The logic of ecological patchiness

Abstract

Most ecological interactions occur in environments that are spatially and temporally heterogeneous-'patchy'-across a wide range of scales. In contrast, most theoretical models of ecological interactions, especially large-scale models applied to societal issues such as climate change, resource management and human health, are based on 'mean field' approaches in which the underlying patchiness of interacting consumers and resources is intentionally averaged out. Mean field ecological models typically have the advantages of tractability, few parameters and clear interpretation; more technically complex spatially explicit models, which resolve ecological patchiness at some (or all relevant) scales, generally lack these advantages. This report presents a heuristic analysis that incorporates important elements of consumer-resource patchiness with minimal technical complexity. The analysis uses scaling arguments to establish conditions under which key mechanisms-movement, reproduction and consumption-strongly affect consumer-resource interactions in patchy environments. By very general arguments, the relative magnitudes of these three mechanisms are quantified by three non-dimensional ecological indices: the Frost, Strathmann and Lessard numbers. Qualitative analysis based on these ecological indices provides a basis for conjectures concerning the expected characteristics of organisms, species interactions and ecosystems in patchy environments.

Keywords: consumer–resource interactions; ecological model; foraging behaviour; patch dynamics; scaling analysis.

Figure 1.

Heuristic scaling of interactions between consumers and patchy resources, in terms of characteristic separation distance between patches, L, and patch duration, T. The critical Frost number, ℱr_critical ≈ 1, quantifies mobile consumers' abilities to move to transient resource patches (table 2). (a) On a plot of log₁₀ L versus log₁₀ T, this threshold translates to lines with slopes 2 and 1, respectively, for consumers with diffusive or directed foraging movements. For a given foraging movement type, patches above the corresponding line are accessible by movement, while those below the line are not. Cross-hatched areas indicate the consequences for patch accessibility of variations in foraging strategy: behavioural changes, e.g. from diffusive to Lévy or directed searches, shift limits from the upper edges of hatched areas to lower edges. Examples are shown corresponding to several hypothetical organisms (bacterium, s = 20 m s⁻¹, τ = 0.5 s; ciliate: s = 200 m s⁻¹, τ = 5 s; copepod: s = 1 cm s⁻¹, τ = 2 s; seabird: s = 10 m s⁻¹, τ = 60 s) illustrating how patches with particular length and timescales L and T may be accessible by movement to some consumers but not others. (b) The critical Strathmann number, 𝒮tr_critical ≈ 1, quantifies consumers' abilities to reproduce explosively within transient resource patches. Consumers with ℱr ≫ 1 and/or 𝒮tr ≫ 1 are likely to be aggregated inside patches, while other consumers are not.

Figure 2.

Schematic of species interactions in a patchy resource environment. (a) For the focal species, grey indicates ℱr ≫ 1, the L–T region where the focal consumer can move to resource patches. Horizontal hatching indicates 𝒮tr ≫ 1, the L–T region where the focal consumer can reproduce in resource patches. Within the union of these two regions, patchiness increases access to resources for the focal consumers compared with a spatially uniform resource distribution; elsewhere, patchiness reduces availability. Vertical hatching indicates ℒe ≫ 1, the L–T region where the focal consumer can remove resource patches. Within the intersection of this region with the ℱr ≫ 1, 𝒮tr ≫ 1 region the focal consumer can be a dominant competitor; outside this intersection, exclusion of other consumers is unlikely. (b) A conjectured requirement is that to persist within this union despite competition from the dominant focal consumer, a subdominant consumer must have higher ℱr (i.e. a shorter T_search, indicated by ℱr_critical occurring at lower L and T) and/or higher 𝒮tr (i.e. a shorter T_reprod, indicated by 𝒮tr_critical occurring at lower T but lower ℒe (i.e. a longer T_consumpt, indicated by ℒe_critical occurring at higher T).