# General mechanisms for a top-down origin of the predator-prey power law

Onofrio Mazzarisi, Matthieu Barbier, Matteo Smerlak

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Onofrio Mazzarisi, Matthieu Barbier, Matteo Smerlak. General mechanisms for a top-down origin of the predator-prey power law. Oikos, 2025, 2025 (9), pp.e11232. 10.1002/oik.11232. hal-05412641

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OIKOS

Research article

# General mechanisms for a top-down origin of the predator-prey power law

Onofrio Mazzarisi and Matteo Smerlak

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Correspondence: Onofrio Mazzarisi (onofrio.mazzarisi@gmail.com)

Oikos

2025: e11232

doi: 10.1002/oik.11232

Subject Editor: Lucas Medeiros

Editor-in-Chief:

Paulo R. Guimaraes

Accepted 09 April 2025

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The ratio of predator-to-prey biomass density is not constant along ecological gradients: denser ecosystems tend to have fewer predators per prey, following a scaling relation known as the 'predator-prey power law'. The origin of this surprisingly general pattern, particularly its connection with environmental factors and predator-prey dynamics, is unknown, but has been proposed to be explainable through density-dependent interactions among prey (what we call a bottom-up origin). In contrast, we combine two complementary theoretical approaches to demonstrate how sublinear predator-prey scaling can also emerge from density-dependent interactions among predators and between predators and prey (a top-down origin). First, we use phenomenological differential equations to explore the role of environmental parameters and dynamical properties behind the predator-prey ratio. Second, we simulate an agent-based model with tunable predator self-regulation to investigate the emergence of predator-prey scaling from plausible microscopic rules. While we cannot rule out alternative explanations, our results show that density-dependent mechanisms relative to predation and intraspecific predator interactions, including prey saturation, predator interference, and predator self-regulation, offer potential explanations for the predator-prey power law.

Keywords: Community dynamics, macroecology, power law, predator-prey

Introduction

General biological patterns offer precious opportunities for synthesis and unification (Brown and Maurer 1989, Levin 1992, Brown et al. 2004, Marquet et al. 2005, Frank 2009, Loreau 2010, Harte 2011, Zaoli et al. 2017). Hatton et al. (2015) put forward evidence of such a general pattern, holding across many terrestrial and aquatic ecosystems, dubbed the 'predator-prey power law'. The analysis by Hatton et al. (2015) reveals a decrease in the ratio of predator to prey when comparing ecosystems at increasingly higher predator and prey densities. Several isolated studies over the

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<footnote n="1">¹The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera, Trieste, Italy</footnote>

<footnote n="2">²National Institute of Oceanography and Applied Geophysics (OGS), Trieste, Italy</footnote>

<footnote n="3">³PHIM Plant Health Institute, University of Montpellier, CIRAD, INRAE, Institut Agro, IRD, Montpellier, France</footnote>

<footnote n="4">⁴Capital Fund Management, Rue de l'Université, Paris, France</footnote>
last 40 years reported evidence for this pattern (McCaulley and Kalff 1981, Messier 1994, Del Giorgio and Gasol 1995, Leibold et al. 1997, Chase et al. 2000, Irigoien et al. 2004, Karanth et al. 2004, Cebrian et al. 2009, Wang et al. 2009, Carbone et al. 2011), hinting at its generality. The law holds for focal predator species as well as for aggregate communities and has been recently shown to be robust in ecosystems with more complex trophic structures than previously analyzed (e.g. including omnivores) (Perkins et al. 2022).

In this paper, we try to understand this observation, focusing on a central dataset: the total of large mammalian herbivores and carnivores showing a large gradient of biomass density across a range of 23 African reserves (Fig. 1, Supporting information). Species composition is similar across the gradient and includes herbivores with a body mass ranging from 5 to 500 kg and large carnivores (20 to 140 kg) (Sinclair et al. 2003, Owen-Smith and Mills 2008, Hatton et al. 2015). A reason for focusing on this dataset is the robustness of the pattern to various factors, including the taxonomy of species included in the communities, variations in species body mass, the possibility of systematic bias in sampling, and alternative regression approaches (Hatton et al. 2015).

Any explanation of this scaling must comprise two parts: what varies between sites to explain that they exhibit different biomass densities and what holds across sites to explain the existence (and exponent) of the scaling law between predator and prey biomass density (cf. Fig. 2 for a summary of the following argument).

We focus first on what could cause the stark differences in density between different sites. Here, we assume that these densities are close to their equilibrium value given the abiotic and biotic environment (rather than, for instance, low and high densities representing early and late stages in a colonization process), as the time series analysis reported by Hatton et al. (2015) suggests populations are at stationarity. Under this assumption, we could start by imagining at least two very distinct explanations.

One possibility is that the gradient is driven by environmental differences among the parks, therefore, by variation in the conditions in which the species find themselves interacting. The other possibility is that, in equivalent environmental conditions, there are differences (phenotypic differences) in the species in the different parks. In the latter case, reaching different equilibria would require having different species or phenotypes, e.g. variation in body mass, size of territory or home range, resource needs, etc. Yet, evidence reported by Hatton et al. (2015) stands against the idea that the different parks have significantly different composition in species or individual size, at least none that would systematically create such a gradient. Therefore, we assume that the biomass density gradient is caused by environmental factors.

