2.1 Microscale dynamics

The IBM accounts for the main processes generally observed across zooplankton and prey types (individual birth and death, zooplankton feeding behavior) and an external flow that aims to mimic the effects of turbulence. Feeding determines prey death rate and zooplankton reproduction rate, and is influenced by the interplay between physics (i.e., flow) and ecology (i.e., predator swimming behavior). At the scale of the IBM, the turbulence flow is represented by a random velocity field that moves individuals; this flow tends to break correlations between prey and zooplankton. In the ocean, zooplankton can perceive visual and physico-chemical cues (e.g., hydrodynamic disturbances and chemical trails), to which they respond in different ways [30]. In our model, we generically captured feeding behavior by defining a perceptual range and two types of responses to cues within that range: active and passive. Active zooplankton individuals sense prey individuals within the perception range, swim towards them at a speed (if speed is strong enough for the predator to overcome the flow) and, with some probability [43] and in a negligible time [44], catch and ingest the target when the prey is within a catching range . Passive zooplankton individuals, on the other hand, do not swim (i.e., ) even if they might sense prey, but they ingest prey if the flow brings both prey and predator within catching range. Active predators in our model would thus resemble cruise feeders that intentionally swim towards prey, while passive predators would encompass zooplankton feeding modes that rely on currents to bring prey within catching range (e.g., ambush feeders) [30]. Finally, the frequent change in orientation and different perception modes observed for zooplankton allow us to assume, for both our active and passive grazers, a circular perception range (instead of a more realistic but technically challenging forward-facing wedge [45]). For further details, see IBM description in Fig 1A, Materials and methods, and S1 Supplementary materials, and parameter values in Table A (S1 Supplementary materials).

Although overall zooplankton behavior depends on various parameters (see above), here we focused on the interplay between flow and swimming behavior. To this end, we varied the small-scale turbulence intensity, , and the swimming speed . Note that, although swimming entails a metabolic cost, we did not include it in this version of the model for the sake of simplicity. We did so, however, in an extension of the model specifically devised to contrast costs and benefits of feeding behavior (see adaptive-feeding model in Materials and methods).

2.1.2 On the short timescale, the movement of active zooplankton induces microscale patchiness.

To understand the role of a given swimming speed in the formation of microscale patchiness we focused on the short timescales, where transport occurs in the absence of demographic processes. Thus, we assumed a constant density of prey and zooplankton (p and z, respectively), initially randomly distributed across space, and we explored a range of values for , leaving the other behavioral traits (e.g., perception range) constant. Fig 2A illustrates the contrast between the typical patterns that emerged, after a short transient, for passive ( cm/s) and active () zooplankton for low values of prey density. For the passive case, the turbulence-controlled transport maintained a random spatial distribution for both species. For active zooplankton, however, the directional movement towards prey increased the correlation between the location of zooplankton (red dots) and prey (green dots) and, unexpectedly, also among zooplankton individuals (for details, see species pair correlation function in Fig G). The latter correlation resulted from predators that were sufficiently close to each other and therefore partially shared the same cues, which led them to move in the same direction.

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Fig 2. Microscale patchiness emerging from the individual-level model.

A) Spatial distribution for prey (green dots) and predators (80%-transparent red dots) for densities (p,z) = (1.28,5); note that this example was chosen to illustrate the emergence of patterns, which are more pronounced at low prey densities. The left half of the picture shows the passive zooplankton case ( cm/s), and the right half the active zooplankton case ( cm/s), for easy visual comparison. B) The patterns from the scenario in panel A) are characterized by the encounter (E) and clumping (C) indices, shown here also for other zooplankton swimming speeds, . C) Encounter index as a function of prey density for different swimming speeds , showing that the benefits of active grazing with respect to the random case increase as prey density decreases. In both B) and C), dots represent the mean across replicates of values reached in the stationary regime, with crosses corresponding to the case depicted in panel A). Solid lines are given by Eq 1. All cases were obtained for moderate turbulence ( cm/s).

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To quantify the formation of microscale patchiness, we defined two spatial indices: the encounter index, E(t), related to prey-zooplankton encounters, and the clumping index, C(t), indicative of zooplankton aggregation (see Materials and methods). We defined both indices to be zero in the well-mixed (random) scenario and exhibit positive values in the presence of microscale patchiness (Eq 6). Figs HA and HB in S1 Supplementary materials illustrate how these spatial indices started at a zero value and, after a short initial transient ( s), reached (quasi-)stationary values, E and C, which are our values of interest henceforth.

