Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.
A field experiment was conducted during rabi season (2020-21) at AICRP on Wheat, College of Agriculture, Jawaharlal Nehru Krishi Vishwa Vidyalaya, Jabalpur (MP), India, to study the effect of pinoxaden on weeds and the yield of wheat. The field experiment was laid out in a randomized block design with seven treatments and replicated thrice. Treatments included applying different doses of pinoxaden at 40, 45, and 90 g a.i. ha-1, clodinafop propargyl at 90 g a.i. ha-1, sulfosulfuron at 25 g a.i. ha-1 as post-emergence along with hand weeding at 30 DAS and weedy check. The experimental field was dominated by Phalaris minor (15.6%) among monocot weeds, while Medicago denticulata (30.82%), Cichorium intybus (29.94%), Chenopodium album (15.32%), and Anagallis arvensis (8.30%) among the dicot weeds throughout the crop growing period. Among the different herbicidal treatments, pinoxaden at 90 g a.i. ha-1 effectively controlled the monocot and dicot weeds and recorded higher weed control efficiency and the lowest weed index. However, the highest value of growth parameters, yield attributes, and grain yield was recorded with the application of pinoxaden at 45 g a.i. ha-1 among all the herbicidal treatments.
Aim: To study the Long-term impact of soil test and targeted yield based nutrient management on vertical variability in carbon fractions of a Vertisol under rice-wheat cropping sequence. Place and Duration of Study: This research trail was conducted during rabi season of 2020-21 in an on-going research programme of AICRP on STCR initiated during 2008 at the Research Farm of Jawaharlal Nehru Krishi Vishwa Vidyalaya, Jabalpur. Study Design: The study has consisted of six treatments of nutrient management practices based on soil test and targeted yields of rice and wheat (T1: Control; T2: GRD; T3: T.Y. 50 and 45 q ha-1 for rice and wheat; T4: T.Y. 60 q ha-1; T5: T.Y. 50 and 45 q with FYM 5 t ha-1 for rice and wheat and T6: T.Y. 60 q with 5 t FYM ha-1) at different soil depths (0-15, 15-30 and 30-45 cm) which were replicated four times in a randomized block design.A total of 72 post- harvest soil samples of wheat were subjected to determination of carbon fractions across the soil depths. Results: Results revealed that Carbon fractions in soil were significantly altered by nutrient management practices over soil depths. However, the highest contents of organic and inorganic carbon fractions in soil were obtained under T6 having highest yield target of 60 q along with FYM 5 t ha-1 and the lowest in control. The results showed that contents of carbon fractions of soil were decreased with consecutive increase in soil depths except less labile carbon and inorganic carbon which increased with soil depths.
Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square-wave to spike-like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase-shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.
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