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.