A comparative study of models for predation and parasitism

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the attack period is now possible with the aid of the generalized LOTKA-VOLTERRA
eqs. (4a. 2a) and (4a. 2b).
The final point of investigation in this section will be concerned with the real
meaning of the 'area of discovery' in the NICHOLSON-BAILEY terminology, i.e. at, or
2RVt in my notation.
In the dynamics of gas molecules, the movement of a molecule can be regarded
as ideally haphazard and independent of other molecules before it collides with another.
Then the number of collisions that will occur during time interval At will be
2RVxAt in which R is the effective radius, V the average speed of each molecule,
and x the population density of the molecules. Clearly, the NICHOLSON-BAILEY model,
as well as the second term in LOTKA-VOLTERRA eq. (4a. la), is an analogy to the
'law of mass-action' in physical chemistry. (This was perhaps the reason by which
VOLTERRA justified his assumption of the linear relationship between the frequency
of encounters and the densities of prey and predator species.) The competition equation
as expressed in eq. (3.8) is in fact identical to what is called in chemistry
the 'velocity equation for a unimolecular reaction'.
It is not difficult to visualize what the effective radius of a gas molecule is,
since it is the radius in which an effective contact with another molecule is made so
that a reaction takes place. However, what is the effective radius for a predator ?
NICHOLSON and BAILEY assumed that this was the radius within which a predator
could recognize the prey. However, for the competition equation to be an exact analogy
to the velocity equation, as the form of the NICHOLSON-BAILEY equation implies, the
path of a predator (a molecule) has to be completely independent of the position of
the prey (other molecules) immediately before the collision. In other words, the
predator's recognition of a prey has to be made, strictly speaking, by bodily contact.
If, however, recognition was made well away from the prey, the predator would have
to approach it, and this immediately means a digression from a free path. The problem
now is, how much deviation from the competition equation would be expected by
the digression from a free path. This degression can be serious under certain circumstances
as will be discussed in w 4e.
Also, if it is assumed that recognition occurs at some distance from the prey,
the predator may see more than one prey individual at a time. Unless the predator
can catch the prey in a sweeping action, each prey must be handled individually.
Under these circumstances, recognition will not result in immediate capture. In other
words, the number of prey that a predator can capture would not increase as fast as
the number of recognitions increases with increasing density of prey population.
This problem will be discussed in ~ 4d.
The last point, which is more important than any other, is the effect of the time
spent catching, killing, digesting, etc., for each victim. In the analogy of unimolecular
reaction, there is no need to think about the time involved in actions taking place
after a collision is made since each molecule ceases to capture more. In the predation