Comparison of a Deterministic and a Stochastic Model for Interaction between Antagonistic Species

There are two fundamental approaches to prey-predator problems which involve the interaction of two or more species. The first is the deterministic approach in which it is assumed that the interaction between two species is proportional to the product of the numbers of participants in the encounter, and that the numbers vary in a deterministic manner specified by some differential equation. The second approach is a stochastic one in which it is recognized that the interplay of the numbers in the prey-predator engagement is essentially a random process and accordingly, that one may associate probability distributions with these numbers. There are few instances in which one can solve in closed form both the deterministic and the stochastic equations for the same situation for all numbers. The same problem arises in several different guises. Perhaps the earliest work on deterministic models was that of Ross [1911], who treated the theory of the spread of malaria. Subsequently Lanchester [1917] proposed the same types of equations to describe combat. Next Volterra [1926] used roughly the same approach to prey-predator problems in a discussion of the upset of the balance between different species of fish in the Adriatic Sea due to the first world war. In 1926 McKendrick proposed a stochastic model of epidemics, which however he could not solve. This was followed by a series of papers by Kermack and McKendrick [1927, etc.] on the deterministic theory of epidemics.