These stationary values, which characterize the microscale patchiness for the given densities (p,z), depended on predator swimming speed (Fig 2B) and turbulence (Figs HC to HF). For high values of , at any fixed , turbulence generated displacements that were large relative to the zooplankton catching range, , which resulted in random mixing (, ). At moderate values of , E and C increased with and eventually reached a plateau when swimming offset turbulence, as predators could reach and stay close to selected prey targets (Fig 2B). At low turbulence, i.e., displacements smaller than the catching range, E and C increased with the swimming speed but showed a maximum at a finite speed (Figs HC and HD). For the remainder of our results, we focus on realistic speeds, which are below (see Materials and methods; we also explored what underlies this interesting peak via a simple model, see S1 Supplementary materials for details). All together, these results show that the tug-of-war between mixing (caused by the turbulent flow) and aggregation (caused by feeding behavior) determines species encounters (Figs HE and HF) and, consequently, potentially influences population-level interactions.

2.1.3 Microscale patchiness is density-dependent.

Because they result from the relocation of individuals, the patterns emerging from the short-timescale processes above influence species interactions and the dynamics of species densities occurring at the longer timescale. In turn, demographic changes locally add and remove individuals and therefore affect short-term spatial correlations, how many individuals are transported, and the directions that active zooplankton will move into on the shorter timescale. This feedback loop shapes the emergent microscale patterns and how they change over time.

To investigate whether and how prey and predator demography influences the dynamics of microscale patchiness, we considered a given turbulence-speed scenario, i.e., fixed , and monitored (p(t),z(t)) over time. Due to the timescale separation, individuals quickly rearranged in space between demographic events, which allowed us to define a stationary encounter and clumping index for each density, i.e., E = E(p,z) and C = C(p,z). We found that, in fact, both indices were significantly correlated only with prey density and not with zooplankton density, i.e., E = E(p) and C = C(p) (Figs I).

Fig 2C shows the emergent correlation between the encounter index and prey density under different swimming speeds. High prey densities led to frequent encounters with zooplankton similarly to the random encounters expected in the well-mixed scenario (i.e., as ); as prey density decreased, however, encounters became rare unless the predator actively swam towards prey, and therefore active feeding substantially increased encounters relative to the passive case. Active feeding also increased the fraction of time predators had prey within reach (see S1 Supplementary materials). Both the encounter and the clumping indices (Fig I) show that active feeding promotes aggregation at low prey densities, but does not provide any significant advantage at high densities.

From Fig 2C we infer that the encounter index scales as a power law with prey density and depends on both small-scale turbulence and zooplankton swimming speed, i.e.,:

(1)

where p₀ = 1 ind./cm² is a (fixed) scaling factor (see Fig 2C) and the coefficients and (unitless), both of which depend on turbulence and swimming speed , can be extracted by fitting Eq 1 to the simulation data in Fig 2C (see Fig J for the fitting procedure). The numerator, , increases with swimming speed but decreases with turbulence intensity, thus encapsulating the aforementioned tug-of-war between turbulence and feeding behavior (Fig JA). The exponent sets the degree of nonlinearity of the density-dependent encounter rates in Eq 1. Overall, could be approximated to (Fig JB), with deviations towards smaller values occurring only when the community is well-mixed (high and low ), a regime in which is negligible and, thus, . This nonlinearity and consequent saturation at low prey densities that emerges from zooplankton feeding behavior alone (Figs JC and JD) provides ecological support for the use of Type III-like functional responses in ecosystem models [46], which typically need to assume low-density saturation for zooplankton consumption to ensure the stability of simulations [47].

Because, instead of making heuristic arguments, we extracted the encounter rate from our simulation data, Eq 1 brings a substantial improvement over previous theoretical proposals [30]. Moreover, Eq 1 allowed us to identify how zooplankton encounters with prey are affected by the environment (i.e., turbulence and prey density) and feeding behavior (i.e., zooplankton swimming speed), key pieces of mechanistic information that allow us to derive next the population-level consequences of microscale patchiness.

2.2 Coarse-graining: From micro to macro

The high computational cost of the IBM makes it suitable only for small systems (on the order of decimeters) and, therefore, it cannot be used to address the kilometer-scale plankton patterns observed in the ocean. Thus, to study the macroscale consequences of the microscale patterns above and their underlying population-level interactions, we implemented a coarse-graining method.

