Results
Theoretical study
As indicated in Table 2, none of the equations currently available in
the literature fulfills these criteria, which casts doubt on their
ability to generate accurate predictions for systems where predator or
prey concentrations deviate significantly from static equilibrium
conditions. The new proposed equation, however, did yield satisfactory
results with regards to all conditions.
[Insert Table 2 here]
Only expectations with regards to equation behaviour under conditions
where prey populations approach 0 were uniformly respected, meaning that
all equations evaluated did accurately predict that per predator and
total predation rates fall to 0 when prey populations disappear.
However, each of the other conditions was violated by one or more
equation.
As could be expected, prey-dependent equations performed poorly in
conditions of high predator densities, while ratio-dependent and
Hassell-Varley equations fared better in these conditions. However, Type
I predation responses caused problematic behaviour under conditions of
high prey densities in all equation types.
Particularly problematic, however, were cases where total predation
rates under very high predator densities either exceeded the total prey
population outright (e.g., \(\operatorname{}\left(g\right)=\infty\))
or else could exceed it under certain parameterisations of the equations
(e.g., \(\operatorname{}\left(g\right)=\frac{\text{αN}}{b}\)). In
fact, only the new proposed Kovai equation showed consistent behaviour
with regards to total predation rates under high predator densities.
Additionally, none of the equations from the literature correctly
predicted the rate of total predation increase upon the addition of an
additional prey. In many circumstances, increasing the prey density by 1
could lead to a much larger increase in total predation rates,
suggesting that the addition of one prey individual to a system could
greatly increase the predation of the other prey individuals
already in the system. Such behaviour seems unrealistic in the real
world and could, in modelling studies, lead to the prediction of higher
predation rates than would be expected. Only the Kovai equation showed
consistent behaviour in this regard.
Empirical study
As can be seen in Table 1, the datasets included in the empirical study
span a wide range of predator-prey relationships as well as equation
types and functional responses (though type II and III responses were
more common). Figure 1 shows the results of different models applied to
predation data from (Kratina et al. 2009), including three
equations recommended by those authors as well as the Kovai equation.
At low values of P, where the prey-predator ratio is high enough to
avoid predator competition, the ratio-dependent model was unable to
model the impact of absolute prey scarcity on predation rates and
therefore confounded the effects of various prey densities (Figure 1,
a). On the other hand, intermediate form equations (Hassel-Varley Type
III, Beddington-DeAngelis, and Kovai; see Figure 1 b, c and d,
respectively), were successful in distinguishing the effects of prey and
predator densities. (Prey-dependent models were overall unsuccessful in
modelling the data, as these models cannot distinguish between different
values of P at all, and are not shown in the figure.)