Lag

Abstract: Nonpoint source (NPS) watershed projects often fail to meet expectations for water quality improvement because of lag time, the time elapsed between adoption of management changes and the detection of measurable improvement in water quality in the target water body. Even when management changes are well-designed and fully implemented, water quality monitoring efforts may not show definitive results if the monitoring period, program design, and sampling frequency are not sufficient to address the lag between treatment and response. The main components of lag time include the time required for an installed practice to produce an effect, the time required for the effect to be delivered to the water resource, the time required for the water body to respond to the effect, and the effectiveness of the monitoring program to measure the response. The objectives of this review are to explore the characteristics of lag time components, to present examples of lag times reported from a variety of systems, and to recommend ways for managers to cope with the lag between treatment and response. Important processes influencing lag time include hydrology, vegetation growth, transport rate and path, hydraulic residence time, pollutant sorption properties, and ecosystem linkages. The magnitude of lag time is highly site and pollutant specific, but may range from months to years for relatively short-lived contaminants such as indicator bacteria, years to decades for excessive P levels in agricultural soils, and decades or more for sediment accumulated in river systems. Groundwater travel time is also an important contributor to lag time and may introduce a lag of decades between changes in agricultural practices and improvement in water quality. Approaches to deal with the inevitable lag between implementation of management practices and water quality response lie in appropriately characterizing the watershed, considering lag time in selection, siting, and monitoring of management measures, selection of appropriate indicators, and designing effective monitoring programs to detect water quality response.

Abstract: 1. Introduction.- a. Discrete and Distributed Lag.- b. Origin of Lags in Biological Models.- c. Lag as an Alternative to Age Structure.- d. Lag as an Alternative to Spatial Structure.- e. The Effects of Lag.- f. Lags and Stochastic Models.- 2. Stability Analysis.- a. The Linear Chain Trick.- b. Instantaneous Models.- c. Models with a Single Discrete Lag.- d. Models with a Single Distributed Lag.- e. An Inequality for Distributed Lag.- f. The Monod Chemostat Model.- g. May's Model of Obligate Mutualism.- 3. Periodic Solutions.- a. Periodic Solutions of the Linear Chain Equations.- b. The Method of Hastings, Tyson and Webster.- c. Hopf Bifurcation.- d. Numerical Integration.- 4. Logistic Growth of a Single Species.- a. Discrete Lag.- b. Distributed Lag in a Model of a Self-poisoning Population.- c. Linear Chain Calculations.- d. Hopf and H.T.W. Methods.- e. Constant Harvesting of a Population in the Presence of Lag.- f. Poincare-Lindstedt Method for Discrete Lag.- g. An Epidemic Model Related to the Logistic Equation.- 5. Biochemical Oscillator Model.- a. The Goodwin Model.- b. Necessary Condition for Instability.- c. Expanding the Set of Equations.- d. A Single Goodwin Equation with Lag.- e. Discrete Lag in the Goodwin Equation.- 6. Models of Haemopoiesis.- a. Wheldon's Model of Chronic Granulocytic Leukemia.- b. Two-lag Models of Cyclical Neutropenia.- c. Time Lag with Attrition a Model of Cyclical Pancytopenia.- 7. Predation Models of the Volterra Type.- 8. Difference Equation Models.- a. Stability Analysis.- b. Conditions under which Spreading the Lag does not affect Local Stability.- c. Chaos in Discrete Dynamical Systems.- d. Extended Diapause in a Single Species Population Model.- e. Analogous Treatment of a Functional Differential Equation.- Supplementary Bibliography.- References.

Abstract: The sources of lag (the delay between input action and output response) and its effects on human performance are discussed. We measured the effects in a study of target acquisition using the classic Fitts' law paradigm with the addition of four lag conditions. At the highest lag tested (225 ms), movement times and error rates increased by 64% and 214% respectively, compared to the zero lag condition. We propose a model according to which lag should have a multiplicative effect on Fitts' index of difficulty. The model accounts for 94% of the variance and is better than alternative models which propose only an additive effect for lag. The implications for the design of virtual reality systems are discussed.