The method utilizes the idea from Renormalization Theory [48,49] that only key components of the microscale dynamics remain relevant at coarser scales and thus are responsible for the resulting large-scale phenomena, as the effects of irrelevant components “average out” at such scales. In other words, a coarser description obtained by looking at the microscale dynamics at larger scales (by, for example, calculating densities at a larger spatial unit of observation) will ultimately only retain the key components of those dynamics. Thus, to obtain the density-field description, we first considered a large square system of linear size composed of smaller sections (pixels) of size L that were computationally suitable for the IBM. Each pixel was assigned a coordinate (i,j) within the larger lattice. and denote respectively the number of prey and zooplankton individuals within pixel (i,j), while and represent their corresponding densities. Thus, instead of tracking individuals within each (IBM-friendly) pixel, we calculated the corresponding pixel-specific densities.

Under this coarse description, we followed a computational approach that borrows concepts from probabilistic simulation methods [50–52] to understand the dynamics of such densities. Specifically, we tracked the changes in species density within each time interval, , for different turbulence-velocity scenarios . In addition to the IBM dynamics within each pixel, individuals can also move between adjacent pixels (either due to flow or behavior, following the IBM rules). Thus, we measured p and z at the focal and adjacent (top, bottom, left and right) pixels at the beginning of the time interval, with and representing increases and decreases in prey abundance within the time interval and and those for zooplankton. See Materials and methods for further details.

Because we tracked the fate of each individual in the whole system, we could keep track of the changes due to birth-death processes (B) and those due to transport (T), i.e., and . We then defined the rate at which density changes due to the former as and , and the latter as and . Importantly, the timescale separation between spatial movement and demographic events further resulted in a Markovian property of the coarse representation such that the changes in the densities of a given pixel, , could be determined by its current densities and those of its adjacent pixels [53]. For pixels large enough for microscale patchiness to emerge (i.e., L > 10 cm; see Fig L), these changes in density obeyed Poisson statistics (Figs MA and MB) with well-defined size scaling laws (Figs MC and MD) and mean determined by species densities, traits, and environmental conditions (Figs ME to MH). Altogether, we obtained a density field description for prey and zooplankton at time t and at every location, , within the larger system, where and translate from pixel integer coordinates to real location within the system:

(2)

These dynamics depend on the IBM parameters and pixel size L, the latter setting the spatial resolution of the description (see Materials and methods).

Macroscale spatial patterns are typically monitored at pixel resolutions for which the IBM is computationally intractable (e.g., for phytoplankton patterns in the oceans, resolutions range from 120m for the high-resolution satellite images from the SeaHawk Cube Sat mission [7] to 1km for the well-known MODIS database [8]. Therefore, in order to understand what our theory would predict when utilizing (or comparing with) available remote-sensing data, we needed to derive Eq 2 explicitly for large pixel sizes. To this end, we simulated the IBM for several sizes up to the upper computational limit (L = 100 cm) and used these simulation data to extrapolate the behavior to larger pixel sizes (see S1 Supplementary materials). This extrapolation was enabled by the well-behaved statistics described above (Figs MA and MB). Ultimately, for pixels of sizes comparable to the typical resolution of empirical data, this procedure yielded the following expressions:

(3)

The first term of each equation represents the reproduction and mortality of prey and zooplankton, respectively; these logistic and linear forms, derived using data from our mechanistic IBM, match the terms typically assumed phenomenologically in population-level feeding models. The last term in each equation accounts for the advection (i.e., directional movement) of individuals due to the flow at the scale of the pixel, which moves around the planktonic predator and prey at the same velocity as the turbulent flow associated with eddies, (see Materials and methods Eq 8). An additional feeding-related directional movement also exists but becomes negligible at the kilometer scale when compared to the advection caused by the flow; its demographic effects, however, remain present for all L, leading to the second term in both equations, where the density-dependent function g encapsulates the microscale patterns induced by active feeding.

As explained in S1 Supplementary materials, further steps in our coarse-graining procedure showed that the per-capita zooplankton feeding function depends on the encounter index E and the catching range ; specifically, . As Fig J shows, this density-dependent term is different from the Holling type I or II functional responses [54] typically used to represent feeding in phenomenological models [55]. For instance, when E = 0 (i.e., well-mixed scenario and/or passive zooplankton), the ingestion rate vanishes as , which allows the recovery of the local prey population. In contrast, when E > 0 (i.e., active zooplankton) feeding behavior compensates for low prey availability, which leads to high ingestion rates, prey population declines and, potentially, extinction.

