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.