We can proceed now to explore the reasons behind the specific form of the scaling along the biomass gradient. We first rule out individual-level metabolic theory (which relates organism mass and metabolism or growth with a 3/4 exponent (Brown et al. 2004, Hatton et al. 2019)) as the explanation for

<figure><img src="image_25.png" /><figcaption>Figure 1. Predator biomass density $B_2^*$ scales sublinearly with prey biomass density $B_1^*$, as $B_2^* \sim (B_1^*)^k$ with $k = 3/4$. Data refers to predator–prey communities in African reserves obtained from the meta-analysis in Hatton et al. (2015) (Supporting information). Each color corresponds to a park, and a table with park information relative to each park code in the figure can be found in the SM. The same pattern is observed in several other terrestrial and aquatic ecosystems (Hatton et al. 2015).</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 2. Summary of potential explanations for the predator–prey power law. We need two parts for any explanation: identify the drivers of biomass variation between samples and describe what holds across samples that give origin to the 3/4 scaling. In this work, we focus on explanations in which the 3/4 exponent stems from the functional form (density-dependence) of population dynamics, and choose to call them ‘bottom–up’ when prey consumption and growth control the exponent, and ‘top–down’ when predator consumption and mortality control it. Typically, the source of variation between sites must then lie in the other equation, meaning that the variation in biomasses is expected to arise from ‘bottom–up’ factors, i.e. prey growth, if the exponent of the scaling law is driven by ‘top–down’ factors, i.e. predator consumption, and conversely. We give some examples of other explanations for the predator–prey power law, and of reasons why we are setting them aside here. For instance, the exponent may come from individual-level metabolic scaling, but average organism size and mass do not differ significantly between sites; the differences in biomasses may stem from sites being out of equilibrium, but these differences seem to be lasting (biomass-poor sites do not appear to be simply early stages in biomass growth); the scaling may be due to several parameters changing in correlated ways, but this would require a precise ‘conspiracy’ of physiological and environmental covariates to lead to such a robust, seemingly unidimensional, gradient across sites.</figcaption></figure>

the community-level 3/4 law, again because there appears to be no systematic variation in individual body mass distribution across the gradient so these metabolic properties should be constant. Turning to other explanations, predator and prey densities could both be controlled entirely by changes in external factors, such as more or less productivity and mortality due to aridity or landscape features, irrespective of any dynamics within and between these trophic levels. In that case, a scaling law between predator and prey density, where both increase in a robust but sublinear relationship, would require a ‘conspiracy’ of parameters (a strong but nonlinear covariation in how the changing environment impacts various species’ productivities and mortalities) for which no ecological mechanism has been proposed yet. Thus we wish to posit that the observed variation of both predator and prey densities stems from the same environmental gradient, but the scaling law between them stems from dynamical mechanisms.

A first explanation of this kind was proposed by Hatton et al. (2015), suggesting that the driving dynamical mechanism is a sublinear relationship between prey density and prey biomass production. As we recall below, the environmental variation implied by their explanation is that different sites must have different interaction strengths between predator and prey (Hatton et al. 2015), e.g. due to different vegetation cover.

We focus on another ecologically-motivated assumption, proposing that the various parks differ most importantly by their primary productivity, which is positively

correlated with herbivore growth (secondary productivity) (McNaughton et al. 1989). Annual precipitation is a good proxy for primary productivity in African parks (Coe et al. 1976). In Fig. 3a, we report predator biomass density vs. precipitation, scaling with an exponent around 5/4 and prey biomass density versus precipitation, scaling with an exponent close to 3/2 (consistent with the work by Coe et al. 1976). The difference between the exponents of these two scaling relations is around 1/4 (the same difference obtained by plotting the two densities as function of the prey density (Fig. 1)) bringing some support to the hypothesis that primary productivity is the main driver. This results in predator–prey biomass ratio versus precipitation scaling as –1/4 (Fig. 3b). Within the assumption of primary productivity as main driver for the gradient, we will show that a range of explanations are possible, and they are all top–down, in the sense of relying on predation dynamics, and specifically on density-dependent predator interactions. Figure 2 summarizes the explanations for the predator–prey power law we consider in this work and clarify our definition of ‘top–down’ and ‘bottom–up’ mechanisms.

## Predator–prey dynamics and macroecological patterns

We aim to compare ecosystems hosting the same predator and prey species pairs at different stationary biomass densities.
<figure><img src="image_27.png" /><figcaption>Figure 3. Predator and prey biomass densities scale differently with respect to park precipitation, with their ratio scaling approximately as −1/4, suggesting that precipitation may be a good proxy for the relevant environmental parameter responsible for the biomass density gradient across different parks. In (a), we report predator and prey biomass density versus park precipitation and, in (b), their ratio, again versus precipitation. We stress that the key patterns are the scaling laws of predators and prey densities considered separately as functions of precipitation. Their ratio, a combined variable, has a lower scaling exponent, and therefore a less pronounced shift in the mean trend, resulting in a worse fit with respect to the separate densities, considering there is only roughly one order of magnitude of precipitation variation.</figcaption></figure>

We encode the parameters responsible for the biomass gradient in the vector $\vec{\rho}$. These parameters can represent biotic and abiotic factors affecting the population dynamics (e.g. amount of primary resources or shelters) but also phenotypic traits, which are sensitive to a change of environment but decoupled from the population densities. We collect parameters of the model that are considered independent of the species' density and do not change across environments in the vector $\vec{\sigma}$. Finally, we define the species density vector $\vec{B}$: all the model ingredients representing the dynamics in a given environment are functions of it. We consider a two-dimensional system with one prey species whose density is denoted by $B_1$, and one predator species, $B_2$, that feeds on it. This coarse-grained setting allows us to highlight how density dependence clearly can shape the trophic structure at different biomass densities, even though it is not suited to describing a realistic food web in detail.

The equations describing the dynamics of the density of prey $B_1$ and of predators $B_2$ can be given, with some degree of generality, by

$$ \frac{dB_1}{dt} = P_1(\vec{\rho}, \vec{\sigma}, B_1) - L_1(\vec{\rho}, \vec{\sigma}, B_1) - C(\vec{\rho}, \vec{\sigma}, \vec{B}), \quad (1) $$

$$ \frac{dB_2}{dt} = P_2(\vec{\rho}, \vec{\sigma}, \vec{B}) - L_2(\vec{\rho}, \vec{\sigma}, B_2), \quad (2) $$

where $P$ stands for production, $L$ for internal losses and $C$ for predation losses. In general, predation losses and predator production depend on both species.

