Sensitivity and uncertainty propagation in coupled models for assessing smallholder farmer food security in the Olifants River Basins, South Africa

Using family balance (i.e., combined net farm and non-farm incomes less family expenses), an output from an integrated model, which couples water resource, agronomic and socio-economic models, its sensitivity and uncertainty are evaluated for five smallholder farming groups (AeE) in the Olifants Basin. The crop management practiced included conventional rainfed, untied ridges, planting basins and supplemental irrigation. Scatter plots inferred the most sensitive variables affecting family balance, while the Monte Carlo method, using random sampling, was used to propagate the uncertainty in the model inputs to produce family balance probability distributions. A non-linear correlation between in-season rainfall and family balance arises from several factors that affect crop yield, indicating the complexity of farm family finance resource-base in relation to climate, crop management practices and environ- mental resources of soil and water. Stronger relationships between family balance and evapotranspira- tion than with in-season rainfall were obtained. Sensitivity analysis results suggest more targeted investment effort in data monitoring of yield, in-season rainfall, supplemental irrigation and maize price to reduce family balance uncertainty that varied from 42% to 54% at 90% confidence level. While sup- plemental irrigation offers the most marginal increase in yields, its wide adoption is limited by avail- ability of water and infrastructure cost

Food insecurity of smallholder farming systems in B72A catchment in the Olifants River Basin, South Africa

Traditional smallholder farming systems are characterized by low yields and high risks of crop failure and food insecurity. Through a biophysical model, PARCHED-THIRST and a socio-economic farming systems simulation model, OLYMPE, we evaluated the performance of farming practices based on maize yield, gross margin and total family balance over a 10-year period in semi-arid Olifants River Basin of South Africa. Farm profitability under scenarios of different maize productions, maize grain and fertiliser price variations were explored for the identified farming systems. Farm types (A to E) were identified from farm surveys, and validated with farmers and extension officers. The order of vulnerability to severe droughts and food insecurity, starting with the most vulnerable is farm Type B, C, D, A and E. Severe drought or flood shock resulted in highest farm gross margin and total family balance reductions, partly due to loss of production for family consumption. Labour returns ranged from US$62/capita.year for crop-based farm types to US$ 363/capita.year for livestock-based farm Type E. Results revealed that livestock and crop diversification are most proficient strategies to ensure stable income and food security for smallholder farmers. Thus, smallholder farming technology innovations and policies should engage in solutions to poor yields and livestock farming

Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients

This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω ) or H˜s−2(Ω ) , 12<s<32, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.EPSR

A time-dependent Green’s function-based model for stream-unconfined aquifer flows

A numerical formulation that is based on the Green element method (GEM), which incorporates a time-dependent Green’s function, is used to solve transient two-dimensional flows of stream-unconfined aquifer interaction. The Green’s function comes from the fundamental solution to the linear diffusion differential operator in two spatial dimensions. In classical boundary element applications, this Green’s function has found use primarily in linear heat transfer and flow problems; its use here for the nonlinear stream-unconfined aquifer flow problem represents the computational flexibility that is achieved with a Green element sense of implementing the singular integral theory. The nonlinear discretised element equations obtained from numerical calculations are linearised by the Picard and Newton-Raphson methods, while the global coefficient matrix, which is banded and sparse, is readily amenable to matrix solution routines. Using four numerical examples, the accuracy of the current formulation is assessed as against an earlier one that incorporates the Logarithmic fundamental solution. It is observed that comparable accuracy is achieved between both formulations, indicating that the current formulation is a viable numerical solution strategy for the stream-aquifer flow problem