Human societies are increasingly altering the water and biogeochemical cycles to both improve ecosystem productivity and reduce risks associated with the unpredictable variability of climatic drivers. These alterations, however, often cause large negative environmental consequences, raising the question as to how societies can ensure a sustainable use of natural resources for the future. Here we discuss how ecohydrological modeling may address these broad questions with special attention to agroecosystems. The challenges related to modeling the two-way interaction between society and environment are illustrated by means of a dynamical model in which soil and water quality supports the growth of human society but is also degraded by excessive pressure, leading to critical transitions and sustained societal growth-collapse cycles. We then focus on the coupled dynamics of soil water and solutes (nutrients or contaminants), emphasizing the modeling challenges, presented by the strong nonlinearities in the soil and plant system and the unpredictable hydroclimatic forcing, that need to be overcome to quantitatively analyze problems of soil water sustainability in both natural and agricultural ecosystems. We discuss applications of this framework to problems of irrigation, soil salinization, and fertilization and emphasize how optimal solutions for large-scale, long-term planning of soil and water resources in agroecosystems under uncertainty could be provided by methods from stochastic control, informed by physically and mathematically sound descriptions of ecohydrological and biogeochemical interactions.
a b s t r a c tSoil salinity and sodicity impose severe constrains to agriculture, especially in arid and semi-arid regions, where good-quality water for irrigation is scarce. While detailed models have been proposed in the past to describe the dynamics of salt and sodium in the soil, they typically require cumbersome calculations and are not amenable to theoretical analysis. Here we present an analytical model for the dynamics of salinity and sodicity in the root zone. We determine the dependence of steady-state salinity and sodicity levels on irrigation water quality and derive the trajectories in the phase space. The only stationary solution the equations admit is a stable node. Through numerical integration and analysis of the eigenvalues of the derived two-dimensional system of equations, the slower time scale associated with sodification is quantified with respect to the faster time scale associated to salinization. The role of different cation exchange equations (Gapon and Vanselow conventions) are shown to be practically the same with regard to the phase-space dynamics and the time scales. The results can be applied in controlling for low levels of salinity and sodicity, and in planning remediation strategies that are timely and economical.
Spatial periodic forcing of pattern-forming systems is an important, but lightly studied, method of controlling patterns. It can be used to control the amplitude and wave number of one-dimensional periodic patterns, to stabilize unstable patterns, and to induce them below instability onset. We show that, although in one spatial dimension the forcing acts to reinforce the patterns, in two dimensions it acts to destabilize or displace them by inducing two-dimensional rectangular and oblique patterns.
We consider simple systems driven multiplicatively by white shot noise, which appear in the modeling of the dynamics of soil nutrients and contaminants. The dynamics of these systems is analyzed in two ways: solving a hierarchy of linear ordinary differential equations for the moments, which gives a time scale of convergence of the stationary probability density function; and characterizing the crossing properties, such as the mean first-passage time and the mean frequency of level crossing. These results are readily applicable to the study of geophysical systems, such as the problem of accumulation of salt in the root zone, i.e., soil salinization.
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