Different choices of what enters in $\vec{\rho}$ have distinct ecological meanings. A change, e.g. of a coefficient in the interaction term could, for example, represent a different concentration of shelter for prey (Berryman et al. 2006). Another common

reason behind different biomass densities of predators and prey in different environments is a variation in the energy influx in the system through primary production, which can be encoded in a changing coefficient in the prey production term. The observed gradient might not be due to variation in one parameter alone, and we must carefully consider how covariance between the relevant parameters may affect our model predictions (Barbier et al. 2021). Consider a choice of the production and losses that correspond to the Lotka–Volterra model with self-regulation,

$$ \frac{dB_1}{dt} = rB_1 - cB_1^2 - qB_1B_2, \quad (3) $$

$$ \frac{dB_2}{dt} = gqB_1B_2 - mB_2 - aB_2^2, \quad (4) $$

where $r$ is the growth rate of the prey, $q$ is the interaction strength, $g$ the conversion rate, $m$ the death rate of the predators, and $c$ and $a$ the self-regulation strength of prey and predators respectively. The inclusion of predator self-regulation can be biologically justified, e.g. when the effect of pathogens is relevant, and simplifies calculations in this pedagogical example. It is instructive to write down the stationary solution for the two species' densities to highlight the explicit dependence on the parameters

$$ B_1^* = \frac{ar}{ca + gq^2}, \quad (5) $$

$$ B_2^* = \frac{gqr}{ca + gq^2}, \quad (6) $$
where we considered negligible predator mortality, $m=0$, for clarity. If the prey growth rate $r$ is the single most relevant parameter that changes along the biomass gradient, i.e. $\rho=r$ and $\bar{\sigma} = (a,g,q,m)$, then linear scaling between predator and prey density $B_2^* \sim B_1^*$ emerges, both being linear in $r$. If, however, the interaction strength $q$ is also relevant and it varies with $r$, for example as $q \sim r^{k-1}$, with $k < 1$, then the observed scaling is sublinear $B_2^* \sim (B_1^*)^k$.

In summary, the choice of the parameters that are supposed to model a biomass gradient has to be informed by observation, and different choices of parameters produce different macroecological patterns starting from the same dynamical model. We want to explore how dynamical, density-dependent effects can be relevant for the scaling. Therefore, we use as a working assumption that only one parameter, possibly different in other datasets, is responsible for the gradient.

Within this working assumption, a linear relation and, therefore, a constant average number of prey per predator in a patch across a biomass gradient would imply that only the stationary number of predators and prey matters, and how densely packed the two species are is irrelevant. If species density matters (Hassell 1986, Hassell et al. 1989, Krebs 1995, Hixon et al. 2002), a sublinear scaling is expected when the net effect of crowding penalizes more the standing biomass of predators than that of prey. In Fig. 4 we pictorially summarize this observation and show examples of mechanisms that could dynamically lead to sublinear scaling. The bottom-up explanation, in the sense of relying on prey dynamics and specifically on density dependence in prey production, emerges from sublinear growth of herbivores (Hatton et al. 2015). The main classes of mechanisms behind the top-down explanation are predator density-dependent self-regulation, predation interference and prey saturation.

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Figure 4. Sublinear scaling reflects the relevance of density-dependent effects for the predator–prey ratio. The figure portrays a thought experiment in which we start with a patch of an ecosystem hosting a given number of predators and prey at stationarity. Then, we change the parameter driving variation in biomass so that the stationary number of prey changes in the same way the two cases, but the number of predators scales in one case linearly, and in the other sublinearly, with the number of prey. Examples of mechanisms that can lead to sublinear scaling are also portrayed, covering bottom-up and top-down explanations. Blue disks represent prey (with small disks representing prey offspring), orange triangles predators. In density-dependent self-regulation (top-down), predators are shown to self-regulate their population size by direct interaction. In predation interference (top-down), on the other hand, this happens because they interfere with each other's hunting. Prey saturation, shows predators unable to consume all the prey available because some are 'shielded' by the others. In sublinear prey production (bottom-up), predators are sustained by prey offspring, and prey are shown to grow sublinearly with their density.
## Bottom-up, sublinear prey production

We summarize here the bottom-up explanation proposed by Hatton et al. (2015), which invokes sublinear prey production to describe the observed predator–prey scaling. The model is defined as

$$ \frac{dB_1}{dt} = rB_1^k - qB_1B_2, \qquad (7) $$

$$ \frac{dB_2}{dt} = gqB_1B_2 - mB_2, \qquad (8) $$

where $r$ is the growth rate of the prey, $q$ is the interaction strength, $g$ the conversion rate, $m$ the death rate of the predators, and $k < 1$ is an exponent characterizing the density-dependent effects on the per capita growth rate of the prey. To be dimensionally consistent, we should multiply the prey production term by a factor $B_0^{1-k}$, where $B_0$ has the dimensions of a density. Here, and mutatis mutandis in the other sections when not otherwise stated, we consider $B_0=1$ without loss of generality for the purpose of this work.

The equilibrium prey density $B_1^*$ and predator density $B_2^*$, are respectively

$$ B_1^* = \frac{m}{gq}, \qquad (9) $$

$$ B_2^* = \frac{rm^{k-1}}{g^{k-1}q^k} = \frac{rg}{m} (B_1^*)^k. \qquad (10) $$

Therefore $rg/m$ has to be constant, and $q$ has to decrease (increase) to move up (down) across the biomass gradient in order to obtain the scaling $B_2^* \sim (B_1^*)^k$. If, instead, $r$ is primarily responsible for the biomass increase, this model predicts a constant prey density while the predators grow. In terms of the general definition above, the two cases correspond, respectively, to $\rho=q$, $\bar{\sigma} = (r, g, m)$ and $\rho=r$, $\bar{\sigma} = (q, g, m)$.

