Scale‐Dependent Feedback and Regular Spatial Patterns in Young Mussel Beds

Theoretical models highlight that spatially self-organized patterns can have important emergent effects on the functioning of ecosystems, for instance by increasing productivity and affecting the vulnerability to catastrophic shifts. However, most theoretical studies presume idealized homogeneous conditions, which are rarely met in real ecosystems. Using self-organized mussel beds as a case study, we reveal that spatial heterogeneity, resulting from the large-scale effects of mussel beds on their environment, significantly alters the emergent properties predicted by idealized self-organization models that assume homogeneous conditions. The proposed model explicitly considers that the suspended algae, the prime food for the mussels, are supplied by water flow from the seaward boundary of the bed, which causes in combination with consumption a gradual depletion of algae over the simulated domain. Predictions of the model are consistent with properties of natural mussel patterns observed in the field, featuring a decline in mussel biomass and a change in patterning. Model analyses reveal a fundamental change in ecosystem functioning when this self-induced algal depletion gradient is included in the model. First, no enhancement of secondary productivity of the mussels comparing with non-patterns states is predicted, irrespective of parameter setting; the equilibrium amount of mussels is entirely set by the input of algae. Second, alternate stable states, potentially present in the original (no algal gradient) model, are absent when gradual depletion of algae in the overflowing water layer is allowed. Our findings stress the importance of including sufficiently realistic environmental conditions when assessing the emergent properties of self-organized ecosystems.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Biogenic gradients in algal density affect the emergent properties of spatially self-organized mussel beds

Liu

Weerman

Gupta

et al. 2014

“…Following Turing's (1952) work, many studies have focused on the mechanism behind the formation of regular patterns throughout science. Today Turing's scenario lies at the heart of many applications attempting to explain patterns in fish skin, mammalian coat markings, phyllotaxis, predator-prey systems, terrestrial vegetation, plankton, intertidal communities and so on (Hassell et al 1991;Murray 1993;Rovinsky et al 1997;Gurney et al 1998;van de Koppel & Crain 2006;Rietkerk & van de Koppel 2008). RD models can exhibit a great variety of spatial patterns (see Cross & Hohenberg (1993), Murray (1993) and Maini et al (1997) for reviews).…”

Section: Introductionmentioning

confidence: 99%

“…Beds of young M. edulis in the Wadden Sea are typically characterized by banded patterns that are aligned perpendicular to the average incoming tidal flow, and have a wavelength of approximately 6 m. Within these bands, mussel can reach high densities of over 30 kg m K2 fresh weight, alternating with almost bare sediment in the interband areas. Van de Koppel et al (2005) provided a potential explanation for the emergence of banded patterns, based on the scale difference in facilitation between neighbouring mussels and competition among the mussels for algal food sources. The scale difference was caused by the flow of the algae in the water with the tidal currents, causing mussels to compete over extensive distances.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

Liu

Sun

et al. 2008

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.

“…Two different mechanisms have been proposed for this type of pattern formation. Van de Koppel et al [85] argue that the binding of mussels to one another, via byssal threads, could cause reduced losses from predation and wave dislodgement at higher mussel densities. Their model has been studied in great detail by Wang et al [86].…”

mentioning

confidence: 99%

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Dagbovie

Sherratt

2014

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package WAVETRAIN (www. ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall-migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.