Such different behaviors can have substantial consequences for the stability of the community dynamics, which can be illustrated by looking at the mean-field limit () of Eq 3. When feeding is passive, we recapitulate the results of the Lotka-Volterra model (Fig NA). However, when feeding is active, trajectories in the (p,z) space do not match the typical expectations for this classic model (compare Fig NA to panels B and C). Specifically, for active feeding we found neither periodic oscillatory dynamics when the carrying capacity was assumed infinite nor inward spirals (i.e., damped oscillations that lead to stationary values) when the carrying capacity was assumed finite. Instead, the data-derived density-dependent interaction term created outward spirals (i.e., unstable oscillations) that can ultimately lead to the extinction of the community. See full linear stability analysis in Figs ND and NE.

In summary, passive feeding leads to qualitatively different population-level dynamics than active feeding, which might ultimately impact the spatial organization of the community at the macroscale. All these results set us up to tackle our original questions: whether and how macroscopic spatial data can be used to identify behavior across space and time in realistic scenarios. With this in mind, in the next section we adapt Eq 3 to study macroscale prey pattern formation.

2.3 Macroscale dynamics

Oceanic blooms are a paradigmatic example of macroscopic phytoplankton patterns. Phytoplankton blooms refer to the massive cell reproduction event that occurs seasonally as days become longer but temperatures are not yet high enough to significantly stratify the water column and therefore nutrients are plentiful in the photic zone. This demographic explosion leads to bursts of biochemical transformations that punctually alter biogeochemical cycles [56]. Blooms therefore co-occur with strong spatiotemporal heterogeneity of abiotic factors at various scales (e.g., nutrient availability and/or large-scale turbulence [57]) which, in combination with dispersal by turbulence, generate complex large-scale spatial patterns of phytoplankton that can be monitored through the remote sensing of chlorophyll.

2.3.1 Grazer behavioral signatures in phytoplankton blooms.

To study whether zooplankton behavioral signatures can be identified in such a dynamic realistic scenario, we simulated upwelling events in the oceans, during which surges of nutrients lead to a wide range of densities for prey targeted by zooplankton. To this end, we modified Eq 3 in two ways. First, to account for the effects of large-scale eddies in the ocean, we added a point-vortex model (see Materials and methods) that set the net flow velocity for each pixel, , (see Fig E) with the intensity parameter, , controlling the angular speed of the eddies. Second, we introduced spatially explicit dynamics for the prey carrying capacity, , to model the heterogeneity of a growth-limiting nutrient that sets the maximum population density in the absence of predation. Specifically, the carrying capacity followed the equation:

(4)

where mimics a continuous input of nutrients occurring during the upwelling event that, if not depleted or perturbed by consumers, forms a horizontal band of width in the center of a 100-km squared domain (Fig OA). The parameter represents the typical time for any initial condition for to converge to the horizontal band in the absence of turbulence; we set days, per [34]. For negligible eddy effects, the velocity of the flow is and thus , but K will deviate from the imposed profile for moderate and strong eddies (Fig OB and first panel of Fig 3A). We initialized this dynamic carrying capacity as with the aim of understanding how prey dynamics and turbulence disrupt such a marked initial pattern (i.e., the band); for robustness, we also explored other carrying capacity profiles (Figs OC and OD), which did not alter our conclusions.

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Fig 3. Zooplankton behavioral signatures in macroscale patterns of prey.

A) Spatial patterns for carrying capacity, prey, and zooplankton (respectively, from left to right) in our simulated upwelling event. Color indicates densities in ind./ml. B) Curvature-density relationship for different swimming speeds, . C) Pattern sharpness (i.e., 1 - slope of curvature) as a function of , with colored dots referring to the cases in panel B). Insets highlight typical patterns for cm/s and cm/s (the latter corresponding to the dashed square in the prey pattern from panel A)). D) Behavioral signature, defined as difference in sharpness relative to the passive case, , for different turbulence levels, . For these simulations, the total domain size was km (100 cells of size L = 120 m) and a horizontal carrying capacity band, , of length 12 km, width m, and amplitude K₀ = 100 was placed around the middle of the vertical axis. We adjusted eddies to introduce a mean flow speed km/day ( day⁻¹) and fixed cm/s to keep the diffusion and advective timescale comparable to experimental values [34,42]. The snapshots used for these figures were taken close to the peak of prey density during the upwelling event (approx. t = 12 days, as seen in Fig PA). We initialized the system using low, homogeneous prey and zooplankton densities (1 and 10⁻² ind./cm², respectively). See Table B in S1 Supplementary materials for parametrization.