This bottom-up explanation is informed by another macroecological observation, namely the sublinear scaling of biomass production with respect to biomass density (Hatton et al. 2015, 2024, Mazzarisi and Smerlak 2024). A discussion on possible mechanistic explanations for sublinear growth can be found in the works by Hatton et al. (Hatton et al. 2015, 2024); it is, however, still an open question. Sublinear production in the dynamical equation for prey does not represent a unique possible explanation for the macroecological pattern observed across ecosystems. However, it can be realistic and parsimonious (Hatton et al. 2024).

Support for $q$ as the relevant parameter to move along the biomass density comes from cross-system studies, which have shown that density increases across prey causes a decrease of nutritional quality and edibility, which may tend, in turn, to weaken interaction strength with consumers (Hairston

and Hairston 1993, Cebrian and Duarte 1994, Leibold et al. 1997, Cebrian and Lartigue 2004, Cebrian et al. 2009, Hatton et al. 2015).

## Top-down, density-dependent predation and predator self-regulation

Here we explore how the predator–prey scaling can have a top-down origin, emerging from density dependence in predation and predator self-regulation. We consider a phenomenological model in which density-dependent effects are important at all scales, following the conceptualization proposed by Barbier et al. (2021). The functional form of the terms involved could, in principle, emerge from underlying spatiotemporal details (Barraquand and Murrell 2013) and should not be necessarily rooted in mechanistic explanations (Novak and Stouffer 2021a). The model reads

$$ \frac{dB_1}{dt} = rB_1 - cB_1^2 - qB_1^\alpha B_2^\beta, \qquad (11) $$

$$ \frac{dB_2}{dt} = gqB_1^\alpha B_2^\beta - mB_2^{1+\gamma}, \qquad (12) $$

where $r$ is the growth rate of the prey, $c$ the strength of prey's self-regulation emerging, e.g. from competition for resources (MacArthur 1970, Chesson 1990), $q$, $g$ and $m$ are, respectively, related to the interaction strength, conversion rate and predator death rate (by properly accounting for dimensionality). The exponent $\alpha$ encodes saturation, $\beta$ is connected to predator interference, and $\gamma$ models the strength of predator self-regulation. The interaction term is scale-free, in the sense that complete saturation with respect to prey density is never achieved. However, this is not unrealistic as evidence suggests that predator feeding rates may often be unsaturated under typical conditions (Coblentz et al. 2023). It is also possible to use other models such as the Hassell–Varley–Holling model (Hassell and Varley 1969, Arditi and Akçakaya 1990, Berec 2009, Přibylová and Berec 2015), especially if the systems are far from complete saturation.

In this work, we are only quantitatively interested in the scaling exponents that emerge from the dynamics. Therefore, Eq. 12 and 11 can be reduced, through non-dimensionalization (Supporting information), to the form

$$ \frac{dB_1}{dt} = \rho B_1 - B_1^2 - \sigma B_1^\alpha B_2^\beta, \qquad (13) $$

$$ \frac{dB_2}{dt} = B_1^\alpha B_2^\beta - B_2^{1+\gamma}, \qquad (14) $$

where we call the remaining parameters $\rho$ (which is responsible for the biomass gradients) and $\sigma$. These are functions of the original parameters, with $\rho \propto r$ and $\sigma$ independent from $r$.
The general stationary solution for predator biomass density as a function of prey reads

$$B_2^* = (B_1^*)^k, \quad \text{with} \quad k = \alpha / (1 + \gamma - \beta), \qquad (15)$$

while the solution for the prey has a closed form only for specific values of the exponents ($\alpha$, $\beta$ and $\gamma$) and must otherwise be solved numerically. Assuming a varying $\rho$, corresponding to a variation of primary productivity, every combination of the exponents resulting in $k=3/4$ predator–prey scaling across ecosystems is viable as long as it produces feasible and stable stationary solutions.

As an example, a meta-analysis of predator–prey pairs of mammals, birds, and reptiles suggests that total predation rates increase as the square root of the product of predator and prey density (Barbier et al. 2021), i.e. $\alpha = \beta = 1/2$. In this case, in order to observe the $k=3/4$ scaling, the exponent associated to predator self-regulation has to be $\gamma = 1/6$. This means that per capita losses for predators must scale like $B_2^{1/6}$. This exponent, incidentally similar to scaling laws found in complex systems outside of ecology (Bettencourt 2013), suggests that even a seemingly weak density-dependence may play a significant role in biomass patterns. In Fig. 5 we report the predator–prey scaling for this choice of parameters.

Differences in primary productivity and herbivore growth are considered among the most common reasons underlying

differences in biomass density in otherwise similar ecosystems. Support for considering $r$ the relevant variable comes from data on the relation between biomass density and precipitation in the parks (Fig. 3) and from previous studies connecting precipitation with primary productivity and herbivore growth (Coe et al. 1976, McNaughton et al. 1989). If precipitation is associated with the relevant variable $\rho$ in our model (Fig. 5b), we expect the scaling in Fig. 3. To rule out an obvious potential confounder, we show in the Supporting information that there is no correlation between park area and precipitation level. This evidence is insufficient to establish primary productivity as the most relevant driver, but it indicates it as plausible.

