Estimating the impact of new high seas activities on the environment: the effects of ocean-surface macroplastic removal on sea surface ecosystems

Assumptions and modelling approach

We consider a deterministic model for the effects of floating macroplastic and ocean cleanup on a single species of neuston in continuous time, ignoring spatial and life history structure and seasonal or other variation in parameter values. Our aim is to provide a qualitative understanding of the system, focusing on equilibrium behaviour in order to inform long-term management strategies for plastic in the oceans. Little is known about interspecific interactions in the neuston, so a multispecies model is currently beyond our capabilities. There is recent evidence of interspecific competition in the neuston from stable isotope studies (Albuquerque et al., 2021). However, the general claim that interspecific interactions are weaker than intraspecific interactions (Mutshinda, O’Hara & Woiwod, 2009) appears to be supported by specific models for aquatic systems (e.g., Lindegren et al., 2009; Forsblom et al., 2021) to the extent that it is built into priors for multispecies models (Ward, Marshall & Scheuerell, 2022). We therefore model only a single species. Additionally, we include only floating macroplastics (particles with size > 0.5 cm; from here on simply plastics), rather than other fractions such as microplastics, because macroplastics are the target of current cleanup efforts.

Our model satisfies the postulate of parenthood, that every living organism has arisen from at least one parent of like kind (Hutchinson, 1978, p. 1), and thus ignores immigration. The neuston is in fact an open system. However, ignoring immigration allows us to frame the problem in terms of the niche structure of a neuston species. The fundamental niche of a species is defined as the set of environmental conditions under which the species can persist indefinitely, and “indefinite persistence” is generally taken to be in the absence of immigration (Holt, 2009). Within the fundamental niche, the proportional population growth rate, ignoring immigration, represents the population-level response of a species to its environment (Maguire, 1973). Such a definition also makes sense for ecosystem functions or services that depend on production, but not those that depend on abundance or biomass. In addition, any cleanup programme aiming to achieve a large reduction in total floating macroplastic would have to operate over a large area, for which it is likely that external inputs would be small compared to the effects of internal dynamics. We focus here on true neuston, which remain at the surface throughout the diurnal cycle. There are also important groups of organisms facultatively associated with the ocean surface, but undergoing diel migration (Hempel & Weikert, 1972). The equilibrium behaviour of a model ignoring diel migration may be a reasonable approximation for the long-term effects of cleanup on such organisms.

We assume that intraspecific interactions can be described by logistic density dependence. The logistic model is widely used, and is the simplest density-dependent model satisfying the postulate of parenthood (Hutchinson, 1978, p. 4). Furthermore, logistic density dependence has the convenient property that we can study effects on equilibrium neuston density relative to its value in the absence of cleanup, without data on the strength of intraspecific density dependence. This is important, given the scarcity of demographic data on neuston populations. We initially describe a model in which plastics can affect the proportional population growth rate of neuston. However, there are very few data on the population-level effects of plastics on ocean organisms. We therefore assume in subsequent analysis of the effects of cleanup (which act through removal of both neuston and plastics) that the effect of plastics on neuston is zero. Assuming no effect of plastics on neuston is conservative with respect to the possible net negative effect of cleanup. Furthermore, the most relevant tradeoff is between negative effects of cleanup on neuston and positive effects on other ocean organisms, rather than between negative and positive effects on neuston.

We model the dynamics of plastic concentration at the ocean surface with a single compartment representing buoyant macroplastics with a constant input rate and a constant natural loss rate per unit plastic concentration. Although models with multiple compartments such as those found in Koelmans et al. (2017) and Lebreton, Egger & Slat (2019) are needed to study the global dynamics of ocean plastic, the buoyant macroplastics compartment is the one most relevant to the effects of ocean cleanup on neuston.

Initial model description

Here, we describe our initial model, including an effect of plastics on the proportional population growth rate of neuston. Let n be neuston density (dimensions ML⁻²; throughout we use the standard symbols M, L and T to refer to the dimensions mass, length and time respectively), let p be plastic density (dimensions ML⁻²) and let t be time (dimensions T). We use a logistic population growth model for neuston, coupled with an input–output model for plastic dynamics:

(1)${\displaystyle \frac{\mathrm{}dn}{\mathrm{}dt}=a_{1}n+a_2{n}^2+a_{3}np-c_{1}kn}$ (2)${\displaystyle \frac{\mathrm{}dp}{\mathrm{}dt}=b_1-b_{2}p-c_{2}kp.}$

