Developing ODE software in new computing environments

Shampine, L. F.; Reichelt, Mark W.

doi:10.1007/978-1-5041-2940-4_17

Cited by 3 publications

(4 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For our numerical method, this is a perturbation limit, thus, we used the absolute error tolerance of approximately 10 −6 to limit how small a δ c (≈ 10 −5 ) we could use. The a(c) curves that appear in the previous section are computed using equation (2.10) and the quadrature algorithm adapt [52]. The algorithm adapt uses three-point Gaussian quadrature to estimate the integrals and the seven-point Kronrod rule to estimate errors.…”

Section: The Numerical Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Analysis and computation of travelling wave solutions of bistable differential-difference equations

Elmer

Vleck

1999

Nonlinearity

View full text Add to dashboard Cite

We consider travelling wave solutions of a class of differential-difference equations. Our interest is in understanding propagation failure, the directional dependence due to the discrete Laplacian, and the relationship between travelling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differential-difference equations that we study include damped and undamped nonlinear wave and reaction-diffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.

show abstract

Section: The Numerical Methodsmentioning

confidence: 99%

“…We are particularly interested in the phenomenon of propagation failure, the existence of a nontrivial interval for a such that the wave speed is zero. Throughout this section we include plots of the a(c) relation calculated from (2.8) using the three-point Gaussian quadrature algorithm adapt [52].…”

Section: Propagation Failurementioning

confidence: 99%

Analysis and computation of travelling wave solutions of bistable differential-difference equations

Elmer

Vleck

1999

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…For each individual in the Genetic Algorithm population, which corresponds to a set of parameters to be evaluated and ranked by the objective function, we have described the series of ODEs with mass transfer terms by taking the PDEs reported in S1 Text, applying a secondorder central approximation to discretize spatial coordinates, giving rise to an ODE system for each compartment. Subsequently, a linear implicit multistep method based on 6 th order finite differences (BDF-6 formula) was used to solve each ODE in time [63,64]. The computation considers, in the beginning, the species cDl 0 with uniform concentration given by the parameter [Dl]tot for each compartment, the species T following a Gaussian curve, as described in the S1 Text, and all other species with zero concentration, and it ended when the system had reached a steady-state for each individual in the GA population.…”

Section: Model Calibration and Simulationmentioning

confidence: 99%

A reaction-diffusion network model predicts a dual role of Cactus/IκB to regulate Dorsal/NFκB nuclear translocation in Drosophila

et al. 2021

View full text Add to dashboard Cite

Dorsal-ventral patterning of the Drosophila embryo depends on the NFκB superfamily transcription factor Dorsal (Dl). Toll receptor activation signals for degradation of the IκB inhibitor Cactus (Cact), leading to a ventral-to-dorsal nuclear Dl gradient. Cact is critical for Dl nuclear import, as it binds to and prevents Dl from entering the nuclei. Quantitative analysis of cact mutants revealed an additional Cact function to promote Dl nuclear translocation in ventral regions of the embryo. To investigate this dual Cact role, we developed a predictive model based on a reaction-diffusion regulatory network. This network distinguishes non-uniform Toll-dependent Dl nuclear import and Cact degradation, from the Toll-independent processes of Cact degradation and reversible nuclear-cytoplasmic Dl flow. In addition, it incorporates translational control of Cact levels by Dl. Our model successfully reproduces wild-type data and emulates the Dl nuclear gradient in mutant dl and cact allelic combinations. Our results indicate that the dual role of Cact depends on the dynamics of Dl-Cact trimers along the dorsal-ventral axis: In the absence of Toll activation, free Dl-Cact trimers retain Dl in the cytoplasm, limiting the flow of Dl into the nucleus; in ventral-lateral regions, Dl-Cact trimers are recruited by Toll activation into predominant signaling complexes and promote Dl nuclear translocation. Simulations suggest that the balance between Toll-dependent and Toll-independent processes are key to this dynamics and reproduce the full assortment of Cact effects. Considering the high evolutionary conservation of these pathways, our analysis should contribute to understanding NFκB/c-Rel activation in other contexts such as in the vertebrate immune system and disease.

show abstract

“…In this section, we investigate the implications of the biological and economic spatial factors on the transitional dynamics. To do this, we numerically calculate the trajectories for the three-patch system using solvers specifically designed for nonlinear large-scale stiff ordinary differential equations (Shampine and Reichelt [1996]). We continue to use the closed case as a benchmark to which to compare the adjustment paths of the fully integrated and cascade systems.…”

Section: Comparisons Of Spatial and Intertemporal Dynamicsmentioning

confidence: 99%

Dynamics of Spatial Exploitation: A Metapopulation Approach

Sanchirico¹,

Wilen²

2001

Natural Resource Modeling

View full text Add to dashboard Cite

ABSTRACT. In this paper we present a bioeconomic model of a harvesting industry operating over a heterogeneous environment comprised of discrete biological populations interconnected by dispersal processes. The model generalizes the Gordon [1954]/Smith [1968] model of open‐access rent dissipation by accounting for intertemporal and spatial “Ricardian” patterns of exploitation. This model yields a simple, but insightful, framework from which one can investigate factors that contribute to the evolution of resource exploitation patterns over space and time. For example, we find that exploitation patterns are driven by biological and fleet dispersal and biological and economic heterogeneity. We conclude that one cannot really understand the biological processes operating in an exploited system without knowing as much about the harvesting system as about the biological system.

show abstract

Developing ODE software in new computing environments

Cited by 3 publications

References 3 publications

Analysis and computation of travelling wave solutions of bistable differential-difference equations

Analysis and computation of travelling wave solutions of bistable differential-difference equations

A reaction-diffusion network model predicts a dual role of Cactus/IκB to regulate Dorsal/NFκB nuclear translocation in Drosophila

Dynamics of Spatial Exploitation: A Metapopulation Approach

Contact Info

Product

Resources

About