Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds

Abstract

Young mussel beds on soft sediments can display large-scale regular spatial patterns. This phenomenon can be explained relatively simply by a reaction-diffusion-advection (RDA) model of the interaction between algae and mussel, which includes the diffusive spread of mussel and the advection of algae. We present a detailed analysis of pattern formation in this RDA model. We derived the conditions for differential-flow instability that cause the formation of spatial patterns, and then systematically investigated how these patterns depend on model parameters. We also present a detailed study of the patterned solutions in the full nonlinear model, using numerical bifurcation analysis of the ordinary differential equations, which were obtained from the RDA model. We show that spatial patterns occur for a wide range of algal concentrations, even when algal concentration is much lower than the prediction by linear analysis in the RDA model. That is to say, spatial patterns result from the interaction of nonlinear terms. Moreover, patterns with different wavelength, amplitude and movement speed may coexist. The results obtained are consistent with the previous observation that self-organization allows mussels to persist with algal concentrations that would not permit survival of mussels in a homogeneous bed.

Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems

“…We therefore based our analysis on estimates that are, for as much as possible, based on experiments in case of mussel movement, and literature evidence in case of growth and mortality processes. Moreover, we provide extensive bifurcation analyses of the models of small-scale and large-scale self-organization in previous papers 13,14,20,25,34,35 .…”

“…domain allowing large-scale pattern formation (changing of the parameter, V or D m is possible 34,35 ). Finally, a completely homogeneous state was obtained by setting both individual mussel movement (b) and sediment diffusion (D S ) to the above-mentioned values.…”

History-dependent patterns of whole ecosystems

“…Note that the information on pattern existence (but not stability) in Fig. 4 has been presented previously by Wang et al (2009). As well as forming one boundary of the parameter region giving patterns, the Hopf bifurcation locus in Fig.…”

“…The key ecological control parameter is d, which reflects the algal concentration in the upper water layers, and Fig. 4 illustrates the part of the d-c parameter plane in which patterns exist, and also where they are stable, for a = 0.6667, b = 41.5692 and g = 0.1333 (these parameter values are taken from Wang et al, 2009). All calculations and plotting were done using WAVETRAIN.…”

“…van de Koppel et al (2005) proposed a model for mussel dynamics in the Wadden Sea, consisting of equations for mussel density m(x, t) and algal concentration a(x, t). When appropriately nondimensionalised (van de Koppel et al, 2005;Wang et al, 2009) …”