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With this setup, we integrated numerically our complete large-scale model following standard schemes (see S1 Supplementary materials) and studied how the spatio-temporal prey pattern depends on predator swimming speed, with the goal of discerning signatures of feeding behavior. In all cases, we initialized both zooplankton and prey populations using small densities. Regardless of the (fixed) value, the initially negligible top-down regulation allowed prey to grow rapidly to a peak of high density (Fig PA and prey panel in Fig 3A). The prey surge then facilitated the growth of predator density (zooplankton panel in Fig 3A), which consequently led to a decrease in prey density (Fig PA). Consistent with our mean-field findings, passive feeding led to stable coexistence of prey and predators, whereas active feeding ultimately led to the extinction of prey.

As this realistic and expected sequence of events unfolded, zooplankton dynamically shaped the emerging prey patterns. Importantly, the wide range of prey densities produced in space and time by these events offered an opportunity to investigate whether the density-dependent feeding effects described in the previous section were present in these macroscale spatial patterns. Based on our results in Fig 2C, we hypothesized that behavioral signatures emerge and should become more evident closer to the expanding edges, where prey density is lower. Specifically, we predicted that active feeding should produce a more noticeable contrast between the interior of the prey patterns and the borders (i.e., sharper boundaries) than passive feeding. Although visually that seemed to be the case (see, e.g., Fig PB), we also quantified the morphological differences caused by different swimming speeds by measuring the dependence between the curvature of the pattern and prey density [58]:

(5)

The function provides a proxy for the curvature of the prey density field p because, for a given location , quantifies density differences in any direction of the two-dimensional system. For smooth patterns, curvature vanishes as prey density asymptotically goes to zero. For sharp boundaries, in contrast, prey density abruptly drops to zero and, consequently, the curvature takes a non-zero value (see Materials and methods for further details and examples).

Using Eq 5, we measured curvature at any given time and every location in our simulated patterns, and extracted a relationship between curvature and prey density. We observed that this relationship approximated a power law throughout the duration of the upwelling event, although its slope varied with time (compare t = 2,12,14 in Fig PC). This power law emerged from the interaction with zooplankton and, at any given time, its slope reflected their feeding behavior (i.e., the magnitude of ; see Fig 3B and Fig PC). In particular, the slope captured clearly the difference between active and passive predators, which we traced to differences in swimming behavior rather than in overall increase in ingestion (passive zooplankton with a higher ingestion rate cannot induce similar signatures, see Fig PD). In other words, this finding confirmed our hypothesis that feeding induces behavioral signatures in prey macroscale patterns, and that these signatures can be inferred by measuring the curvature of boundaries or, more specifically, the difference in its slope. Thus, to quantify the sharpness of the boundary at a given time we used the slope of the curvature-density relationship: , where is the pattern curvature averaged over pixels with prey density p.

With the expression above, we could infer a sharpness for any given swimming speed, (see slopes in Fig 3B, and summary panel for as a function of and time in Fig PE). An ideal observation period encompasses the peak of prey density during the upwelling event because that window exhibits the widest range of densities that result from model dynamics. We indeed observed fingerprints of feeding behavior in sharpness (Fig 3C), as showed a very low value for simulations with passive zooplankton (), and increased smoothly but steeply when we instead considered active zooplankton (). A similar phenomenology was observed at other times across the duration of the upwelling event (Fig PE). Therefore, sharpness reflects behavioral signatures, and this abrupt change in value provides a way to discern between passive and active predators. The ability to notice these differences depended on turbulence: as one might expect, at high turbulence (large ) the strong eddies blurred behavioral signatures (Fig 3D).

2.3.2 Analysis of real blooms.

Motivated by our findings above, we analyzed the sharpness of real blooms across the globe in search of behavioral signatures (see example in Fig 1B and snapshot locations in Fig SA). Specifically, we processed high-resolution images [7] and, assuming a reasonable conversion factor between chlorophyll-a and phytoplankton density [59,60], we obtained the corresponding curvature-density relationship (see Materials and methods for the protocol that allowed us to select snapshots around the peak of blooms). Whether we look at one or all locations, the data reveal a richer scenario than the one produced by our model for any given . Differently from, e.g., Fig 3B, real-world blooms exhibit a curvature function with a density-dependent slope, which yields a density-dependent sharpness function that transitions abruptly from to as phytoplankton density decreases (black dots Fig 4A). Since our theoretical results indicate that active and passive behavior are associated with high and low sharpness, respectively, the real-bloom data suggest the existence of behavioral segregation, with passive grazers dominating at high densities and active grazers at low densities.

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Fig 4. Spatial niche partitioning.

