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I've often heard that a population, human or otherwise, will continue to grow as long as there is food available (assuming nothing else is killing them off). It makes sense: if you have food you can live, and if nothing is hunting you you'll survive to reproduce.

I recently designed a piece of software to simulate an ecosystem, with groups of creatures of different species eating and hunting and reproducing alongside each other. It was very simplified (each animal had simple attack/defense/speed/stealth values, etc), but something became rapidly apparent: in every simulation the predators overwhelmed the prey, reproducing until their numbers could not be sustained by the herbivores, and leading to an inevitable die-off of both groups. I could delay the die-off by adjusting different values and initial population counts, but it would always happen eventually. The predators would eat and breed and eat and breed until the entire system collapsed.

At first I thought it was just the product of my over-simplified system, but it got me thinking: what prevents predators from overpopulating in real life?

It seems like the natural tendency would be for (for example) the sharks to continue breeding and eating until all the fish are gone, or the wolves to eat all the deer, etc. Obviously some predators have predators of their own, but that's just putting off the question: if the hyenas don't overpopulate because the lions eat them, then what's keeping the lions from overpopulating? I can't come up with anything that would prevent the apex predators from growing too numerous, then fighting each other over a dwindling prey population, then dying off entirely when there was no more food to find.

Do predator populations self-regulate to prevent putting undo stress on their prey populations? Or is there some other mechanism to keep the predator hierarchy from becoming top-heavy?

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As a supplement to Remis answer, here's a link to the Nature Knowledge Project page on population dynamics: nature.com/scitable/knowledge/library/… –  jarlemag Mar 5 at 23:08
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aka - they tend to starve to death when there are too many –  shigeta Mar 6 at 20:34

4 Answers 4

up vote 10 down vote accepted

No, I don't think there is any kind of auto-regulation. You'll be interested in various model of prey-predator or of consumer-ressource interactions.

For example the Lotka-Volterra equations describe the population dynamics of two co-existing species where one is the prey and the other is a predator. Let's first define some variables…

  • $x$ : Number of preys
  • $y$ : number of predators
  • $t$ : time
  • $\alpha$, $\beta$, $\delta$ and $\gamma$ are parameters describing how one species influence the population size of the other one.

The Lotka-Voltera equations are:

$$\frac{dx}{dt} = x(\alpha - \beta y)$$ $$\frac{dy}{dt} = -y(\gamma - \delta x)$$

You can show that for some parameters the matrix for these equations have a complex eigenvalue meaning that the long term behavior of this system is cyclic (periodic behavior). If you simulate such systems you'll see that the population sizes of the two species fluctuate like this:

enter image description here

where the blue line represents the predators and the red line represents the preys.

Representing the same data in phase space, meaning with the population size of the two species on axes $x$ and $y$ you get:

enter image description here

where the arrows shows the direction toward which the system moves. If the population size of the predators ($y$) reaches 0 (exctinction), then $\frac{dx}{dt} = x(\alpha - \beta y)\space$ becomes $\frac{dx}{dt} = x\alpha \space$ (which general solution is $x_t = e^{\alpha t}x_0$) and therefore the populations of preys will grow exponentially. If the population size of preys ($x$) reaches 0 (extinction), then $\frac{dy}{dt} = -y(\gamma - \delta x)\space$ becomes $\frac{dy}{dt} = -y\gamma \space$, and therefore the population of predators will decrease exponentially.

Following this model, your question is actually: Why are the parameters $\alpha$, $\beta$, $\delta$ and $\gamma$ not "set" in a way that predators cause the extinction of preys (and therefore their own extinction)? One might equivalently ask the opposite question? Why don't preys evolve in order to escape predators so that the population of predators crushes?

As showed, you don't need a complex model to allow the co-existence of predators and preys. You could describe your model a bit more accurately in another post and ask why in your model the preys always get extinct. But there are tons of possibilities to render you model more realistic such as adding spatial heterogeneities (places to hide for example as suggested by Audrius Meškauskas). One can also other trophic levels, stochastic effects, varying selection pressure through time, age- or sex-specific mortality rate due to predation etc..

I would also like to talk about other things that might be of interest in your model (two of them need you to allow evolutionary processes in your model):

1) lineage selection: predators that eat too much end up disappearing because they caused their preys to get extinct. This hypothesis has nothing to do with some kind of auto-regulation for the good of species. Of course you'd need several species of predators and preys in your model.

2) Life-dinner principle. While the wolf runs for its dinner, the rabbit runs for its life. Therefore, there is higher selection pressure on the rabbits which yield the rabbits to run in average slightly faster than wolves. This evolutionary process protects the rabbits from extinction.

3) You may consider..

  • more than one species of preys or predators
  • environmental heterogeneity
  • partial overlapping of distribution ranges between predators and preys
  • When one species is absent, the model behave just like an exponential model. You might want to make a model of logistic growth for each species by including $K_x$ and $K_y$ the carrying capacity for each species.

    and you might get very different results.

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Another factor that the asker's model might not have considered is predator-against-predator action. Territorial conflicts would become more common as the predator-prey ratio increases. Another possible factor might be reduced hunting ability from malnutrition. –  Paul A. Clayton Mar 6 at 16:20

One of the possible adjustments of these mathematical models is to introduce a "place to hide", making some (small) percent of the prey population not accessible (or much more difficult to access) for predators. After the number of predators decreases from starvation, prey individuals are relatively safer outside the "place to hide" and can grow over this limit before the number of predators increases again.

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An excellent notion! –  Alexander Winn Mar 13 at 5:25

Remi.b's answer is an excellent one, and this should be taken as a supplement to it:

It's possible your simulation is correct

The Lotka-Volterra equations are what is known as a deterministic model, and it describes the behavior of predator-prey systems (in a somewhat simplified fashion) in large populations. Small populations are subject to what is known as stochastic extinction - as the predator and prey curves approach their minimums, they may predict populations less than 1, which in reality are either 0 or 1, and when they're 0...well, someone's gone extinct.

Odds are your simulation is on a small population, and if its a simulation, rather than calculus, you should be seeing those stochastic effects (to be sure - if your simulation keeps track of integer animals, rather than continuous animals, and random chance is involved, this is going to be something you have to worry about).

In a similar model I've been working with, that's a pretty simple adaptation of a L-V model that should, deterministically, result in a stable system like in Remi.b's picture, the predators go extinct 20% of the time, and the prey 80% of the time.

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In history, there are also mass extinctions very well outlined by this video: http://www.youtube.com/watch?v=FlUes_NPa6M. They "reset" the world and allow everything to start over again.

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