A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase.  Every being, which during its natural lifetime produces several eggs or seeds, must suffer destruction during some period of its life, and during some season or occasional year; otherwise, on the principle of geometrical increase, its numbers would quickly become so inordinately great that no country could support the product.  Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species, or with the individuals of distinct species, or with the physical conditions of life.  It is the doctrine of Malthus applied with manifold force to the whole animal and vegetable kingdoms; for in this case there can be no artificial increase of food, and no prudential restraint from marriage.  Although some species may be now increasing, more or less rapidly, in numbers, all cannot do so, for the world would not hold them....

The amount of food for each species of course gives the extreme limit to which each can increase; but very frequently it is not the obtaining food, but the serving as prey to other animals, which determines the average numbers of a species.¹

Vito Volterra and Alfred Lotka translated Darwin's concepts for predatory prey interactions into a mathematical model.  The model is actually a first order system, called the predator-prey (or Lotka-Volterra) system:

This model is the one of the simplest of predator and prey interaction models.  x represents the predator population and y represents the prey population; these variables are defined only in the first quadrant, called the population quadrant.  The parameters, called rate constants, a, b, c and d are all positive.  Without the cross terms (x y), we can see that the predator population decays exponentially, x' = - a x, and that the prey population grows exponentially, y' = c y.

The cross terms, b x y and d x y, represent the interaction of the two species.  Notice that, clearly, the predator population is affected positively and that the prey population is affected negatively by interaction.  In other words, food promotes the predator's growth rate while serving as food diminishes the prey's growth rate.

The predator-prey system is a result of the Balance Law:

Supposing that migration into and out of the community is negligible, the rate of change is simply the difference between the birth and death rates.  These rates must be proportional to the population size.  In other words:

where R₁ and R₂ are the coefficients of proportionality and a measure of the contribution of the average individual of a species to the overall growth rate of that species.  If R₁ and R₂ are constant, this setting would represent the growth or decay, depending on the signs of R₁ and R₂, of each of the species.  Various choices of R₁ and R₂ determine different types of models of interactions.  In the case of the predator-prey system, R₁ = - a + b y and R₂ = b - c x.

Listed below, for comparison, are some other interacting species models.

Overcrowding:

Cooperation:

Competition:

¹Charles Darwin, "Struggle for Existence," The Origin of Species, new ed., Chap. 3 (from 6th English ed.) (New York: Appleton, 1882)

References:
Borrelli, Robert and Coleman, Courtney, Differential Equations: A Modelling Perspective, pp. 276-286, first edition, John Wiley & Sons, 1998

Farlow, Hall, McDill, West, Differential Equations and Linear Algebra, pp. 98-104, Prentice Hall, 2002