A) Curvature-density relationship extracted from our adaptive-feeding model with cm/s (colored striped band) and from SeaHawk data (dots and error bars considering blooms across different locations and time of the year, see Table C in S1 Supplementary materials). The simulation estimates result from averaging over 50 snapshots (to account for the different sources of variability such as initial condition and eddy distribution) taken within a 4-day time-window right after the upwelling event; the vertical line indicates the density threshold identified in panel A for , and colors (see colorbar in panel D) indicate the dominant behavior predicted by the adaptive-feeding model for each prey density. B) Relative difference, , between fitness for active (, with f₁) and passive (, with f₀) zooplankton for different swimming speeds. Vertical dashed lines highlight, for each swimming speed, the prey density threshold below which being active is advantageous (i.e., value of p below which ). C) Prey density and D) zooplankton behavior patterns for the adaptive-feeding model with cm/s, for a snapshot close to the peak of the upwelling event. To match realistic concentrations, we set and remaining parameters as in Fig 3A. For all panels, q = 4, day⁻¹, .

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2.4 Conclusions

Our multiscale framework provides a previously computationally inaccessible bridge across scales, from the micro- and millimeter-scale of individuals to the centimeter- and meter-scale of species interactions, and ultimately to the kilometer-scale of macropatterns. Built on a mechanistic individual-based model and scaled via a rigorous coarse-graining procedure, our framework is tractable without having to sacrifice crucial biological detail. This allowed us to understand how prey spatial patterns at different scales may be regulated by zooplankton behavioral traits.

At the microscale, we found that patchiness is characterized by two emergent spatial features: an increase in interspecific encounters and an aggregation of zooplankton, both of which are regulated by turbulence, predator swimming speed, and prey density. These findings bring mechanistic insight to existing empirical work. Our framework provides theoretical support for the field and experimental studies on the impact of turbulence on zooplankton feeding behavior and capture success, which have shown that encounters with prey are affected positively by active swimming and negatively by turbulence [30,72,73]. Additionally, previous work observed that reproducing the different capture success at low densities of passive and active feeding behavior requires the use of qualitatively different feeding response functions [61,74]; in our framework, such functions emerged from the analysis of our IBM, thus allowing us to trace mechanistically their origin. Further, our results suggest that the small-scale zooplankton aggregation and its negative dependence on turbulence observed via towed video microscopy [75] can be explained simply from emergent, indirect correlations between zooplankton individuals in active search and feeding of prey; previous models assumed ballistic or random zooplankton swimming behavior, and thus could not reproduce this pattern, or they did so only after invoking attractive social forces between zooplankton individuals (e.g., [31]).

Analyzing simulation data statistically to infer population-level expressions enabled a rigorous mapping that connected scales. Thus, our coarse-graining method allowed us to obtain a density field description that was fully parameterized by species traits and the physical environment. Because, according to Renormalization Theory, “irrelevant” details are lost during coarsening, these equations represent the fundamental mechanisms underlying the interactions between zooplankton and their prey, with density-dependent feeding playing a particularly relevant role in the emergent large-scale pattern. Previous attempts to connect individual and population levels either estimated population-level encounter rates by, for example, resorting to heuristic, geometry-based, arguments [30]; calculated feeding flow using hydrodynamical models [36,76]; or developed analytical approximations to account for attraction-repulsion between individuals [31]. However, in those approaches the effects of collective emergent phenomena (e.g., density-dependent encounters) were not accounted for, mostly due to the intrinsic complications when dealing with spatially-extended systems [77,78]. Therefore, existing models could not provide a mechanistic link between individual and population levels.

Overcoming this theoretical “problem of scale” allowed us to access valuable ecological information about plankton communities across scales. In particular, we were able to map the curvature of macroscale prey patterns to the swimming speed of the grazing zooplankton, which revealed spatial changes in dominant zooplankton behavior driven by the relative costs and benefits of passive versus active feeding. Our approach thus provides what, to our knowledge, is the first bottom-up framework to identify predator behavior from macroscale prey patterns. Establishing a two-way connection across scales could, for example, be used to monitor which zooplankton (or, at least, which feeding behavior) dominate during the different stages of an upwelling event at given spatial locations. This information can be helpful for understanding not only the dynamics of the microbial loop and oceanic biogeochemistry, but also any associated ecosystem services (e.g., fisheries with preference for particular zooplankton types). Note that, although nutrients and other sources of regulation, such as viruses, are also expected to affect the shape of oceanic macroscopic patterns such as those associated with upwelling events, the robustness of our results against, e.g., different nutrient spatial profiles emphasizes the importance of feeding behavior as a key driving factor for the curvature of the pattern.