As an additional test, we parameterize this model with realistic values for the focal species in the African parks considered in this work. The scaling relation at stationarity considering the full dimensional model is (Supporting information)

$$B_2^* = \left( \frac{gq}{m} \right)^{k/\alpha} B_0^{1+k/\alpha-k} (B_1^*)^k. \qquad (16)$$

Starting from independent estimations of the parameters $g$ (Hatton et al. 2015), $q$ (Pawar et al. 2012), $m$ (Hatton et al. 2015) and the biomass threshold for nonlinearity $B_0$ (Hatton et al. 2024), we can estimate the coefficient predicted

<figure><img src="image_29.png" /><figcaption>Figure 5. Predator biomass density $B_2^*$ scales sublinearly with prey biomass density $B_1^*$ for given exponents for saturation $\alpha$, interference $\beta$ and self-regulation $\gamma$ (a) when herbivore growth is varied to simulate a biomass gradient, shown as a straight line on log–log axes. By solving numerically Eq. 13 and 14, here we show, for $\alpha = \beta = 1/2$, $\gamma = 1/6$, $\sigma = 1/10$ and $\rho \in [1,100]$, (b) the predator stationary biomass density against prey stationary biomass density ($k=3/4$) and (c) their ratio with respect to the environmental parameter $\rho$ (Eq. 13), which is proportional to the herbivore growth rate $r$.</figcaption></figure>
by our model and confront it with the fitted coefficient from the data $C_{fit}$=0.075, with dimensions $[C_{fit}]=(\text{kg km}^{-2})^{1-k}$ and 95% confidence interval (C. I.) $C_{fit}$=0.038, 0.15. If we assume $\alpha$=1, we get

$$C_{est} = \left( \frac{gq}{m} \right)^{k/\alpha} B_0^{1+k/\alpha-k} = 0.041 (\text{kg / km}^2)^{1-k}, \quad (17)$$

which is within the confidence interval of $C_{fit}$. In general $k/\alpha < 1$ is favorable to improve the fit unless $B_0$ is large. As there is a significant margin for error in all these estimates, this calculation is only meant as a ‘sanity check’ to verify that our model predictions are at least compatible with some empirical evidence.

## Agent-based model as a proof of concept for the top-down case

Predator–prey agent-based (ABM) models (Bénichou et al. 2007, 2011, Gurarie and Ovaskainen 2013, Barbier and Watson 2016, Salahshour 2023) can provide a bridge between mathematical models and more realistic behavioral rules, providing more intuition on the dynamics underlying our patterns. We present a simple ABM that produces sublinear predator–prey scaling across a biomass gradient through top-down effects. We focus, as an example, on density dependence in predator self-regulation.

We consider a $L \times L$ grid on which the agents move with Moore connectivity (each site has eight neighbors) and periodic boundary conditions. No occupancy restriction is enforced: there is no upper bound to the number of agents a grid point can host. There are two species: prey and predators. Agents of both species possess an internal energy that changes during the dynamics; metabolism decreases it, feeding increases it, and during a reproduction event, the parent donates half of its energy to the offspring. Prey move on an adjacent grid point, and their energy changes of an amount $\Delta E = -\mu$, where $\mu$ is the metabolic rate. Then, they eat a fraction of the resources present on the grid, $R$, which is converted into energy $\Delta E = R/N_1$ where $N_1$ is the number of prey present in the grid point. Finally, if their energy is $E \le 0$ they die, otherwise, with probability $P=\omega_1$ they reproduce. The dynamics of predators are slightly more involved. They move and their energy changes as $\Delta E = -\mu$ due to metabolism. If they are alone in the grid point and if their energy reached zero or less, they die. Otherwise, with probability $P=N_1/(N_1+N_2)$ they engage a prey while with probability $1-P$ engage a predator, where $N_2$ is the number of predators in the grid point. In the first scenario, with probability $P=\delta$, they eat the prey, gaining $\Delta E = \eta$, where $\eta$ is the predation efficiency, and subsequently, with probability $P=\omega_2$, they reproduce. In the case in which they do not catch the prey, and if their energy is zero or less, they die. In the scenario in which they engage a predator, they kill the competitor with probability $P=f$. Finally, if their energy is $E \le 0$ they die of starvation.

At the end of the time step the amount of resources at each grid point increases by $\rho$, the resource growth rate. The functioning of the model is summarized in Fig. 6.

A biomass gradient in the model is obtained by tuning the resource growth rate $\rho$: the higher $\rho$, the higher the biomass density at stationarity. The key element of this model is the density dependence in the probability that a predator has of attacking and killing another predator when encountered. We call it aggressiveness and denote it by $f(N_2, \xi)$, where $\xi$ is a parameter that establishes a threshold over which the attack is certain. It controls the sensitivity of this mortality with respect to the predator density (i.e. number at a given grid point). In order to explore more regimes, we study the form

$$f(N_2, \xi) = \min\{(N_2 / \xi)^\nu, 1\}, \quad \text{with } \nu \ge 0, \quad (18)$$

where the exponent $\nu$ modulates the dependence of the aggressiveness on predator density. We choose for our simulations a threshold $\xi=100$ that allows for a large enough range of prey densities.

Figure 7 shows the equilibrium biomass density of predators $B_2^*$ as a function of the equilibrium biomass density of prey $B_1^*$ (equivalent to the numerical density upon choosing unitary mass for both species without loss of generality) for different values of the resource growth rate $\rho$. If the aggressiveness does not depend on the number of predators in the grid point ($\nu=0$) a linear scaling is observed. Any positive value of $\nu$ implies sublinear scaling. The general asymptotic expression of the density of predators as a function of the density of prey reads (Supporting information)

$$B_2^* = (\xi^\nu \omega_2 \delta)^k (B_1^*)^k, \quad \text{with } k = 1/(\nu+1), \quad (19)$$

where $\delta$ is the probability of a catch when a predator engages with a prey.

The model presented here serves as a proof of concept that density-dependent deleterious effects for predators (in this case, self-regulation) can induce sublinear scaling across a biomass gradient. It also shows that the scaling relationship may break even with the right ingredients. In this case, the predator population cannot be sustained below a given resource growth rate, a feature not uncommon in predator-prey models with explicit resource dynamics (Supporting information).

