mathematician with interests in biological interactions and fisheries, developed a
simple, differential-equation model of the interaction between a prey population and a
predator population. The classic example of such an interaction is the snowshoe hare–
Canadian lynx oscillation (MacLulich 1937; Stenseth et al. 1997). The model is
usually called the Lotka–Volterra predator–prey model, and it provides a simple
example of the steps in a process model. Let the size of the prey population be N1,
that of the predator population N2. The following two equations might describe the
interdependent changes in these two quantities:
(^)
(^) The * denotes multiplication (as in computer languages), b is the net [birth − death]
rate of the prey in the absence of predation, d is the death rate of the predator, and K1
and K2 are constants. In modeling terminology, N1 and N2 are “state variables” of the
system and b, K1, K2, and d are system “parameters”.
(^) Some assumptions behind this model are realistic, some are not:
(^1) In the absence of predation the prey species would increase exponentially
at rate b (an unlimited habitat apart from predation is unrealistic).
2 The rate at which prey are eaten is proportional to the product of the
densities of prey and predator, an “encounter” model (moderately realistic).
3 Time spent by predators consuming prey and converting them to new
predators is negligible (no time lags, not realistic).
4 And still other assumptions.
(^) Despite the unrealistic aspects, these equations are instructive in many ways and
have been much studied. They are typical of differential equation systems describing
ecological interactions in that they have no explicit solutions. That is, you cannot
rearrange them (separate the variables), integrate and find functional representations
for the time courses of N1 and N2 (which would be called a “solution” to such a
system). So, we turn to other approaches to study them further.
(^) Most often the interactions are approximated by finite-difference equations, and
then some finite time increment is repeatedly applied to make sequential calculations
for N1 and N2. The resulting “implicit” solution is a time-series of the variables, and it
can be studied for its characteristics. This is called Euler’s method, among other
names. For the Lotka–Volterra equations the finite-difference version is:
(^)