Geometry

Figure 5.4: Banding in the presence of a geographic feature that does not al- low vegetation to grow. Although here the competition kernel is offset along the diagonal of the system, the direction of the vegetation (shown in green) locally is more strongly affected by the geometry of the geographic feature (shown in grey). The other parameters of the system are K= 1, r0 = 0, µ= 0.1, l1 = 1, l2= 2.

In the previous section a number of nu- merical experiments were ran where the boundary of the domain was fixed with occupied vegetation sites along the East boundary. For each simulation the off- set of the competition kernel was per- pendicular to the line of constant vege- tation. However, there can of course be situations where the offset of the compe- tition kernel is not perpendicular to the geometry of the boundary, which can lead to frustration between the band of constant vegetation and the direction of the offset (Fig. 5.4). This can provide some insight into expected spatial pat- terns along coastlines, where there is a strong tendency for bands of vegetation to align along the coast, but dominant currents may flow in a different direction.

5.7

Conclusion

A variety of models of spatial vegetative processes have been presented and analysed. Of specific interest is the interaction between environment and vegetation that leads to regular spatial patterns such as banding. Whilst investigating these specific features, we have also tried to keep the resulting equations at the fullest generality possible. This has included keeping the number of parameters in the model to a minimum and also proposing a general model of vegetation with interaction from its environment.

The interaction between environment and vegetation is in particular a spatial one. As such, the first step in the analysis of mechanisms that generate spatial pat- terns was to generate a number of reaction-diffusion models that explicitly model the interaction between vegetation and its environment. Marine vegetation is af- fected by a number of non-local processes such a currents and wave actions, however these were ignored in the initial approach. Instead soft sediment was modelled via

a transport-diffusion process, where both the transport term and diffusion term are affected by the density of vegetation. The growth of vegetation itself depends on both the current density of vegetation and the presence of soft sediment that allows the vegetation to root and provides protection from other environmental forces. With these considerations and for specific growth terms regular patterns can form via a symmetry breaking Turing bifurcation. Although the reaction-diffusion ap- proach has some appeal, it lacks a mechanistic underpinning for seagrass ecology in particular.

The limitations of the reaction-diffusion approach can be overcome by considering a form of vegetative growth involving an integro-differential equation, where kernels mediate both the growth and competition felt by the vegetation species. We have shown that this equation is the limiting case of a system where both environment and vegetation are explicitly modelled via integro-differential equations. The result- ing model overcomes the previous limitations by allowing arbitrary choice over the spatial extent of the growth and competition terms.

The integro-differential equation provides the necessary broad spatial distribution as observed, however fails to capture the boundaries between vegetation and bare sea floor. These are due to small numbers of vegetative units near the boundaries leading to the continuous assumption breaking down. The integro-differential model was therefore converted into a discrete-time Markov process. This model still has the regular pattern formation associated with the integro-differential equation and also had the stochastic properties of the boundaries between vegetation and empty states. This model then qualitatively fits the properties of the seagrass-environment system with plausible underlying mechanisms. We may therefore use this model to explore the relationship between spatial pattern and persistence (Chapter 6); how we may validate the model by fitting to spatial pattern data (Chapter 7) and how disease impacts the dynamics and spatial pattern of vegetation (Chapter 9).

Bibliography

Heiko Balzter, Paul W Braun, and Wolfgang K¨ohler. Cellular automata models for vegetation dynamics. Ecological modelling, 107(2):113–125, 1998.

Ulf Dieckmann, Richard Law, and Johan AJ Metz. The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, 2000.

G Bard Ermentrout and Leah Edelstein-Keshet. Cellular automata approaches to biological modeling. Journal of theoretical Biology, 160(1):97–133, 1993.

Mark S Fonseca and Susan S Bell. Influence of physical setting on seagrass land- scapes near beaufort, north carolina, usa. Marine Ecology-Progress Series, 171: 109, 1998.

Reinier Hille Ris Lambers, Max Rietkerk, Frank van den Bosch, Herbert HT Prins, and Hans de Kroon. Vegetation pattern formation in semi-arid grazing systems.

Ecology, 82(1):50–61, 2001.

P Hogeweg. Cellular automata as a paradigm for ecological modeling. Applied mathematics and computation, 27(1):81–100, 1988.

Elizabeth E Holmes, Mark A Lewis, JE Banks, and RR Veit. Partial differential equations in ecology: spatial interactions and population dynamics. Ecology, pages 17–29, 1994.

Sonia K´efi, Maarten B Eppinga, Peter C de Ruiter, and Max Rietkerk. Bistability and regular spatial patterns in arid ecosystems.Theoretical Ecology, 3(4):257–269, 2010.

John Kuo and C Den Hartog. Seagrass taxonomy and identification key. InGlobal seagrass research methods, volume 33, pages 31–58. Elsevier, 2001.

Ren´e Lefever and Olivier Lejeune. On the origin of tiger bush. Bulletin of Mathe- matical Biology, 59(2):263–294, 1997.

Nuria Marba, Just Cebrian, Susana Enriquez, and Carlos M Duarte. Growth pat- terns of western mediterranean seagrasses: species-specific responses to seasonal forcing. Marine ecology progress series. Oldendorf, 133(1):203–215, 1996.

Kenneth A Moore and Frederick T Short. Zostera: biology, ecology, and manage- ment. InSeagrasses: Biology, Ecology and Conservation, pages 361–386. Springer, 2006.

Kenneth A Moore, Hilary A Neckles, and Robert J Orth. Zostera marina(eelgrass) growth and survival along a gradient of nutrients and turbidity in the lower chesa- peake bay. Marine Ecology Progress Series, 142(1):247–259, 1996.

Jonathan A Sherratt. An analysis of vegetation stripe formation in semi-arid land- scapes. Journal of mathematical biology, 51(2):183–197, 2005.

Frederick T Short. The seagrass,Zostera Marina l.: Plant morphology and bed structure in relation to sediment ammonium in izembek lagoon, alaska. Aquatic Botany, 16(2):149–161, 1983.