The impact of rheotaxis and flow on the aggregation of organisms

Abstract

Dispersed populations often need to organize into groups. Chemical attractants provide one means of directing individuals into an aggregate, but whether these structures emerge can depend on various factors, such as there being a sufficiently large population or the response to the attractant being sufficiently sensitive. In an aquatic environment, fluid flow may heavily impact on population distribution and many aquatic organisms adopt a rheotaxis response when exposed to a current, orienting and swimming according to the flow field. Consequently, flow-induced transport could be substantially different for the population members and any aggregating signal they secrete. With the aim of investigating how flows and rheotaxis responses impact on an aquatic population's ability to form and maintain an aggregated profile, we develop and analyse a mathematical model that incorporates these factors. Through a systematic analysis into the effect of introducing rheotactic behaviour under various forms of environmental flow, we demonstrate that each of flow and rheotaxis can act beneficially or detrimentally on the ability to form and maintain a cluster. Synthesizing these findings, we test a hypothesis that density-dependent rheotaxis may be optimal for group formation and maintenance, in which individuals increase their rheotactic effort as they approach an aggregated state.

Keywords: Keller–Segel; biological aggregations; chemotaxis; currents; rheotaxis.

Figure 1.

Schematic illustrating the principle components of the model. (a) Under chemotaxis alone, organisms are directed towards high concentrations of a secreted aggregating cue. (b) Under positive rheotaxis, organisms are transported by the flow, but also orient and swim against the current. (c) Under both rheotaxis and chemotaxis, the organisms must balance their overall movements according to both the attractant distribution and the flow field.

Figure 2.

Aggregation in a uniform flow. (a) Autoaggregation parameter space (α - U), with parameters for plots in (b,c) indicated by dot. Note that here we assume negligible population growth (ρ = 0). (b) Autoaggregation in a still environment, ω = 0. Population density u represented as (b)(i) space–time map (white u(x, t) = 0, black u(x, t) ≥ 4U) and snapshots in (b)(ii–v). (c) As (b), but under constant flow (ω = 1). (d) Autoaggregation parameter space (α - ϕω, coloured region) under rheotaxis and flow (zero population growth, ρ = 0). Colour (see inset of (e)(i)) indicates the cluster growth rate predicted by LSA (deep red = faster growth). Space–time density maps for marked points: (d)(ii) α = 60, ω = 1, ϕ = 1; (d)(iii) α = 60, ω = 5, ϕ = 1. (e) As (d) but including population growth (ρ = 1). (e)(ii) α = 150, ω = 1, ϕ = 1; (e)(iii) α = 150, ω = 5, ϕ = 1. For all simulations, we have set initial densities as dispersed (u₀(x) = U, v₀(x) = U + ε(x), for small random (specifically, random white noise) perturbations ε(x)). Non-specified parameters are U = 0.05 and δ = 1. Details of the LSA are provided in the appendix.

Figure 3.

(a,b) Accelerated unification through the action of rheotaxis. (a) Space–time population density maps showing cluster evolution under a constant flow (ω = 1) and (i) negligible (ϕ = 0), (ii) under-compensating (ϕ = 0.5), (iii) compensating rheotaxis (ϕ = 1). (b) Boxplot showing the mean time to reach a unified cluster (averaged across 100 randomized initial data). For (a,b), initial distributions are as in figure 2. (c) Evolving density for a population initialized as three separated Gaussian shaped clusters of masses m = 0.5, 0.75 and 1, centred at x = 40, 10 and 70, respectively. (d) Computed cluster speed, c, as a plot of rheotaxis strength, ϕ, for isolated clusters of mass m; dotted vertical line indicates compensating rheotaxis and horizontal solid line indicates holding position. (e) Travelling pulse profile for a cluster of mass (i) m = 0.5 and (ii) m = 4 under compensating rheotaxis. Density u(x, t) shown at t = t*, t* + 10, t* + 20, with arrow indicating movement direction. In (a–e), ρ = 0, δ = 1 and α = 80.

Figure 4.

