When are bacteria really gazelles? Comparing patchy ecologies with dimensionless numbers

Abstract

From micro to planetary scales, spatial heterogeneity-patchiness-is ubiquitous in ecosystems, defining the environments in which organisms move and interact. However, most large-scale models still use spatially averaged 'mean fields' to represent natural populations, while fine-scale spatially explicit models are mostly restricted to particular organisms or systems. In a conceptual paper, Grünbaum (2012, Interface Focus 2: 150-155) introduced a heuristic, based on three dimensionless ratios quantifying movement, reproduction and resource consumption, to characterise patchy ecological interactions and identify when mean-field assumptions are justifiable. We calculated these dimensionless numbers for 33 interactions between consumers and their resource patches in terrestrial, aquatic and aerial environments. Consumers ranged in size from bacteria to whales, and patches lasted from minutes to millennia, with separation scales from mm to hundreds of km. No interactions could be accurately represented by naive mean-field models, though 19 (58%) could be partially simplified by averaging out movement, reproductive or consumption dynamics. Clustering interactions by their non-dimensional ratios revealed several unexpected dynamic similarities. For example, bacterial Pseudoalteromonas exploit nutrient plumes similarly to Mongolian gazelles grazing on ephemeral steppe vegetation. We argue that dimensional analysis is valuable for characterising ecological patchiness and can link widely different systems into a single quantitative framework.

Keywords: comparative ecology; dimensional analysis; interactions; model systems; patchiness; scale.

FIGURE 1

Conceptual space‐time depictions of patchy consumer‐resource interactions. (a) Hypothetical interaction between consumers (solid coloured lines) feeding on resource patches (grey regions). As time (horizontal axis) progresses, consumers move on a simplified one‐dimensional spatial domain (vertical axis), seeking resource patches. When they enter a patch, they begin to deplete it, indicated by lightened shading to the right of consumer trajectories (i.e., after their passage). Black dots represent reproduction events, and hence the birth of new consumers. Brackets show patch duration (T) and the separation distance (L), as well as the time for consumers to locate a patch (T _search), reproduce (T _repro) and consume a patch (T _cons, assuming they are at a ‘typical’ density within it). The box in the upper right gives the ratios of the interaction's time scales as its dimensionless Frost (Fr), Strathmann (Str) and Lessard (Le) numbers. While this schematic shows a single value for each of these quantities, in nature they will of course be variable, so in practice, the three ratios are calculated using ‘characteristic’ order‐of‐magnitude values for L, T, T _search, T _repro and T _cons. Panels (b), (c) and (d) depict the different physical scales of various resource patches. (b) Forest stands, dune grass patches and whale falls persist for tens or hundreds of years and are spaced at kilometre scales. (c) Trees bear ripe fruit, dung piles remain favourable for dung beetles and newly settled cockles are small enough for crabs to eat for weeks, with separation scales of metres. (d) Phytoplankton blooms persist for days to weeks, while zooplankton and fish aggregations last from hours to days, with separation scales of km

FIGURE 2

The sizes of consumer and resource organisms, and the spatio‐temporal scales of resource patches, spanned many orders of magnitude. (a) Body size of consumer organisms (vertical axis) plotted against the size of an individual resource item (horizontal axis). Numbered labels refer to the interaction IDs in Table 3, each containing a consumer organism and a corresponding resource. (Note that the two bacterial interactions are excluded, since ‘organism size’ is poorly defined for a dissolved nutrient resource.) (b) Resource patches spanned approximately eight orders of magnitude in their spatial and temporal scales. Each point represents the resource patch within an interaction, labelled as in (a). The horizontal axis represents the typical distance scale separating patches in space, ranging from ~1 mm (microscale nutrient plumes, interaction 1) to ~250 km (caribou herds, interaction 27). The vertical axis represents the patches’ typical duration, ranging from ~5 min (seal groups, interaction 31) to ~1,500 years (whale falls, 8)

FIGURE 3

All consumer‐resource interactions plotted on logarithmic axes in dimensionless Frost‐Strathmann (a), Frost‐Lessard (b) or Strathmann‐Lessard (c) space, showing the relative importance of movement (Fr), reproduction (Str) and depletion (Le) in driving patch dynamics. In (a) and (b), two points, connected by a line, are plotted for each consumer‐resource pair: the lighter point is the Frost number assuming diffusive (i.e. random walk) movement, while the filled point is the Frost number for directed movement. The horizontal and vertical grey lines show the critical values, where Fr, Str and Le equal 1. The shaded regions on each plot indicate interactions which are not functionally patchy with respect to each pair of dimensionless ratios. For interactions in these regions, the mean‐field simplification may be (partially) justifiable. Point shapes and colours represent clusters of similar interactions identified via a k‐medoids analysis; see Figure 4 and text for details

FIGURE 4

Clustering based on the log‐transformed directed Frost, Strathmann and Lessard numbers identified five groups of dynamically similar consumer‐resource interactions. The three left columns in the heatmap (red–blue colour scale) show the logarithmic values of Fr, Str and Le for each interaction. The four right columns display the ecosystem in which the interaction takes place, the consumer and patch types and the log‐ratio of body mass between resource (R) and consumer (C). These latter four variables are shown for additional context but did not influence the clustering, which was driven entirely by Fr, Str and Le. Each cluster is labelled with a descriptive name; see text for details

FIGURE 5

Patch exploitation by the marine bacterium Pseudoalteromonas haloplanktis and the Mongolian gazelle Procapra gutturosa are dynamically similar, despite temporal scales that differ by a factor of ~10³ and spatial scales that differ by a factor of ~10⁶. (a) Conceptual space‐time diagram showing several bacteria (black lines) foraging for nutrient plumes from sinking phytoplankton cells (blue patches) in a simplified one‐dimensional seascape. Black dots mark cell divisions, and hence the birth of new bacteria. (b) Similar diagram showing hypothetical paths of several gazelles browsing on patches of steppe vegetation which bloom following localised rains. As in a), black dots mark births of new organisms. The spatial and temporal scales of the patches are approximately accurate for each system, as are the speeds and generation times of the organisms. Note that for simplicity, only a few trajectories are plotted; in reality, the density of both organisms within their patches would be significantly higher