## Prey Predator System

Investigation of dynamics of one population can be considered as a forerunner and not yet a part of systems ecology in the narrow sense. The first example of a mathematical model of a system of populations in ecology is the famous Lotka-Volterra model. Lotka in 1924 formulated it to analyze host-parasite system, while Volterra in 1926 independently proposed it to analyze a prey-predator system. The corresponding system of differential equations is:

where N is the number of prey organisms, rc is the familiar instantaneous biotic potential of the prey species, a measures the specific efficiency with which the predator captures prey, b is the efficiency of conversion of prey into predator numbers, and m is the specific mortality rate of the predator. The solution to the system of eqn [4] with appropriate initial conditions is a pair of periodic functions in which the peak of the prey precedes in time the peak of the predator population.

Volterra used it to find out how indiscriminate fishing by fishermen affects the prey and the predator population. Let the fishermen have an equal fishing effort e on the prey and the predator population. The fishing rate of prey is assumed proportional to prey population N, while the fishing rate of predator is assumed proportional to the predator population, P. The equations with fishing become:

There are two equilibrium solutions, that is, solutions when dN/dt = dP/dt = 0. We denote these by (N*, P*). The first equilibrium (or steady state) is the total extinction, that is, the extinction of prey and predator: N* = 0, P* = 0. The second equilibrium is the persistence of both prey and predator:

Assume that rc > e, otherwise the predator population tends to extinction and the model reduces to the exponential growth of prey population [2].

It is clear that if the fishing effort e decreases, the equilibrium concentration of prey population will decrease and the equilibrium concentration of predator population will increase. This counterintuitive conclusion is known as the Volterra principle. The principle explained why when fishing in the Adriatic Sea ceased during World War I, the concentration of prey decreased and the concentration of predators (such as sharks) increased.

The Volterra principle is perhaps the first systems ecology principle. Volterra stated the principle about 40 years prior to the formulation of systems ecology as a discipline. The discipline was formed in the 1960s when theoretical research on ecosystems started to be explored by the help of computers, first analog and soon after digital. The first papers on theoretical ecosystem models were published by Odum in 1960, Olson in 1961, Garfinkel in 1961, Van Dyne in 1966, and Patten and Witkamp in 1967. The book by Watt in 1968 has also been very influential. The series of four volumes edited by Patten in 1971, 1972, 1973, and 1976, and the book by May in 1973 finally established the field of theoretical systems ecology.

Today the Lotka-Volterra model is mostly of pedagogic importance and it is used as such in courses of mathematical ecology. Namely, in the above system, in the absence of predator population, the prey population grows according to the first law of population growth which is characteristic of an infinite and not a finite environment. In the finite environment this term must be replaced by the second law of population growth. Furthermore, capturing rate of prey is proportional to prey population (concentration or density, N) and predator population P. This term is called a bilinear collision, and it works fine only in the case that the predator population is always equally hungry, that is, when the prey population is in very short supply. Since this is not always the case, the term must be replaced by a saturation of feeding rate of predators. The simplest and the most often used term is the Michaelis-Menten interaction: vNP/(h + N). Here v is the maximum specific feeding rate of predators and h is the prey concentration at which the specific feeding rate of predator is v/2. When in this term N is very small, as it has been assumed in the binary collision interaction, a is approximately equal to v/h. With these changes, the above Lotka-Volterra prey-predator model becomes Rosenzweig-McArthur model (1963):

dN/dt = rcN(1 - N/K) - vNP/(h + N) dP/dt = bvNP/ (h + N) - mP

Several properties of the model have been investigated. First, for this system too, the Volterra principle holds. Second, when the flow of food to the prey increases and hence the carrying capacity, K, rises, the stable equilibrium point may turn into a stable limit cycle. Rosenzweig in 1971 termed this phenomenon the paradox of enrichment.

Limit cycle may also occur in biological control of a pest population by a predator population: if predator is very efficient at catching the prey and if it is able to drive prey population to a low number in comparison to K. In other words, large variations of prey and predator populations are possible and persistent.

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