## Rr

dt 0R V

which implies that function V is constant along the trajectories of the Lotka-Volterra model [1]. Moreover, V(R, C) > 0 for positive population abundances (because the inequality x — ln x > 1 holds for every x > 0) and function V minimizes at the equilibrium point (R*, C*). Thus, solutions of the equation V(R, C) = const, which are closed curves in the prey-predator phase space (Figure 1b), correspond to solutions of model [1]. This analysis shows that both prey and predator numbers will oscillate periodically around the equilibrium with the amplitude and frequency that depend on the initial prey and predator densities. Moreover, the average values of

Figure 1 Solutions of the Lotka-Volterra model [1] in time domain (a, solid line shows prey abundance, dashed line predator abundance) and in the prey-predator abundance phase space (b). Dashed lines are the isoclines. Parameters: r = 1, X = 1, e = 0.2, m = 1.

prey and predator densities over one period coincide with their equilibrium densities R* and C*. Indeed, the equation for prey can be rewritten as

Integration of this equation over one population cycle of length T time units gives ln(R(T)) - ln(R(0)) = rT - A i C(t)dt

Since T is the period, the left hand side of the above equality is zero (because R(0) = R(T)) and

where C denotes the average predator density. Similarly, the average prey density over each cycle equals the prey equilibrium density.

The above analysis shows that the prey-predator equilibrium is Lyapunov stable (i.e., after a small perturbation the animal abundances stay close to the equilibrium, Figure 1b), but it is not asymptotically stable because the population abundances do not return to the equilibrium. This particular type of equilibrium stability is sometimes called the neutral stability. The eigenvalues of the Lotka-Volterra model evaluated at the equilibrium are purely imaginary (±i^/-m) which implies that the period of prey-predator cycles with a small amplitude is approximately 2-k/\J—rn.

The mechanism that makes prey-predator coexistence possible in this particular model is the time lag between prey and predator abundances, with the predator population lagging behind the prey population (Figure 1a). The Lotka-Volterra model shows that (1) predators can control exponentially growing prey populations (this type of regulation is called the top-down regulation), (2) both prey and predators can coexist indefinitely, (3) the indefinite coexistence does not occur at equilibrium population densities, but along a population cycle. Can this model explain the question about the observed changes in predator and prey fish abundances during World War I? Volterra hypothesized that fishery reduces the prey per capita growth rate r and increases the predator mortality rate m, while the interaction rates e and A do not change. Thus, ceased fishery during World War I should lead to a decrease in average prey fish population R* and to an increase in the average predator fish population C*, exactly as observed.