A comparative study of models for predation and parasitism

course an instantaneous hunting function which has been referred to in w 3 as f(x) Y.
The expression f(x, y) is a general one, and f(x) Y more specific. Whichever expression
is convenient will be used in this paper.
The above form of presentation was in fact used by GAUSE (1934) in his explana-
tion of VOLTERRA'S theory, though GAUSE did not explicitly explain why the linear
term for x in the second equation was excluded.
The following two points were raised by GAUSE (1934). First, the assumption
of a geometric increase in the prey population in the absence of predators is not
correct, since the population growth in any animal species, in the absence of natural
enemies, would normally follow the logistic law, and so a population would not grow
indefinitely. That is to say, function g~(x) in eq.
23
(4a. 2a) would not be a linear
function of x but should be an instantaneous form of the logistic law, and this sugges-
tion sounds reasonable. GAUSE'S second point is that in the predator population the
rate of increase per predator at different densities of prey would not be a linear
function of the prey density either ; i.e. (dy/dt)/y is not a linear function of x.
This conclusion of GAUSE'S was based on an observation by SMIRNOV and WLADIMI-
gOW (see GAUSE 1934, p. 139), which showed that the rate of increase of a parasite
population, Morrnoniella vitripennis, in relation to the density of its host, Phorrnia
groenlaudica, was not linear, and an exponential function of x was more appropriate
for gs(x, y). This suggestion, however, is not immediately acceptable for reasons
discussed in detail in w 4d.
My last point is concerned with the interpretation of t. If g~ is a non-zero positive
function for all x's~0, (x of course is never negative), it means that progeny are
produced in the prey population, and at the same time these progeny are susceptible
to predation during t. Similarly, if g2 takes at times a positive value, it means that
the predator population must also produce their progeny which attack the prey during
t. Hence, there is no clear distinction between generations ; generations are continuous
as in protozoa.
and (4a. lb)
Under these circumstances, the solution of simultaneous eqs. (4a. la)
generates the predator-prey oscillations that were actually shown by
both LOTKA and VOLTERRA.
However, generations can be discrete, as in many insect species, in which case
the progeny of prey produced during the time vulnerable to predation in the present
generation may be attacked only in the following generation. Also, the progeny of
predators produced in the present generation may not attack the prey in this generation.
The populations in the present generation are then only subject to decrease during t
(within the generation), in which case both functions g~ and g2 will never become
positive. Under these circumstances, a solution of the two simultaneous equations
gives changes in numbers in both populations of the present generation, only during
the period of predation within the generation (see w 4b). Hence, in this case, separate
equations are required to compute the number of progeny to be produced to form
the next generation by the survivors of each population in the previous generation (s).