3.1. A Simple Predator–Prey Model 87
from the interaction, then PQ is large. If one of P and Q is small, and the
other is large, then (at least for some values of P and Q) the product PQ
will be midsize. Most importantly, if either P or Q is increased, so that we
would expect the interaction to be greater, then PQ increases. The product
PQ, then, behaves roughly as we would want to give a good description of
the amount of interaction we might expect between the populations.
We model the two populations by:
P = rP(1 − P/K ) − sPQ,
Q =−uQ + v PQ.
Here, s and v both denote positive constants. The term “−sPQ” describes a
detrimental effect of the predator–prey interaction on the prey, and the term
“v PQ” describes a beneficial effect of the interaction on the predator. There
is no reason to expect that the values of s and v need to be of the same size,
since the predator may well benefit more than the prey is harmed, or the prey
may be harmed more than the predator benefits.
The use of a term such as kPQ to model population interactions is some-
times called a law of mass action. One way to justify it is to imagine individu-
als in two populations of size P and Q moving around at random and mixing
homogeneously. Then, over a certain time interval, we might expect the num-
ber of chance meetings between individuals in the different populations to be
PQ. A fraction of these meetings will be significant enough to result in kPQ
predation interactions during a time step. Note that a mass action term in a
model means the equations are nonlinear; even though this is a very simple
interaction term, we should perhaps expect complicated dynamics.
Rewriting our simple predator–prey model in terms of populations, rather
than changes in populations, gives:
P
t+1
= P
t
(1 + r (1 − P
t
/K )) − sP
t
Q
t
,
Q
t+1
= (1 − u)Q
t
+ v P
t
Q
t
.
with r , s, u, v, and K all positive constants, and u < 1.
With any model, getting accurate ideas of parameter values from data
collected on experimental study populations can be quite difficult. The ap-
propriate values of the constants appearing in the species–interaction terms
are perhaps even harder to have an intuitive feel for than the r and K of the
logistic model. For the time being, it’s enough to know that, the larger s and
v are, the stronger the effect of the predator–prey interaction.