## The Lotka Volterra Model

Why did a complete closure of fishery during World War I cause an increase in predatory fish and a decrease in prey fish in the Adriatic Sea? This was the question that led Vito Volterra to formulate a mathematical conceptualization of prey-predator population dynamics. In his endeavor to explain mechanisms by which predators regulate their prey, he constructed a mathematical model that describes temporal changes in prey and predator abundances. The model makes several simplifying assumptions such as (1) the populations are large enough so it makes sense to treat their abundances as continuous rather than discrete variables; (2) the populations are well mixed in the environment (which is the reason why this type ofmodel is sometimes called mass action model in an analogy with chemical kinetics); (3) the populations are closed in the sense that there is no immigration or emigration; (4) the population dynamics are completely deterministic, that is, no random events are considered;

(5) in absence of predators, prey grow exponentially;

(6) the per predator rate of prey consumption is a linear function of prey abundance; (7) predators are specialists and without the prey their population will decline exponentially; (8) the rate with which consumed prey are converted to new predators is a linear function of prey abundance; (9) both populations are unstructured (e.g., by sex, age, size, etc.); and (10) reproduction immediately follows feeding etc.

If R(t) and C(t) are the prey and predator abundance, respectively, then under the above assumptions the population dynamics are described by two differential equations dR , , x

dc dt

where r is the per capita prey growth rate, A is the rate of search and capture (hereafter search rate) of a single predator for an individual prey item so that AR is the per predator rate of prey consumption (i.e., the functional response), e is the rate with which consumed prey are converted into predator births, and m is the per capita predator mortality rate. Model , which was independently formulated by Alfred Lotka, is today known as the Lotka-Volterra prey-predator model. For initial population abundances R(0) and C(0), this model predicts future abundance of prey R(t) and predators C(t) (Figure 1a).

From the ecological point of view, the important information such a model can provide is whether or not population abundances tend to an equilibrium at which both species will coexist. At the equilibrium, predator and prey abundances do not change (i.e., dR/dt = dC/dt = 0), which gives m , „ r R* = — and C* = -e A A

This equilibrium (shown as the solid dot in Figure 1b) is at the intersection of the prey and predator isoclines; they are the lines in the phase space along which dR/dt= 0

and dC/dt = 0 (shown as dashed lines in Figure 1b). Interestingly, the prey equilibrium depends only on parameters that describe population growth of predators whereas the predator equilibrium depends on the prey per capita growth rate r. Thus, increasing the prey growth rate r (which is sometimes called enrichment in the ecological literature) does not change the prey equilibrium density, but it increases the predator equilibrium abundance.

Knowing the interior equilibrium does not tell us whether this equilibrium is stable with respect to perturbations in population abundances or not. In other words, we want to know if after some (random) perturbation from the equilibrium, population abundances will return to this equilibrium or not. For the Lotka-Volterra model  this question is easy to solve because the model is an example of a conservative system with the first integral

V(R, C) = m(R/R* -1 -lnR/R*) + r(C/C* - 1 -ln C/C*) 

which is constant along the trajectories of the model (here ln denotes the natural logarithm). Indeed, the time derivative of V along a trajectory of model 