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.)