Mathematical modeling of population dynamics in predator-prey systems began with the pioneering efforts of A. J. Lotka (1925) and V. Volterra (1926). Their equations described the changes with time of both prey and predator population, as a function of a minimum number of parameters. A simple way to deduce equilibrium conditions is the following.
In the absence of the predator, the prey will exponentially grow according to dty/dt — ri Ni 
where N1 is the size of the prey population and r1 is its intrinsic rate of growth.
The rate at which the prey will be consumed by the predator will depend on the frequency of their encounters (N1N2, where N2 is the size of the predator population), multiplied by the predator efficiency p, a measure of the percentage of encounters actually ending with the capture and consumption of the prey by the predator. Thus, in the presence of a predator, the growth rate of prey population will turn into dN1 / dt = r1N1 - pN1N2 
As for the predator, its population is expected to vary according to the efficiency a at which a consumed prey is converted into the production of a predator, less the natural decrease of predator population due to its intrinsic mortality rate d2. Thus, dN2 /dt = apN1 N2 - d2 N2 
In this simple model, parameters such as the mortality of prey population due to factors other than predation are ignored. More complex models incorporate additional parameters to obtain a closer approximation to real systems. Equations (2) and (3) are nevertheless sufficient to derive a first-approximation set of equilibrium conditions:
dN1/dt = 0 when r1N1 = pN1N2, i.e., N2 = r1 /p  dN2/dt = 0 when apN1N2 — d2N2, i.e., N1 = d2/ap 
Mathematical modeling generates expectation of cyclic fluctuations in population size of both prey and predator, a fall in the number of preys determining a subsequent fall in the number of predators; this, in turn, will allow the prey population to recover, thus providing predators with a new opportunity to grow in number; and so on. In fact, cyclic fluctuations have been repeatedly documented, from field data as well as from laboratory experiments. Actual population fluctuations may depart more or less conspicuously from perfect regularity. Regularity will be higher if the predator is not very efficient in its action, if the habitat is heterogeneous and offers good shelters to the prey, and if alternative preys are available to the predator, when the preferred kind is at very low density.
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