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.
a b s t r a c tHow the properties of ecosystems relate to spatial scale is a prominent topic in current ecosystem research. Despite this, spatially explicit models typically include only a limited range of spatial scales, mostly because of computing limitations. Here, we describe the use of graphics processors to efficiently solve spatially explicit ecological models at large spatial scale using the CUDA language extension. We explain this technique by implementing three classical models of spatial self-organization in ecology: a spiral-wave forming predator-prey model, a model of pattern formation in arid vegetation, and a model of disturbance in mussel beds on rocky shores. Using these models, we show that the solutions of models on large spatial grids can be obtained on graphics processors with up to two orders of magnitude reduction in simulation time relative to normal pc processors. This allows for efficient simulation of very large spatial grids, which is crucial for, for instance, the study of the effect of spatial heterogeneity on the formation of self-organized spatial patterns, thereby facilitating the comparison between theoretical results and empirical data. Finally, we show that large-scale spatial simulations are preferable over repetitions at smaller spatial scales in identifying the presence of scaling relations in spatially self-organized ecosystems. Hence, the study of scaling laws in ecology may benefit significantly from implementation of ecological models on graphics processors.
Abstract. We present an implementation of a Two-Level Preconditioned Conjugate Gradient Method for the GPU. We investigate a Truncated Neumann Series based preconditioner in combination with deflation. This combination exhibits fine-grain parallelism and hence we gain considerably in execution time when compared with a similar implementation on the CPU. Its numerical performance is comparable to the Block Incomplete Cholesky approach. Our method provides a speedup of up to 16 for a system of one million unknowns when compared to an optimized implementation on one core of the CPU.
Abstract-Solving an ill-conditioned linear system with a two level preconditioned Conjugate Gradient method on the GPU presents many options. The viability of these options is studied for different bubbly flow problems. On the basis of experiments conducted, we propose strategies that make our approach computationally suitable. We use the Truncated Neumann series based preconditioning scheme in combination with Deflation for implementing the two-level preconditioned Conjugate Gradient method and test different configurations on a unit cube with varying number of bubbles. Our results exhibit up to an order of magnitude speedup on the GPU. Our preconditioning scheme combined with deflation proves competitive (in terms of computation time and convergence) when compared to deflation with Incomplete Cholesky preconditioning.
SUMMARYIn both bubbly and porous media flow, the jumps in coefficients may yield an ill-conditioned linear system. The solution of this system using an iterative technique like the conjugate gradient (CG) is delayed because of the presence of small eigenvalues in the spectrum of the coefficient matrix. To accelerate the convergence, we use two levels of preconditioning. For the first level, we choose between out-of-the-box incomplete LU decomposition, sparse approximate inverse, and truncated Neumann series-based preconditioner. For the second level, we use deflation. Through our experiments, we show that it is possible to achieve a computationally fast solver on a graphics processing unit. The preconditioners discussed in this work exhibit fine-grained parallelism. We show that the graphics processing unit version of the two-level preconditioned CG can be up to two times faster than a dual quad core CPU implementation.
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