We focused here on the perception and active swimming of a generic zooplankton to demonstrate the power of a framework that accounts for individual-level detail; in the future, extensions of our framework could easily be made to incorporate more specific aspects of feeding that have been shown to be important for pattern formation at small scales, such as social interactions [31] and gyrotaxis [27]. A further developed version of our framework that captures how behavior varies across taxa could explain mechanistically patterns of structural variability that have been recently cataloged [12]. Because the expectation is that changes in environmental conditions will alter feeding behavior [61], our framework provides a diagnostic (if not predictive) tool to identify those behavioral changes through the analysis of macroscale patterns. For example, climatic changes affecting zooplankton physiologically and/or the strength of turbulence will alter the costs and benefits of active feeding: if, for instance, short- or long-term climatic changes increase turbulence, higher mixing will make the investment in active feeding less beneficial for zooplankton, and therefore curvature will transition from non-zero to zero slope at smaller p values. This means that sharp edges will be found only in more prey-depleted areas of a pattern, and therefore overall patterns will look more generically smooth. Nonetheless, although our methodology suggests a non-invasive way to make inferences regarding prey and zooplankton interactions and behavior (that is, by measuring the sharpness of remote-sensing images of prey macroscopic patterns), the validation of such inferences will still require a combination of such remote-sensing images with species-specific parametrization of behavioral and physiological responses to environmental conditions [61,79].

Beyond the marine context, our computational multiscale framework can be easily applied to and be useful when analytical and computational approaches to scale up individual-level behavior are challenging: for example, when organisms are moving based on complex decision-making strategies [80] that account for the different costs and benefits of individual actions (e.g., fish schools and starling flocks aiming to avoid predatory attacks) [81]. For such cases, coupling with our coarsening method either a mechanistic IBM tailored to the focal system or empirical individual-level data enables an exact and mechanistic connection between scales. This approach provides an alternative to the strong mathematical barriers that arise in analytical methods [31,77] with which to check for potential behavioral signatures in large-scale patterns.

3.1 Individual-based model simulations

Individuals exist within a square domain of size L with periodic boundary conditions (i.e., torus topology). The individual-based model (IBM) accounts for prey and zooplankton birth-death processes, zooplankton feeding behavior, and the influence of an external flow (Sect 2.1). We implemented these processes using a time-dependent Gillespie algorithm [82] for the birth-death events, in combination with a continuous advection process (see S1 Supplementary materials for detailed description).

We parameterized our model using values that are generically applicable to zooplankton and their prey (see Table A in S1 Supplementary materials). For the former, we aimed to capture the range of behaviors observed from microzooplankton to fish larvae, which is therefore also reflected in the choice of parameter values for the predator and the prey. Thus, we set the prey reproduction rate and the zooplankton mortality rate . For computational convenience, we set the prey carrying capacity at K = 100 individuals/ml. Zooplankton catching range, , and perceptual range, , were set as a function of body size (i.e., the effective radius, set to for our generic zooplankton [30]) as , . For the chosen body size, zooplankton velocities can reach values around cm/s [30], which will be our ‘focal’ feeding speed for active zooplankton.

With this parameterization and considering a search rate , ingestion rates ranged from a couple of prey per minute to a prey individual every few seconds depending on prey availability, which is within observations (see experimental reference values in, e.g., [83,84]). For more details see Fig F. Other parameterizations did not alter qualitatively our conclusions (see Table A in S1 Supplementary materials).

3.2 Spatial characterization of the microscale patchiness

To quantify the interspecific correlations of the emergent patterns, we monitored through time the number of prey that were accessible to each zooplankton individual (i.e., within its catching range, ). For a fixed pair of prey and predator densities, (p, z), the per-predator average number of potential targets reached a constant value after a short transient (Fig HA). Analogously, to quantify predator aggregation we measured the stationary value of the number of zooplankton individuals that were within a focal predator’s catching range, (Fig HB). Using these two measures, we calculated the encounter index E, which quantifies how much prey availability there is per predator, and the clumping index C, which quantifies predator aggregation, both relative to the values expected for the well-mixed scenario [31]:

(6)

where the denominators represent the expected prey targets and zooplankton individuals, respectively, for the well-mixed case. With the definitions above, E = C = 0 when the community is well-mixed (i.e., individuals are randomly distributed in space); a positive spatial correlation between prey and zooplankton is indicated by E > 0; and correlation among zooplankton, by C > 0.