A comparison with the phenomenological model in Eq. 13 and 14 is possible through the two expressions for the scaling exponent $k$ (Eq. 15 and 19) and thanks to the fact that in both models predator density depends on the environmental parameter only through prey density. A parameterization of the phenomenological model that captures the ABM dynamics is $\alpha=1, \beta=1$ and $\gamma=\nu+1$.

## Discussion

In this work, we showed that top-down mechanisms represent a viable explanation for the origin of the predator-prey power
Agent-based model

<figure><img src="image_30.png" /><figcaption>Figure 6. Visual description of the agent-based model. The general setting and the parameters are defined, as well as predator and prey dynamics.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>Figure 7. In the ABM, predator biomass density $B_2^*$ scales more and more sublinearly with respect to prey biomass density $B_1^*$ for increasing levels of aggressiveness (encoded in the exponent $v$ and described by Eq. 18). Each point is a time average of the densities over simulations that reached stationarity (in particular from step 50 to step 500), for each value of the aggressiveness exponent $v$ the resource growth rate $\rho$ varies between 1 and 100. The grid area is $4^2$, the initial density of prey and predators is 1, the aggressiveness threshold is $\xi=100$, the reproduction probabilities are $\omega_1=\omega_2=1/2$, the catch probability is $\delta=1/3$, the predation efficiency is $\eta=10$, and the metabolic rate $\mu=1$. The resulting scaling exponents depend only on $v$ and are robust to changes in the other parameters. The pink, red, and dark red lines are theoretical predictions from Eq. 19.</figcaption></figure>

law, alongside the bottom-up explanation in (Hatton et al. 2015). Under the assumption of prey productivity as the primary driver for the biomass density gradient across ecosystems, we demonstrated that combinations of nonlinearities in the interaction term and predator self-regulation can produce the predator-prey power law. While there is no single model that can characterize predator-prey interactions in general (Skalski and Gilliam 2001, Novak and Stouffer 2012a, 2012b, Stouffer et al. 2021), we considered a scale-free model (Barbier et al. 2021) which capture the relevant dynamical trends that could be behind the macroecological pattern. We also devised and analyzed an agent-based model to provide an example of how predator-prey scaling can emerge as a top-down effect starting from plausible microscopic rules.

We are not able to rule out alternative explanations. Extensive field studies are needed to identify the most relevant factors (and their covariance) behind differences in equilibrium biomass density in ecosystems with similar species composition. This identification is crucial to understand if dynamical ingredients and which ones determine the emergence of predator-prey scaling. We hope to stimulate field work and analysis of existent data sets to test the plausibility of the top-down explanation presented here and, possibly, identify the specific density-dependent mechanisms behind it. Cross-systems studies on predator self-regulation analogous to the one performed for the interaction term in the work by Barbier et al. (2021) would be particularly valuable.

It is also possible to put to the test different models by requiring their consistency with other macroecological

patterns, such as the sublinear scaling of production with respect to biomass density (Hatton et al. 2015, 2024). The scaling exponent is approximately the same as in the predator-prey scaling and is recovered for prey production in the bottom-up framework. The top-down model predicts, by the equilibrium requirement on the prey (Eq. 11), that prey production scales as $B_1^{\alpha+\beta k}$ for a range of densities and then as $B_1^2$ for higher densities, depending on the value of $\sigma$. For $\alpha=\beta=1/2$ and $\gamma=1/6$ the exponent is 0.875, a bit higher than 3/4. If $\gamma=0$, we have the same exponent $k=\alpha/(1-\beta)$ for both patterns until production scales as $B_1^2$ for high enough densities. From this perspective, the bottom-up model is more appropriate to capture the general dynamics. Production scaling with biomass density, however, is not recovered for predators; and although available data are still fragmentary, they are indicative that the pattern should hold at every trophic level (Hatton et al. 2015, 2024).

Establishing if either of these two classes of dynamical explanations for the predator-prey power law is close to a correct description of reality, at least at large scales, may also impact how we can act upon these ecosystems. Indeed, the predator-prey scaling pattern could be considered a reference for a healthy (or at least natural) trophic structure in the ecosystem, indicating the expected predator biomass given the prey biomass and vice versa (Karanth et al. 2004, Hatton et al. 2021). Acting on a driver of biomass variation would keep the proportions in check, i.e. consistent with the pattern, only if the underlying dynamics is the one matching that driver. For example, immunocontraception is employed as a management
tool to reduce population growth of wild populations, e.g. ele-
phants in reserves (Del Giorgio and Gasol 1995, Druce et al.
2011). This means acting on prey productivity *r*, and therefore,
the predator population is expected to reach densities predicted
by the predator–prey scaling law only in the top–down case.

Finally, it remains a challenge to establish to what extent
scaling patterns at the ecosystem level such as the predator–
prey power law are connected to other levels of organization.
For example, macroecological scaling patterns in plankton
communities can emerge from body size scaling through eco-
evolutionary dynamics (Wickman et al. 2024). This is a deep
result, but it requires producer (phytoplankton) mean body
size variation with density, which is not observed in other
systems, such as the mammalian herbivores and carnivores.
On the other hand, recent work on marine phytoplankton
competing for light and nutrients suggests that organismal
metabolism can be affected by community biomass indepen-
dently of body size (Fant and Ghedini 2024). This shows how
in competitive communities density-dependent processes can
affect physiological response at the organismal level, opening
the possibility that similar effects can emerge also in the case
of trophic interactions.

*Acknowledgements* – We thank J. Arnoldi, A. Bideault, L. Fant, B. Girardot, I. Hatton and M. Novak for fruitful discussions and critical feedback on the work. A preprint version of this article has been peer-reviewed and recommended by PCIEcology (https://doi.org/10.24072/pci.ecology.100692). Open access publishing facilitated by Istituto Nazionale di Oceanografia e Geofisica Sperimentale, as part of the Wiley – CRUI-CARE agreement.

**Funding** – Funding for this work was provided by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research, by the Trieste Laboratory on Quantitative Sustainability - TLQS and by NSF BIO OCE grant 2023473.

