Quick addendum: Lotka-Volterra equations

In my previous post about the simulation of a simple virtual prey-predator  ecosystem on the world of Wa-Tor, I have mentioned Lotka-Volterra equations as another simple model for such relations. I think it would be interesting to observe how this model performs - is there something curious these equations can show us?

Just to remind - Lotka-Volterra equations describe a relationship between the sizes of two populations: prey (denoted as x) and predators (denoted as y). The equations are:

ẋ  = αx - βxy 
ẏ  = δxy - γy

The rate of change of the prey population (ẋ) depends on the natural rate of reproduction (αx) and the rate of killing by predators (-βxy). The latter is modelled based on the chance that the animals of two species encounter each other - the more of the either, the higher the chance! The predators benefit from those encounters, and thus gain numbers at the rate described by the term δxy. The predators, being at the top of the food chain can only die to accidents, sickness and the old age. This is modelled by the term proportional to the current predator population (-γy).

In this interactive chart below you can check the various outcomes for such an ecosystem by varying the parameters of the equation (α, β, γ, δ, and the initial populations x0 and y0):

L-V parameters	Initial conditions	sim. parameters
α:	x0:	tend:
β:	y0:	Δt:
γ:
δ:

The Lotka-Volterra model, besides having quite a bad-ass name, is quite powerful. Still, it suffers from the problems caused by its simplicity. One of them is known as the atto-fox problem. Notice how the population numbers do not actually describe the size of population in the units of individuals, but rather in terms of relative sizes. Yet, it often happens that due to a choice of parameter, one of the population sizes (say that of the predators) dwindles to obscenely low values - there would be less than one animal left! As an example, only 10⁻¹⁸ of a fox may have been left alive at some point in time. Obviously, any real population cannot bounce back from such a disaster. Thus the model reaches its boundaries.