Introduction
Correctly representing the relationship between predators and prey is a
key challenge to all attempts at modelling trophic interactions and
population dynamics (Kratina et al. 2009). A wide range of
predation equations is currently available in the ecological literature
to calculate predation rates (prey consumed per predator per unit time)
based on an independent variable (prey density, predator density, or
both) and one or more empirical parameters (Abrams & Ginzburg 2000).
These equations can be, very generally, classified according to the
independent variable and the mathematical form (functional response) of
the equation that links this variable to predation rates.
According to the independent variable, predation equations are often
divided into prey-dependent (where the prey population (N) is the only
independent variable) and ratio-dependent equations (where the ratio
between prey and predator (P) populations, \(NP\), is used as an
independent variable). A wide variety of intermediate forms have also
been proposed. Of these, we here mention some of the most commonly used,
the Hassell-Varley forms (Hassell & Varley 1969) and the
Beddington-DeAngelis equation (Beddington 1975; DeAngelis et al.1975). The former represent a compromise between prey and ratio
dependence, where the \(\frac{N}{P}\) prey-predator ratio is replaced by\(\frac{N}{P^{m}}\), where m is a parameter in the range\(\left[0,1\right]\) that allows the function to
transition smoothly between either prey or predator dependence. The
Beddington-DeAngelis equation, on the other hand, incorporates both N
and P, but not in pure ratio form.
Functional responses, on the other hand, are generally categorised as
Type I (linearly increasing, \(y=\alpha x\)), Type II (logistic-type
growth; \(y=\frac{\text{αx}}{x+b}\)), Type III (sigmoidal type
growth; \(y=\frac{\alpha x^{2}}{x^{2}+b}\)), and Type IV (increasing
to a peak, followed by a decreasing response) functional responses
(Holling 1959), where x is the independent variable (N in the case of
prey dependence and\(\ NP\) in the case of ratio dependence). Type I
rarely occurs (with the notable exception of filter feeders) (Jeschkeet al. 2004), Type II indicates predator saturation, and type III
combines predator saturation at high prey densities with sigmoidal
growth in predation at lower prey densities (which could be caused by
the saturation of limited prey refuges in the habitat, or be indicative
of preferential prey switching or learning by the predator). Type IV
would occur in the case of dangerous prey (Líznarová & Pekár 2013),
where higher prey densities actually diminish the predator’s capacity to
feed.
Some equations (Hassell-Varley, prey-dependent, and ratio-dependent
forms) can be combined with all response types, while others (such as
Beddington-DeAngelis) do not lend themselves to such modifications but
intrinsically generate functional forms similar to one or more of the
Holling forms.
None of these equations, however, is completely satisfactory from a
theoretical standpoint. Ratio dependence is unrealistic at low absolute
prey densities, when the physical scarcity of prey is likely to decrease
predation rates as predators spend most of their time searching, even if
the predator population is itself very low and the prey-predator ratio
correspondingly high (Abrams & Ginzburg 2000), and also produces what
have been described as “unusual” results (namely, infinite prey
availability per predator) as predator density declines towards 0
(Gleeson 1994). On the other hand, prey dependence is unrealistic at low
prey-predator ratios, when predators must necessarily divide available
prey amongst themselves (Abrams & Ginzburg 2000), and studies have
found experimental data partially between ratio and prey dependence
(Schenk et al. 2005).
There is therefore a need for an equation that encompasses both
processes; while alternatives include equations of intermediate form
partially between ratio and prey dependence, their formulations, as will
be discussed later, do not necessarily guarantee mathematically sound
resolution of the different issues faced by each extreme. For instance,
the transition between prey and predator dependence in the
Hassell-Varley forms is dependent on a parameter, not on prey or
predator populations, while the theoretical considerations mentioned
above suggest that population levels are important in determining when
either relationship is more appropriate. Switching equations based on
the nature of predator interference in the system has also been proposed
(Skalski & Gilliam 2001).
There is therefore a need for clear criteria and expected behaviours for
all useful predation equations, which may promote a more standardised
development of predation equations that are widely useful across various
population densities and ratios. Such a list of criteria has been
previously proposed by (Berryman et al. 1995), but most relate to
complete predator-prey models and not to the predation function itself.
Of the criteria that do apply are that predators “must have finite
appetites” and that the potential for intraspecific competition should
be included, criteria which disqualify prey-dependent equations.
However, these criteria apply only to per predator feeding rates and do
not include important conditions regarding total predation rates under
extreme (high or low) prey or predator populations.
Despite the long-standing debate, many of these issues have still not
been satisfactorily resolved (Abrams 2015), and these equations are
widely used in theoretical and practical research, despite their
previously identified shortcomings. Even prey dependence and Type I
functional responses, which have been shown to be in most ecological
systems generally inferior to other (though imperfect) alternatives
(Abrams & Ginzburg 2000; Jeschke et al. 2004) have been recently
used as part of larger ecological and population dynamics models
(Chesson & Kuang 2008; Sanchez et al. 2018; Sanders et
al. 2018; Imbert et al. 2020).
These shortcomings occur because most of these equations are built upon
assumptions (mainly mostly stable prey and predator population
densities) that are unlikely to hold true under natural conditions.
These are structural equation (model) errors, which cannot be
circumvented through parametrisation and calibration. As such, the
predictions of these equations under extreme conditions (such as those
often seen in agriculture or other human-managed systems) may be
invalid. For instance, equations whose assumptions do not hold under
conditions of very high predator density are unlikely to give correct
predictions with regards to augmentative biocontrol efforts, where large
numbers of predators or parasitoids are released into the field. This is
a major impediment to the successful application of predator-prey
equations to ecological, and, especially, agroecological modelling. As
(Kratina et al. 2009) have observed, “choosing appropriate
functional responses is crucial for adequate predictions of food web
dynamics. When predator dependence is incorporated into predator–prey
models their stability is usually enhanced,” highlighting the
importance of correct functional response equations to ecological model
behaviour and reliability.
This paper addresses this gap in the literature by 1) proposing a
standard set of criteria that all useful predation equations should
observe, 2) showing that the majority of equations available in the
literature fail to meet one or several of these conditions, and 3)
proposing a new equation (“Kovai equation”) for modelling predation
rates that does meet these criteria. The paper concludes with 4) a
comparative evaluation of the performance of all reviewed equations by
applying them to a range of predation datasets from the literature to
demonstrate the practical improvements offered by the improved predation
equation formulation. This is the first research article to
comprehensively address limitations in existing predation equations and
to offer a mathematically sound improvement to these equations that
performs well both theoretically and empirically.