Author contributions

**Onofrio Mazzarisi:** Conceptualization (lead); Data cura-
tion (lead); Formal analysis (lead); Funding acquisition
(supporting); Investigation (lead); Methodology (lead);
Project administration (lead); Software (lead); Visualization
(lead); Writing – original draft (lead); Writing – review
and editing (lead). **Matthieu Barbier:** Conceptualization
(supporting); Data curation (supporting); Formal analysis
(supporting); Investigation (supporting); Validation (equal);
Visualization (equal); Writing – original draft (supporting);
Writing – review and editing (supporting). **Matteo Smerlak:**
Conceptualization (supporting); Funding acquisition (lead);
Investigation (equal); Methodology (supporting); Project
administration (supporting); Validation (equal); Writing
– original draft (supporting); Writing – review and editing
(supporting).

Data availability statement

Data are available in the Supporting information and in
Hatton et al. (2015). The Julia code used for analysis and

simulations is available at https://github.com/onofriomazz
arisi/top-down-vs-bottom-up (Mazzarisi et al. 2025).

**Supporting information**

The Supporting information associated with this article is
available with the online version.

References

Arditi, R. and Akçakaya, H. R. 1990. Underestimation of mutual interference of predators. – Oecologia 83: 358–361.

Barbier, M. and Watson, J. R. 2016. The spatial dynamics of predators and the benefits and costs of sharing information. – PLOS Comp. Biol. 12: e1005147.

Barbier, M., Wojcik, L. and Loreau, M. 2021. A macro-ecological approach to predation density-dependence. – Oikos 130: 553–570.

Barraquand, F. and Murrell, D. J. 2013. Scaling up predator–prey dynamics using spatial moment equations. – Methods Ecol. Evol. 4: 276–289.

Bénichou, O., Loverdo, C., Moreau, M. and Voituriez, R. 2007. A minimal model of intermittent search in dimension two. – J. Phys. Condens. Matter 19: 065141.

Bénichou, O., Loverdo, C., Moreau, M. and Voituriez, R. 2011. Intermittent search strategies. – Rev. Mod. Phys. 83: 81–129.

Berec, L. 2009. Impacts of foraging facilitation among predators on predator–prey dynamics. – Bull. Math. Biol. 72: 94–121.

Berryman, A. A., Hawkins, B. A. and Hawkins, B. A. 2006. The refuge as an integrating concept in ecology and evolution. – Oikos 115: 192–196.

Bettencourt, L. M. A. 2013. The origins of scaling in cities. – Science 340: 1438–1441.

Brown, J. H. and Maurer, B. A. 1989. Macroecology: the division of food and space among species on continents. – Science 243: 1145–1150.

Brown, J. H., Gillooly, J. F., Allen, A. P., Savage, V. M. and West, G. B. 2004. Toward a metabolic theory of ecology. – Ecology 85: 1771–1789.

Carbone, C., Pettorelli, N. and Stephens, P. A. 2011. The bigger they come, the harder they fall: body size and prey abundance influence predator–prey ratios. – Biol. Lett. 7: 312–315.

Cebrian, J. and Duarte, C. M. 1994. The dependence of herbivory on growth rate in natural plant communities. – Funct. Ecol. 8: 518–525.

Cebrian, J. and Lartigue, J. 2004. Patterns of herbivory and decom-
position in aquatic and terrestrial ecosystems. – Ecol. Monogr.
74: 237–259.

Cebrian, J., Shurin, J. B., Borer, E. T., Cardinale, B. J., Ngai, J. T., Smith, M. D. and Fagan, W. F. 2009. Producer nutritional quality controls ecosystem trophic structure. – PLoS One 4: 1–5.

Chase, J. M., Leibold, M. A., Downing, A. L. and Shurin, J. B. 2000. The effects of productivity, herbivory and plant species turnover in grassland food webs. – Ecology 81: 2485–2497.

Chesson, P. 1990. MacArthur's consumer-resource model. – Theor. Popul. Biol. 37: 26–38.

Coblentz, K. E., Novak, M. and DeLong, J. P. 2023. Predator feed-
ing rates may often be unsaturated under typical prey densities.
– Ecol. Lett. 26: 302–312.
Coe, M. J., Cumming, D. H. and Phillipson, J. 1976. Biomass and production of large African herbivores in relation to rainfall and primary production. – *Oecologia* 22: 341–354.

Del Giorgio, P. A. and Gasol, J. M. 1995. Biomass distribution in freshwater plankton communities. – Am. Nat. 146: 135–152.

Druce, H. C., Mackey, R. L. and Slotow, R. 2011. How immuno-contraception can contribute to elephant management in small, enclosed reserves: Munyawana population as a case study. – PLoS One 6: e27952.

Fant, L. and Ghedini, G. 2024. Biomass competition connects individual and community scaling patterns. – Nat. Commun. 15: 9916.

Frank, S. A. 2009. The Common patterns of nature. – J. Evol. Biol. 22: 1563–1585.

Gurarie, E. and Ovaskainen, O. 2013. Towards a general formalization of encounter rates in ecology. – Theor. Ecol. 6: 189–202.

Hairston, N. G. J. and Hairston, N. G. S. 1993. Cause–effect relationships in energy flow, trophic structure and interspecific interactions. – Am. Nat. 142: 379–411.

Harte, J. 2011. Maximum entropy and ecology: a theory of abundance, distribution, and energetics. – Oxford Univ. Press.

Hassell, M. P. 1986. Detecting density dependence. – Trends Ecol. Evol. 1: 90–93.

Hassell, M. P. and Varley, G. C. 1969. New inductive population model for insect parasites and its bearing on biological control. – Nature 223: 1133–1137.

Hassell, M. P., Latto, J. and May, R. M. 1989. Seeing the wood for the trees: detecting density dependence from existing life-table studies. – J. Anim. Ecol. 58: 883–892.

