Root growth models: towards a new generation of continuous approaches

Models of root system growth emerged in the early 1970s, and were based on mathematical representations of root length distribution in soil. The last decade has seen the development of more complex architectural models and the use of computer-intensive approaches to study developmental and environmental processes in greater detail. There is a pressing need for predictive technologies that can integrate root system knowledge, scaling from molecular to ensembles of plants. This paper makes the case for more widespread use of simpler models of root systems based on continuous descriptions of their structure. A new theoretical framework is presented that describes the dynamics of root density distributions as a function of individual root developmental parameters such as rates of lateral root initiation, elongation, mortality, and gravitropsm. The simulations resulting from such equations can be performed most efficiently in discretized domains that deform as a result of growth, and that can be used to model the growth of many interacting root systems. The modelling principles described help to bridge the gap between continuum and architectural approaches, and enhance our understanding of the spatial development of root systems. Our simulations suggest that root systems develop in travelling wave patterns of meristems, revealing order in otherwise spatially complex and heterogeneous systems. Such knowledge should assist physiologists and geneticists to appreciate how meristem dynamics contribute to the pattern of growth and functioning of root systems in the field.

Natal nest locations of the Asian house rat (Rattus tanezumi) in lowland rice–coconut cropping systems: a coconut penthouse or rice bunds with water frontage?

CONTEXT. Rattus tanezumi is a serious crop pest within the island of Luzon, Philippines. In intensive flood-irrigated rice field ecosystems of Luzon, female R. tanezumi are known to primarily nest within the tillers of ripening rice fields and along the banks of irrigation canals. The nesting habits of R. tanezumi in complex rice–coconut cropping systems are unknown. AIMS. To identify the natal nest locations of R. tanezumi females in rice–coconut systems of the Sierra Madre Biodiversity Corridor (SMBC), Luzon, during the main breeding season to develop a management strategy that specifically targets their nesting habitat. METHODS. When rice was at the booting to ripening stage, cage-traps were placed in rice fields adjacent to coconut habitat. Thirty breeding adult R. tanezumi females were fitted with radio-collars and successfully tracked to their nest sites. KEY RESULTS. Most R. tanezumi nests (66.7%) were located in coconut groves, five nests (16.7%) were located in rice fields and five nests (16.7%) were located on the rice field edge. All nests were located above ground level and seven nests were located in coconut tree crowns. The median distance of nest sites to the nearest rice field was 22.5m. Most nest site locations had good cover of ground vegetation and understorey vegetation, but low canopy cover. Only one nest location had an understorey vegetation height of less than 20 cm. CONCLUSIONS. In the coastal lowland rice–coconut cropping systems of the SMBC, female R. tanezumi showed a preference for nesting in adjacent coconut groves. This is contrary to previous studies in intensive flood-irrigated rice ecosystems of Luzon, where the species nests mainly in the banks of irrigation canals. It is important to understand rodent breeding ecology in a specific ecosystem before implementing appropriate management strategies. IMPLICATIONS. In lowland rice–coconut cropping systems, coconut groves adjacent to rice fields should be targeted for the 20 management of R. tanezumi nest sites during the main breeding season as part of an integrated ecologically based approach to rodent pest management.

Tillage and herbicide treatments with inter-row cultivation influence weed densities and yield of three industrial crops

Field experiments were conducted in northern Greece in 2003 and 2004 to evaluate effects of tillage regimes (moldboard plowing, chisel plowing, and rotary tilling), cropping sequences(continuous cotton, cotton-sugar beet rotation,and continuous tobacco) and herbicide treatments with inter-row hand hoeing on weed population densities. Total weed densities were not affected by tillage treatment except that of barnyardgrass (Echinochloa crus-galli), which increased only in moldboard plowing treated plots during 2003. Redroot pigweed (Amaranthus retroflexus)and black nightshade (Solanum nigrum) densities were reduced in continuous cotton, while purple nutsedge (Cyperus rotundus), E. crus-galli, S. nigrum, and johnsongras(Sorghum halepense) densities were reduced in tobacco. A. retroflexus and S. nigrum were effectively controlled by all herbicide treatments with inter-row hand hoeing,whereas E. crus-galli was effectively reduced by herbicides applied to cotton and tobacco. S. halepense density reduction was a result of herbicide applied to tobacco with inter-row hand hoeing. Yield of all crops was higher under moldboard plowing and herbicide treatments. Pre-sowing and pre-emergence herbicide treatments in cotton and pre-transplant in tobacco integrated with inter-row cultivation resulted in efficient control of annual weed species and good crop yields. These observations are of practical relevance to crop selection by farmers in order to maintain weed populations at economically acceptable densities through the integration of various planting dates, sustainable herbicide use and inter-row cultivation; tools of great importance in integrated weed management systems. Keywords: cropping sequence, herbicide, integrated weed management, inter-row cultivation,tillage.

On variational data assimilation in continuous time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalization of what is known as weakly constrained four-dimensional variational assimilation (4D-Var) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two-point boundary value problem. For imperfect models, the trade-off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. This trade-off turns out to be trivial if the model is perfect. However, even in this situation, allowing for minute deviations from the perfect model is shown to have positive effects, namely to regularize the problem. The presented formalism is dynamical in character. No statistical assumptions on dynamical or observational noise are imposed.

Sensitivity and out-of-sample error in continuous time data assimilation

Data assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time–tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade-off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade-off. A relation between the sensitivity and the out-of-sample error is established, which allows the latter to be calculated under operational conditions. A minimum out-of-sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade-off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, also known as synchronization. Numerical examples demonstrate the feasibility of the approach.

Evaluating raw ensembles with the continuous ranked probability score

The continuous ranked probability score (CRPS) is a frequently used scoring rule. In contrast with many other scoring rules, the CRPS evaluates cumulative distribution functions. An ensemble of forecasts can easily be converted into a piecewise constant cumulative distribution function with steps at the ensemble members. This renders the CRPS a convenient scoring rule for the evaluation of ‘raw’ ensembles, obviating the need for sophisticated ensemble model output statistics or dressing methods prior to evaluation. In this article, a relation between the CRPS score and the quantile score is established. The evaluation of ‘raw’ ensembles using the CRPS is discussed in this light. It is shown that latent in this evaluation is an interpretation of the ensemble as quantiles but with non-uniform levels. This needs to be taken into account if the ensemble is evaluated further, for example with rank histograms.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.