range and toward the edge, the harvest, the drought, the road, and the hard winter lowering the numbers, the density falling without anyone in the throng having caused it.

Direction

Toward the density lowered and the line neared. Because attrition removes members and the benefit is per-capita, thinning lowers both the numbers and the gain each draws from them, so the same reliance pays less as the density drops, the mate harder to find and the dilution weaker, and the population is carried toward the critical line unremarked, the count still reading survivable.

Lever

The invisibility of the line from above. Because the threshold carries no marker and a number just over it reads as a number far above it, thinning lowers the density without any signal that a line is near, so the crossing is assembled from no alarm, only from ordinary losses that reach a critical point the headcount does not show.

Echo

A throng thinned, and a line drawn nearer. The attrition has lowered the density, so what reads as a smaller but surviving population is, in truth, a benefit reduced with the numbers and a line approached unseen, the count reassuring at the very density where the safety is failing.

Direction

Gather and keep the density high. Confer on each member the dilution, the mobbing, and the mate-finding that only numbers give, and be the safety the crowd makes, because the more of the crowd there is, the more each within it draws.

Orders

(The Throng gathers and by gathering confers a per-capita benefit that rises with its density, the safety made of bodies that holds only while the bodies are there.)

Implementation

(The Range; the aggregating process; the density confers the benefit on each member, the crowd the safety no single member carries and none can keep once thinned below the critical number.)

Targets

• [ ] the throng gathers and keeps the density high
• [ ] the density confers a per-capita benefit that rises with the numbers
• [ ] the safety is genuinely the crowd's, real while the crowd is dense
• [ ] the benefit falls with the numbers, the safety lost as the throng thins

Direction

Live the way the crowd rewards. Keep to the many, trust the throng to find the mate, dilute the predator, and raise the alarm, and do the same at low density as at high, because the behaviour is sound and nothing about the member has changed, only the number around it.

Orders

(The Member keeps to the behaviour the throng rewarded, drawing the benefit only a crowd confers, and cannot behave its way out of a fewness the behaviour did not cause.)

Implementation

(The Range; the spiralling process below the line; the benefit each member draws from the density falls as the numbers fall, the same reliance sustaining fewer once past the critical number.)

Targets

• [ ] the member keeps to the behaviour the throng rewarded, at low density as at high
• [ ] the benefit it draws is the density's, rising and falling with the numbers
• [ ] nothing about the member changes, only the crowd around it thins
• [ ] below the line the sound behaviour cannot sustain the few, and the member is drawn down by the loss of the crowd

Direction

Keep the density above the critical line before the attrition can carry it across. Protect the range, restrain the harvest, and supplement the throng, and treat an ample headcount as suspect for showing the danger only after it is past, so the spiral is never entered and the safety never lost.

Orders

(The Keeper holds the density over the line and makes the guarding continuous, at the cost of spending on a population that reads ample and restraining a harvest that reads sustainable.)

Implementation

(The Refuge; the holding process; the density is kept above the critical number, holding the benefit and keeping the population off the spiral that the count would not warn of.)

Targets

• [ ] the density is held above the critical line before the crossing
• [ ] the range is protected, the harvest restrained, the throng supplemented
• [ ] an ample headcount is treated as suspect, the danger unseen until past the line
• [ ] the spiral is never entered, at the continuous cost of spending on a population that reads ample

Direction

Lower the density by the losses any population takes, carry the numbers toward the critical line, and past it turn the per-capita growth negative. Give no warning, because the attrition is ordinary and the line is invisible from above.

Orders

(The Threshold thins the ranks by attrition no member authored, the harvest, the drought, the road, the hard winter, carrying the density toward and across a line that carries no marker.)

Implementation

(The Range into the edge; the thinning process; the attrition lowers the density toward the critical number, while the count still reads survivable and the line beneath it gives no sign.)

Targets

• [ ] the density is lowered by ordinary attrition no member authored
• [ ] the numbers are carried toward the critical line unremarked
• [ ] past the line the per-capita growth turns negative
• [ ] the crossing is marked by nothing the headcount shows, the count reading survivable to the last