The structure of the model is summarized in Fig. 1. In the neuston dynamics Eq. (1), a₁ denotes neuston proportional population growth rate at low density (dimensions T⁻¹) and a₂ denotes the effect of neuston density on neuston proportional population growth rate (dimensions M⁻¹L²T⁻¹). We write the logistic neuston population growth equation as a second-order Taylor polynomial approximation around zero (Lotka, 1956, p. 65) with a₁ > 0 and a₂ < 0. In the absence of plastic and cleanup the population will increase when rare, and will have carrying capacity −a₁/a₂. The parameter a₃ denotes the effect of plastic on neuston proportional population growth rate (dimensions M⁻¹L²T⁻¹). The sign of this parameter is unknown: it is possible that plastic has a positive effect on neuston proportional population growth rate (for example, some forms of plastic may provide substrate for attachment of eggs of some neuston species) (Goldstein, Rosenberg & Cheng, 2012). The positive parameter k denotes the effort devoted to ocean cleanup, measured in some convenient way such as energy, money or area swept per unit time (denoted [effort]T⁻¹), and the positive parameter c₁ denotes the rate of neuston removal per unit effort of cleanup (dimensions [effort]⁻¹). We do not include an external input of neuston, as explained above.

In the plastic dynamics Eq. (2), the positive parameter b₁ denotes external input of macroplastics into the open ocean (dimensions ML⁻²T⁻¹), through routes such as transport from rivers via coastal waters (Lebreton, Egger & Slat, 2019). The positive parameter b₂ denotes the natural loss rate of macroplastics from the layer of the ocean affected by cleanup (dimensions T⁻¹). This is thought to occur mainly through fragmentation into microplastics (Koelmans et al., 2017; Lebreton, Egger & Slat, 2019). The positive parameter c₂ denotes the rate of macroplastic removal per unit effort of cleanup (dimensions [effort]⁻¹).

Full details of model analysis are given in the Supplemental Information Section S1.

Relationship between equilibrium scaled plastic and neuston densities under cleanup when plastic has no direct effect on neuston

We now make the simplifying assumption (justified in the ‘Assumptions and modelling approach’ section) that plastic has no effect on neuston proportional population growth rate (i.e., a₃ = 0) and study the relationship between scaled plastic and neuston densities at equilibrium, relative to their values in the absence of cleanup. We treat scaled plastic density as under our control through some management strategy that determines cleanup effort, and examine how this will affect neuston. Let n^∗ denote neuston concentration as a fraction of its equilibrium value in the absence of cleanup, and p^∗ denote plastic concentration as a fraction of its equilibrium value in the absence of cleanup.

Under the assumption of no plastic effect on neuston, we can write the scaled equilibrium neuston density as a function of scaled equilibrium plastic density: (3)${\displaystyle n^{\ast }\left(p^{\ast }\right)=max\left\{0,1-\left(\frac{1}{p^{\ast }}-1\right)\Pi \right\},}$

where the dimensionless parameter $\Pi =\frac{b_2}{a_1}\frac{c_1}{c_2}$ is the ratio of natural loss rate of macroplastics to neuston proportional population growth rate at low density, times the ratio of cleanup efficiencies. Thus a neuston population will be most affected if it has slow growth relative to the natural plastic loss rate (b₂/a₁ large), and if the cleanup strategy removes neuston at a high rate relative to plastic (c₁/c₂ large).

Parameter values

Here, we summarize the plausible ranges of the parameters b₂, a₁ and c₁/c₂ that we considered. Full details are given in the Supplemental Information. Estimates of the natural loss rate of plastic b₂ vary widely, with differences in model assumptions making an important contribution to this variation. We considered the range 0.03a⁻¹ to 1.26 a⁻¹ (throughout, we use a⁻¹ to denote units of per year). There is little information on proportional population growth rates at low density (a₁) for neuston. We therefore used an allometric approach based on body size, which suggested the range 1.08a⁻¹ to 63.52 a⁻¹ for small neuston species, and the range 0.08a⁻¹ to 4.75 a⁻¹ for large neuston species. Little is known about the efficiency of neuston removal relative to plastic removal (c₁/c₂). Since neuston and floating plastic overlap in size and occur in the same location, 1 is a plausible value for this ratio. However, other values are not implausible, and we therefore considered the range [1/10, 10].

Visualization of model behaviour

Equation (3) shows that the relationship between scaled equilibrium neuston density and scaled equilibrium plastic density is determined entirely by the dimensionless parameter $\Pi =\frac{b_2}{a_1}\frac{c_1}{c_2}$. We therefore calculated the range of possible values of Π for small and large neuston species from the ranges for b₂, a₁ and c₁/c₂ (‘Parameter values’ section). We plotted the envelope of possible relationships between the proportion of neuston remaining (n^∗) and the proportional reduction in plastic (1 − p^∗) for small and large neuston species. To understand how this relationship depends on the underlying parameters b₂, a₁ and c₁/c₂, we plotted the relationship between n^∗ and 1 − p^∗ for five logarithmically-spaced values of one parameter at a time, spanning the plausible range of values, and holding the other two parameters at their geometric midpoints. We show in Supporting Information S3, that effects of cleanup on neuston density are likely to occur on a time scale of months to decades after the start of a cleanup programme.