A comparative study of models for predation and parasitism

17
of social interactions among predators and the effect of hunger. One way to incorpo-
rate these factors and~ their influence on the form of an overall hunting equation will
be shown in the following paragraphs.
Social interactions among predators may be classified into two major categories,
social interference and facilitation. These cause a reduction or increase, respectively,
in the instantaneous hunting efficiency of each predator as compared to what it would
potentially exhibit if these factors were not operating (for a detailed discussion, see
w 4i). Let S be the factor by which f(x) is reduced or increased. Then an effective
instantaneous hunting function will be Sf(x). Also the effect of social interaction
must vary as the densities of both predators and prey vary. For instance, too many
predators hunting too few prey would experience more intense interference, than other-
wise, among the predators. Therefore, S must at least be a function of both Y and x,
which will be written as S (Y, x). Incorporating the factor S (Y, x) into eq. (3.4),
we have
and so
dx/dt= -S(Y, x)f(x) Y (3. 14),
Yt = _ ~,jx dx (3.15).
xo S (Y, x)f(x)
However, the intensity of social interaction might change with time, in which case, S
may also be a function of t. The complex function Sf in eq. (3.14) is a generalized
instantaneous hunting function and can be written as f(x, Y), and for further gener-
alization as above it may be written as f(x, Y, t). But I shall avoid such complica-
tions at the moment.
The integral on the right-hand side of eq. (3.15) generally involves Y but not
t. This suggests that if z(=xo-x) is evaluated in eq. (3. 15), it would be a function
of xo, Y, and t, rather than one of x0 and Yr. Here, Y and t no longer form a single
complex variable. Thus the overall hunting equation becomes
z =F(x0, Y, t) (3.16).
Equation (3. 16) has three independent variables, and so it can be presented only
in a four-dimensional coordinate system, or more practically in a series of three-dimen-
sional coordinate systems;if, for instance, z, x0, and Y formed the three axes of a
graph, separate graphs would be needed for each l. This means that if any social
interaction is involved, different results should be expected between observations with
different values of t.
The effect of hunger can be incorporated in much the same way as is that of
social interactions. Suppose f(x) is an instantaneous hunting function of an individual
predator when it can potentially exert its maximum output in hunting. If the predator
is partially satiated, the maximum performance will only be partially realized.
partial realization will be expressed by a factor H, which is an index of the hunger
level and is naturally defined between 0 and 1 ; H may also be less than unity when
the animal has been so starved that it cannot exert its full potential effort. Under
This