Dynamics under non-uniform flow. (a) The flow field, w. (b–e) density u (colourscale in (b)(i)). (b) Excluding chemotaxis and rheotaxis (α = ϕ = 0), the population stays unclustered, though non-uniform flow leads to moderate accumulations. (c) Under weak chemotaxis and no rheotaxis (α = 10, ϕ = 0), flow-induced accumulations form into clusters that are sustained following removal of the vortex flow. (d) Addition of rheotaxis (α = 10, ϕ = 1) suppresses the clusters that formed in (c). In (a–d), the population is initially dispersed, u₀(x) = U, v₀(x) = U + ε(x). (e–g) Bifurcation diagrams of numerically determined steady states, u_s(x), represented via $\hat{u}={\text{max}}_{x\in \Omega }(u_s)-{\text{min}}_{x\in \Omega }(u_s)$. Bifurcation parameter is α. Solid branches indicate numerically stable steady states, dotted lines indicate unstable steady states. Representative steady states for the locations indicated by a square shown in below panels. (e) Uniform flow, no rheotaxis (ϕ = 0). (f) Uniform flow, interrupted by a region of slower flow, no rheotaxis (ϕ = 0). (g) As (f), but with rheotaxis (ϕ = 1). Simulations in (a–d) use $\Omega =[-20,20]\times [-20,20]$ and w(x, y, t) = (1, 0) + 0.005(− x − 10y, 10x − y)[tanh (t − 100) − tanh (t − 300)][1 − tanh (0.1x² + 0.1y² − 10)]. Simulations in (e–g) use $\Omega =[-50,50]$ and w(x) = 1.0 + ɛ(1.0 − tanh (x) + tanh (x − 10)), where (e) ɛ = 0, (f–g) ɛ = 0.01. In all simulations, U = 0.05, δ = 1 and ρ = 0.

Figure 5.

Non-uniform, non-constant flows disintegrate clusters, but they can be stabilized by rheotaxis. (a–c) Non-uniform, non-constant flow and zero rheotaxis (ϕ = 0). Each frame shows the instantaneous flowfield (arrows), population density (colourscale) and initial cluster location (dotted blue circle) under: (a) weak flow, ω = 1; (b) moderate flow, ω = 2; (c) strong flow, ω = 4. (d) Non-uniform, non-constant flow and compensating rheotaxis (ϕ = 1), under strong flow, ω = 4. (e) Non-uniform, non-constant flow and undercompensating rheotaxis (ϕ = 0.75), under strong flow, ω = 4. The final column plots two measures: the proportion clustered (PC) and the proportion localized (PL), both normalized against their values at t = 0. Solid bar in top left panel represents a length scale of 10. avg shows densities averaged across time from t = 0 to 80. Initial circular cluster centred on (0, 0) and given by u₀(x, y) = v₀(x, y) = 0.5(1 − tanh (10(x² + y² − 25))) and we set ρ = 0, δ = 1, α = 15 (weak chemotaxis regime). $PC(t)={\int }_{\Omega }H(u(\mathbf{x},t)-4U)u(\mathbf{x},t)\hspace{0.17em}\mathrm{d}\mathbf{x}$, where $\Omega$ is the full spatial region and H( · ) denotes the Heaviside function. $PL(t)={\int }_{{\Omega }_i}u(\mathbf{x},t)d\mathbf{x}$, where ${\Omega }_i$ denotes the region inside the dotted blue line. Flowfield uses the hycom dataset described in methods.

Figure 6.

Dynamics of an initially dispersed populations exhibiting density-dependent rheotaxis. Each panel shows the instantaneous flow field (arrows) and population density u, (colourscale: top right). Strong flow from figure 5. Rheotaxis coefficient ϕ = ϕu^p/(κ^p + u^p), for: (a) κ = 0 (constant compensating rheotaxis); (b) κ = 1; (c) κ = 4; (d) κ = ∞ (zero rheotaxis). Initial distributions u₀(x) = U, v₀(x) = U + ε(x) and we use ϕ = 1, p = 2, ρ = 0, δ = 1, U = 0.05, α = 15 (weak chemotaxis regime). Flowfield uses the hycom dataset described in methods.

Figure 7.

Neutral stability curves (NSC), indicating the range of unstable wavenumbers for particular (ω,ϕ) combinations: non-zero length ranges indicate the possibility of autoaggregation from the USS. (a) Zero population growth (ρ = 0), for three choices of α. (b) Logistic population growth (ρ = 1), for three choices of α. Under zero population growth a non-zero length range is found for all (ω,ϕ) combinations, although of restricting length as these parameters increase. Including population growth leads to a critical threshold, such that for large flow/rheotaxis autoaggregation is suppressed. Non-specified parameters are δ = 1 and U = 0.05.