Hatton, I. A., Dobson, A. P., Storch, D., Galbraith, E. D. and Loreau, M. 2019. Linking scaling Laws across eukaryotes. – Proc. Natl Acad. Sci. USA 116: 21616–21622.

Hatton, I. A., McCann, K. S., Fryxell, J. M., Davies, T. J., Smerlak, M., Sinclair, A. R. E. and Loreau, M. 2015. The predatorprey power law: biomass scaling across terrestrial and aquatic biomes. – Science 349: aac6284.

Hatton, I. A., Mazzarisi, O., Altieri, A. and Smerlak, M. 2024. Diversity begets stability: sublinear growth and competitive coexistence across ecosystems. – Science 383: eadg8488.

Hatton, I. A., Heneghan, R. F., Bar-On, Y. M. and Galbraith, E. D. 2021. The global ocean size spectrum from bacteria to whales. – Sci. Adv. 7: eabh3732.

Hixon, M. A., Pacala, S. W. and Sandin, S. A. 2002. Population regulation: historical context and contemporary challenges of open vs closed systems. – Ecology 83: 1490–1508.

Irigoien, X., Huisman, J. and Harris, R. P. 2004. Global biodiversity patterns of marine phytoplankton and zooplankton. – Nature 429: 863–867.

Karanth, K. U., Nichols, J. D., Kumar, N. S., Link, W. A. and Hines, J. E. 2004. Tigers and their prey: predicting carnivore densities from prey abundance. – Proc. Natl Acad. Sci. USA 101: 4854–4858.

Krebs, C. J. 1995. Two paradigms of population regulation. – Wildl. Res. 22: 1–10.

Leibold, M. A., Chase, J. M., Shurin, J. B. and Downing, A. L. 1997. Species turnover and the regulation of trophic structure. – Annu. Rev. Ecol. Syst. 28: 467–494.

Levin, S. A. 1992. The problem of pattern and scale in ecology: the Robert H. MacArthur award lecture. – Ecology 73: 1943–1967.

Loreau, M. 2010. From populations to ecosystems: theoretical foundations for a new ecological synthesis (MPB-46). – Princeton Univ. Press.

MacArthur, R. 1970. Species packing and competitive equilibrium for many species. – Theor. Popul. Biol. 1: 1–11.

Marquet, P. A., Quiñones, R. A., Abades, S., Labra, F., Tognelli, M., Arim, M. and Rivadeneira, M. 2005. Scaling and power-laws in ecological systems. – J. Exp. Biol. 208: 1749–1769.

Mazzarisi, O. and Smerlak, M. 2024. Complexity-stability relationships in competitive disordered dynamical systems. – Phys. Rev. E 110: 054403.

Mazzarisi, O., Barbier, M. and Smerlak, M. 2025. Data from: General mechanisms for a topdown origin of the predator–prey power law. – Github, https://github.com/onofriomazzarisi/top-down-vs-bottom-up.

McCauley, E. and Kalff, J. 1981. Empirical relationships between phytoplankton and zooplankton biomass in lakes. – Can. J. Fish. Aquat. Sci. 38: 458–463.

McNaughton, S. J., Oesterheld, M., Frank, D. A. and Williams, K. J. 1989. Ecosystem-level patterns of primary productivity and herbivory in terrestrial habitats. – Nature 341: 142–144.

Messier, F. 1994. Ungulate population models with predation: a case study with the North American moose. – Ecology 75: 478–488.

Novak, M. and Stouffer, D. B. 2021a. Geometric complexity and the information-theoretic comparison of functional-response models. – Front. Ecol. Evol. 9: 740362.

Novak, M. and Stouffer, D. B. 2021b. Systematic bias in studies of consumer functional responses. – Ecol. Lett. 24: 580–593.

Owen-Smith, N. and Mills, M. G. L. 2008. Predator–prey size relationships in an African large-mammal food web. – J. Anim. Ecol. 77: 173–183.

Pawar, S., Dell, A. I. and Savage, V. M. 2012. Dimensionality of consumer search space drives trophic interaction strengths. – Nature 486: 485–489.

Perkins, D. M., Hatton, I. A., Gauzens, B., Barnes, A. D., Ott, D., Rosenbaum, B., Vinagre, C. and Brose, U. 2022. Consistent predator–prey biomass scaling in complex food webs. – Nat. Commun. 13: 4990.

Přibylová, L. and Berec, L. 2015. Predator interference and stability of predator–prey dynamics. – J. Math. Biol. 71: 301–323.

Salahshour, M. 2023. Predator–prey interactions give rise to scale-invariant ecosystems. – N. J. Phys. 25: 123012. DOI 10.1088/1367-2630/ad0d37.

Sinclair, A. R. E., Mduma, S. and Brashares, J. S. 2003. Patterns of predation in a diverse predator–prey system. – Nature 425: 288–290.

Skalski, G. T. and Gilliam, J. F. 2001. Functional responses with predator interference: viable alternatives to the Holling type II model. – Ecology 82: 3083–3092.

Stouffer, D. B. and Novak, M. 2021. Hidden layers of density dependence in consumer feeding rates. – Ecol. Lett. 24: 520–532.

Wang, G., Hobbs, N. T., Twombly, S., Boone, R. B., Illius, A. W., Gordon, I. J. and Gross, J. E. 2009. Density dependence in northern ungulates: interactions with predation and resources. – Popul. Ecol. 51: 123–132.

Wickman, J., Litchman, E. and Klausmeier, C. A. 2024. Eco-evolutionary emergence of macroecological scaling in plankton communities. – Science 383: 777–782.

Zaoli, S., Giometto, A., Maritan, A. and Rinaldo, A. 2017. Covariations in ecological scaling laws fostered by community dynamics. – Proc. Natl Acad. Sci. USA 114: 10672–10677.