Mathematical background

Predation equations may be defined as the predation rate per predator per unit time, \(f\left(N,P\right)\), or as the total predation rate per unit time, \(g\left(N,P\right)=P*f\left(N,P\right)\) whereN and P are the prey and predator densities, respectively. The following sections present a proposed set of standard criteria for predation equations, followed by a proposed equation that satisfies them. In the following sections, \(\alpha\) denotes the maximum feeding rate for the predator, or the amount of prey per unit time that an individual predator can consume if no time is wasted searching for prey.
As comparison, commonly used equations from the literature were evaluated according to the same criteria, including prey-dependent, ratio-dependent, and the intermediate Hassel-Varley equation (Hassell & Varley 1969), each combined with Type I, II and III Holling functional response forms (Holling 1959; Abrams & Ginzburg 2000). A final equation, the Beddington-DeAngelis equation (which does not lend itself to combination with Holling functional response forms) was also tested (Beddington 1975; DeAngelis et al. 1975). See Table 2 for a full definition of each equation.

Behaviour at high prey density

As prey density increases, predation per predator should asymptotically approach the maximum feeding rate for the predator. In mathematical terms, \(\operatorname{}{f\left(N,P\right)}=\alpha\), and, by extension, \(\operatorname{}{g\left(N,P\right)}=\alpha P\).

Behaviour at low prey density

As prey density approaches 0, both total as well as per predator predation should tend towards 0, that is,\(\operatorname{}{f\left(N,P\right)}=\operatorname{}{g\left(N,P\right)}=0\).

Behaviour at high predator density

As predator density increases, overall predation should increase asymptotically towards the total prey population available (or perhaps slightly less, if it is assumed that some habitat niches offer complete predation protection to a limited number of prey), while prey availability per predator decreases as predators begin competing for prey. In mathematical terms,\(\operatorname{}{g\left(N,P\right)}\leq N\), though\(\operatorname{}{f\left(N,P\right)}=0\).

Behaviour at low predator density

On the other hand, as predator density decreases, the total predation should also approach 0: \(\operatorname{}{g\left(N,P\right)}=0\).

Other considerations

Finally, many equations consider the functional response of predation rates to a measure of prey abundance (often prey density or else the prey-predator ratio), based on the assumption that predation rates are not a constant function of prey abundance and will instead likely change nonlinearly along with prey abundance (for instance, due to predator saturation, or else to the existence of niches where a subpopulation of prey may be partially shielded from predation). Two of the three types (Type II and Type III) of functional response proposed by Holling (Holling 1959), and widely used since, were designed to model such effects.
However, although one may reasonably expect that additional prey added to a system may be more vulnerable to predation than already present prey (due, for instance, to the saturation of protected niches), the addition of new prey should in no circumstance increase the instantaneous predation risk for prey that was already present. In other words, adding 1 unit of prey cannot increase total predation by more than 1 unit of prey, or, mathematically,\(0\leq\frac{\text{dg}}{\text{dN}}\leq 1\).

The Kovai equation

As can be later seen in Table 2, most equations currently used for modelling predation fail in at least one of the above-mentioned criteria. We here describe a new equation, which we propose to call by the name of the city where research leading to its development was conducted.
\begin{equation} f\left(N,P\right)=\alpha\left(1-e^{-\frac{\text{Nu}}{\left(\text{αP}\right)}}\right)\nonumber \\ \end{equation}\begin{equation} u=1-\frac{b}{N}\left(1-e^{-\frac{N}{b}}\right)\nonumber \\ \end{equation}
The first part of the Kovai equation takes the ratio between predator and prey into account, partitioning available prey amongst the predators present. In this sense, the Kovai equation is similar to a ratio-dependent equation, but instead of using the raw prey density to calculate the prey-predator ratio, a measure of effective prey availability, u , is used that takes the scarcity of the prey (and prey refuges) into account and ranges between 0 and 1. This performs a similar purpose as the Holling Type III functional response, but retains the notable advantage that\(\frac{d}{\text{dN}}uN=1-e^{\frac{-N}{b}}\), which is in the range\(\left[\left.\ 0,\ 1\right)\right.\ \) for non-negativeb , thereby ensuring that the effective prey availability will never increase faster than the prey density itself does. b may be interpreted as the maximum capacity of prey refuges (\(\operatorname{}\left(N-Nu\right)=\lim_{N\rightarrow\infty}\left[N-N+b\left(1-e^{-\frac{N}{b}}\right)\right]=b\)), or, alternatively, as the prey density at which\(e^{-1}\approx 36.8\%\) of the prey is accessible to the predator.
At low values of b , the equation presents a functional response form similar to the Holling Type II form, allowing this parameter to control the functional response form of the equation. In addition, the fact that u is dependent on N and not on the ratio N/P enables the Kovai equation to account for prey scarcity in a similar manner to prey-dependent equations.