3.3 Point-vortex model

In order to account for the key elements of turbulence at large-scales, we introduced a fixed number of vortices (or eddies) in our spatial domain. Setting the variability of the size and rotation direction of the eddies in a realistic way leads to a stirring process with scale-free velocity spectrum [34,85]. We used the following stream function to capture these features:

(7)

where controls the angular speed of eddies (in units of day⁻¹), sets the rotation direction, while and are the radius and center of each eddy i (for ). Due to the imposed periodic boundary conditions, the influence of the vortices looped around the domain, which was accounted for by the sum over n; since this effect decays exponentially fast, however, considering only already achieved a precision . The flow velocity, , is the curl of the stream function:

(8)

We sampled vortex sizes from a power-law probability distribution, (for ) and obtained a scale-free transversal velocity spectrum , where k is the spatial frequency (Fig E). This probability distribution generated a power-law spectrum with exponent , characteristic of geostrophic turbulence occurring at large scales [34].

3.4 Adaptive-feeding model

The adaptive-feeding model modifies Eq 3 to account for a metabolically mediated switch between passive and active behavior. Specifically, the model introduces two modifications that aim to reflect the costs and benefits of each of the two feeding strategies.

The first change consists in replacing the feeding function, g(p), by:

(9)

where c₀ > 1 reflects the reduced energy expenditure and reduced risk of feeding-related predation associated with passive feeding [63,86]. To assess the sensitivity of our findings to this new parameter, we explored a broad range for , which can result from a variety of factors; for example, passive zooplankton are 2–8 times less likely to be predated upon than active zooplankton [87], which motivated the moderate value c₀ = 10 chosen for illustration purposes in Fig 4A [88].

The second change consists in modifying the mortality rate m for active predators which, since the term reduces growth, can be seen also as a proxy for metabolic costs:

(10)

where q is a fixed hydromechanical factor and reflects the energy dissipated due to drag, consistent with Stokes’ Law [63].

The behavioral switch is controlled by the relative difference in associated “fitness” , where and are simply defined as the ratio between the benefits and costs of the corresponding behavior [63].

Altogether, these modifications extend our previous model by introducing a trade-off between the two behaviors: the energy-saving but potentially unproductive passive behavior versus the expensive but targeted active behavior.

3.5 Measuring the curvature-density relationship

For simulations, we selected hourly snapshots following the peak of prey density during the upwelling event. Immediately after the peak is when the widest range of prey densities are observed, and feeding starts significantly shaping the (very abundant) prey community. For these reasons, and although the curvature curves looked approximately as power laws at any time along the duration of the upwelling event, our analysis of the curvature-density relation (Eq 5) focused on averages obtained within 4 days following the peak.

For satellite images, we first transformed chlorophyll-a observations into phytoplankton density, assuming a ratio [Chl-a mg/m³]:[ind./ml]=1.25, which in turn assumes averaged values for carbon content ( mg per cell) [60] and a chlorophyll-to-carbon conversion factor ( gChla/gC) [59] across phytoplankton species. This transformation establishes a common quantitative basis to compare the curvature–density relationships obtained from modeled population density and chlorophyll-a concentration, without affecting the qualitative features relevant for behavioral inference. Then, we obtained the distances between neighboring pixels by means of a geodesic under the WGS84 ellipsoid, which is necessary to calculate the spatial derivatives involved in the curvature expression (Eq 5). These spatial derivatives were calculated using a linear interpolation of the non-uniform grid. To ensure a wide range of phytoplankton densities, we selected snapshots provided by satellite sensors [7] at locations across the globe where chlorophyll-a activity was consistent with a bloom, favoring observations clear of clouds. Specifically, we selected locations where a spring-summer or fall-winter bloom peak was (or had recently) occurred, which typically led to Chlorophyll-a readings ranging from the low to the high extremes of the detection limit.

For each snapshot (either from simulations or real blooms), we obtained the associated local curvature and density for every pixel; we used a log (base 10) scale, , to better elucidate a potential linear (or non-linear) relationship. We then binned the data according to prey density, and obtained the expected value of the logarithm of the curvature within each bin, generating the curve vs. p (Fig PC).

For smooth patterns, the profile becomes flat as prey density goes to zero at the borders, and thus the curvature vanishes; for example, if we had an ideal circular pattern and the boundary of the pattern were described by the smooth function , where r represents the distance to the center of the pattern, the associated curvature would be , which vanishes as , exhibiting zero sharpness, . For sharp boundaries, in contrast, the density profile drops abruptly to zero, and concurrently so does the curvature to a non-zero value; as an example, a sharp radial density profile given by if r < 1 and p = 0 if r > 1 leads to a constant curvature yielding even at the pattern boundary, where .