Optimized CUDA-Based PDE Solver for Reaction Diffusion Systems on Arbitrary Surfaces

Partial differential equation (PDE) solvers are commonly employed to study and characterize the parameter space for reaction-diffusion (RD) systems while investigating biological pattern formation. Increasingly, biologists wish to perform such studies with arbitrary surfaces representing ‘real’ 3D geometries for better insights. In this paper, we present a highly optimized CUDA-based solver for RD equations on triangulated meshes in 3D. We demonstrate our solver using a chemotactic model that can be used to study snakeskin pigmentation, for example. We employ a finite element based approach to perform explicit Euler time integrations. We compare our approach to a naive GPU implementation and provide an in-depth performance analysis, demonstrating the significant speedup afforded by our optimizations. The optimization strategies that we exploit could be generalized to other mesh based processing applications with PDE simulations.

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.

The cultivation of Castanea sativa (Mill.) in Europe, from its origin to its diffusion on a continental scale

The history of Castanea sativa (sweet chestnut) cultivation since medieval times has been well described on the basis of the very rich documentation available. Far fewer attempts have been made to give a historical synthesis of the events that led to the cultivation of sweet chestnut in much earlier times. In this article we attempt to reconstruct this part of the European history of chestnut cultivation and its early diffusion by use of different sources of information, such as pollen studies, archaeology, history and literature. Using this multidisciplinary approach, we have tried to identify the roles of the Greek and Roman civilizations in the dissemination of chestnut cultivation on a European scale. In particular, we show that use of the chestnut for food was not the primary driving force behind the introduction of the tree into Europe by the Romans. Apart from the Insubrian Region in the north of the Italian peninsula, no other centre of chestnut cultivation existed in Europe during the Roman period. The Romans may have introduced the idea of systematically cultivating and using chestnut. In certain cases they introduced the species itself; however no evidence of systematic planting of chestnut exists. The greatest interest in the management of chestnut for fruit production most probably developed after the Roman period and can be associated with the socio-economic structures of medieval times. It was then that self-sufficient cultures based on the cultivation of chestnut as a source of subsistence were formed.