Consider the nonlinear predator-prey system

where u=u(t) and v=v(t) designate the population density of prey and predator, respectively, at time t. The parameters , and are proportionality constants for birth, mortality, and predation rates. Note that the prey only give birth onto empty space, while the predator gives birth only by taking over space occupied by prey.

To begin suppose that there are only prey of type i. Setting we find that if this species obtains an equilibrium of

Now consider two species u₁ and u₂. The equations for equilibrium are

It is natural-and often helpful-to think of these equations as lines in the phase space. For a concrete example consider , , the first equation is given by the blue line, the second, red.

Since the lines are parallel, an equilibrium (point of intersection) occurs only when the death to birth ratios are the same. Therefore, without a predator, no matter how many prey species we start out with, all but the one with the largest birth to death ratio will go extinct.

It is known that adding a predator can cause the coexistence of two prey species (an effect referred to as Predator-Mediated Coexistence). This is achieved when the predator's feeding rate on the stronger species keeps the latter in check without totally wiping it out. This is where our research began, the question being: How many species can coexist in this way?

On a computer we ran a number of simulations where prey species with random parameters were successively dropped into an environment that contained a predator. What we hoped to see was a gradual build up in the number of prey coexisting with the predator. Instead, as the following chart indicates, we were never able to get more that two prey species at one time.

To explain the chart: the first species, denoted by the black cross, coexisted with the predator until the species denoted by the blue asterisk was introduced. This species was eventually displaced by the species represented by a green circle, etc. As you can see the dominant species in the long run is that which has the largest birth to predation ratio.

The next graph shows an example of how three prey species and a predator typically evolved. The predator, here and throughout the text, is represented by a red broken line.

In an attempt to increase the number of prey coexisting, we added a second predator to the model.

In this way we increased the prey to 3 (in the above picture all species settle down on a static equilibrium in the long run), but were now unable to get 4 prey to coexist. This led to the conjecture that r predators could mediate at most r+1 prey, which, a short time later, we found had been proved by Levin (1970):

Theorem 1. If the differential equations for the prey densities have the form

where is open space, then r+2 prey species can not coexist.

In search of the coexistence of more prey, we adopted the following model

The prey now compete only amongst themselves for space. A more striking change occurs in the predation term. In the ecological literature this is known as type II feeding, where the amount of prey consumed by 1 predator has a finite limit as the density of the prey tends to infinity. When g _I=0 the system reduces to one similar to that of the first model and, as might be expected, 2 prey is again the limit. When all parameters are varied over a wide range of values, our experiments could do no better than 1 predator - 2 prey.

Our next model sought a periodic environment to generate more coexistence.

The idea behind the periodic function was to allow for the incorporation of definite mating seasons which could vary from species to species. With regards to Levin's theorem space is still a single limiting resource, but now the prey are exploiting it at different times, making it effectively several resources. Upon further investigation, however, we realized that this model did not require a predator for the coexistence of many prey. The next figure shows five prey species coexisting without a predator.

Having tried type 2 feeding and failed we generalized the nonlinear term

In words, when the density of drops below the critical density c the predator virtually ignores .

Concerned that the predator might not have enough food to sustain it we included u₁

Species 1 became known as "macaroni and cheese" because of its role as the prey's basic food source. The following table displays the results of repeating our very first experiment with this latest model.

It appears that we will be able to add as many prey as we want.

A later experiment showed that, by lowering the birth rate and increasing the predation rate of species 1, macaroni and cheese could be phased out and shown to be unnecessary to the coexistence of many species.

Note that removing species 1, the black cross, did not result in a collapse of our system.

Our final model takes the form

where ,.

serves in a similar capacity as it did in the last model, reducing the rate at which the predator feasts on as the density of shrinks (this is known in the ecological literature as the Allee effect).

The final term in the ODE for the prey is a sort of "social friction," its ultimate effect being to prevent over crowding. Because of this intraspecies competition more than 1 prey species can exist without mediation by a predator, but unlike the model with a periodic environment, the number of prey which can coexist without a predator appears to be limited. Why this term allows for two prey species to coexist in the absence of a predator can be seen in the phase diagram

The solid lines represent the nullclines, as they did earlier, for two prey with no social friction term. Including the social friction term results in the curved dotted lines, which have a good chance of intersecting if the birth to death ratios for the two species are close and/or q is chosen small (greater curvature). A similar analysis with three prey is much more difficult, but we did run simulations on a computer to get some idea of what we should expect. In the following graph distinct species are introduced at times t=0, 400, 800, 1200 and 1600 with an initial density of 0.1. The exponents are set at p=1/10, q=6.

It appears from the graph that species 1 (yellow) and species 2 (cyan) would coexist, and indeed that is the case. However, you might also reasonably infer from the graph that the first 3 species may have reached a steady equilibrium had they been given more time, but that is not the case. Curious of this result, we ran an additional 2,000 trials with the above parameters,never getting more than 2 species to exist at once.

In this last graph the same prey as above are introduced in the same sequence but this time a predator is present from the outset. As you can see, all the prey survive. To emphasize the importance of the predator in this coexistence, at t=2700 the predator is eradicated, causing the system to collapse to one prey species.