Modeling eutrophic lakes: From mass balance laws to ordinary differential equations

2017 ◽  
Vol 14 (11) ◽  
pp. 1750151
Author(s):  
Addolorata Marasco ◽  
Luciano Ferrara ◽  
Antonio Romano
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Homogeneous Model ◽  
Stable Steady State ◽  
Model Parameters ◽  
Balance Laws ◽  
State Variables ◽  
Phosphorus Cycle ◽  
Eutrophic Lakes ◽  
The Mean

Starting from integral balance laws, a model based on nonlinear ordinary differential equations (ODEs) describing the evolution of Phosphorus cycle in a lake is proposed. After showing that the usual homogeneous model is not compatible with the mixture theory, we prove that an ODEs model still holds but for the mean values of the state variables provided that the nonhomogeneous involved fields satisfy suitable conditions. In this model the trophic state of a lake is described by the mean densities of Phosphorus in water and sediments, and phytoplankton biomass. All the quantities appearing in the model can be experimentally evaluated. To propose restoration programs, the evolution of these state variables toward stable steady state conditions is analyzed. Moreover, the local stability analysis is performed with respect to all the model parameters. Some numerical simulations and a real application to lake Varese conclude the paper.

scholarly journals Exploitation of a Productive Asset in the Presence of Strategic Behavior and Pollution Externalities

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1682
Author(s):  
N. Baris Vardar ◽  
Georges Zaccour
Keyword(s):  
Strategic Behavior ◽  
Piecewise Linear ◽  
Growth Function ◽  
Piecewise Linear Approximation ◽  
Stable Steady State ◽  
Model Parameters ◽  
State Variables ◽  
Long Run ◽  
Productive Asset ◽  
Asset Growth

We study the strategic behavior of firms competing in the exploitation of a common-access productive asset, in the presence of pollution externalities. We consider a differential game with two state variables (asset stock and pollution stock), and by using a piecewise-linear approximation of the nonlinear asset growth function, we provide a tractable characterization of the symmetric feedback–Nash equilibrium with asymptotically stable steady state(s). The results show that the firm’s strategy takes three forms depending on the pair of state variables and that different options for the model parameters lead to contrasting outcomes in both the short- and long-run equilibria.

Parametrizations of sub-attractors in hyperbolic balance laws

2012 ◽  
Vol 142 (3) ◽  
pp. 563-583
Author(s):  
Julia Ehrt
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Global Attractor ◽  
Balance Laws ◽  
Number Of Zeros ◽  
Hyperbolic Balance Laws ◽  
All Solutions ◽  
Finite Dimensionality

We investigate the properties of the global attractor of hyperbolic balance laws on the circle, given byut + f(u)x = g(u).The new tool of sub-attractors is introduced. They contain all solutions on the global attractor up to a given number of zeros. The paper proves finite dimensionality of all sub-attractors, provides a full parametrization of all sub-attractors and derives a system of ordinary differential equations for the embedding parameters that describe the full partial differential equation dynamics on the sub-attractor.

scholarly journals On deterministic approximation of Markov processes by ordinary differential equations

1998 ◽  
Vol 4 (2) ◽  
pp. 99-114 ◽  
Author(s):  
L. I. Rozonoer
Keyword(s):  
Differential Equations ◽  
Markov Model ◽  
Ordinary Differential Equations ◽  
Chemical Reaction ◽  
Markov Processes ◽  
Random Variables ◽  
Mean Values ◽  
Multidimensional Lattice ◽  
The Mean

For a class of Markov processes on the integer multidimensional lattice, it is shown that the evolution of the mean values of some random variables can be approximated by ordinary differential equations. To illustrate the approach, a Markov model of a chemical reaction is considered

scholarly journals MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS

2009 ◽  
Vol 19 (11) ◽  
pp. 3593-3604 ◽  
Author(s):  
CRISTINA JANUÁRIO ◽  
CLARA GRÁCIO ◽  
DIANA A. MENDES ◽  
JORGE DUARTE
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Control Technique ◽  
Stable Steady State ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Firm Model ◽  
One Dimensional Maps

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

scholarly journals Resilience Analysis for Competing Populations

2019 ◽  
Author(s):  
Artur César Fassoni ◽  
Denis de Carvalho Braga
Keyword(s):  
Differential Equations ◽  
Basins Of Attraction ◽  
Ecological Resilience ◽  
Model Parameters ◽  
Reproduction Rate ◽  
State Variables ◽  
Mathematical Methods ◽  
Theoretical Concepts ◽  
Critical Transitions ◽  
Competing Populations

AbstractEcological resilience refers to the ability of a system to retain its state when subject to state variables perturbations or parameter changes. While understanding and quantifying resilience is crucial to anticipate the possible regime shifts, characterizing the influence of the system parameters on resilience is the first step towards controlling the system to avoid undesirable critical transitions. In this paper, we apply tools of qualitative theory of differential equations to study the resilience of competing populations as modeled by the classical Lotka-Volterra system. Within the high interspecific competition regime, such model exhibits bistability, and the boundary between the basins of attraction corresponding to exclusive survival of each population is the stable manifold of a saddle-point. Studying such manifold and its behavior in terms of the model parameters, we characterized the populations resilience: while increasing competitiveness leads to higher resilience, it is not always the case with respect to reproduction. Within a pioneering context where both populations initiate with few individuals, increasing reproduction leads to an increase in resilience; however, within an environment previously dominated by one population and then invaded by the other, an increase in resilience is obtained by decreasing the reproduction rate. Besides providing interesting insights for the dynamics of competing population, this work brings near to each other the theoretical concepts of ecological resilience and the mathematical methods of differential equations and stimulates the development and application of new mathematical tools for ecological resilience.

scholarly journals Systems biology informed deep learning for inferring parameters and hidden dynamics

2020 ◽  
Vol 16 (11) ◽  
pp. e1007575 ◽  
Author(s):  
Alireza Yazdani ◽  
Lu Lu ◽  
Maziar Raissi ◽  
George Em Karniadakis
Keyword(s):  
Deep Learning ◽  
Systems Biology ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Learning Algorithm ◽  
Optimization Procedure ◽  
System Level ◽  
Benchmark Problems ◽  
Model Parameters ◽  
Unknown Parameters

Mathematical models of biological reactions at the system-level lead to a set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a new systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, external forcing, and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems.

Computer Programming

SIMULATION ◽  
1965 ◽  
Vol 4 (5) ◽  
pp. 317-323 ◽  
Author(s):  
Joseph L. Hammond
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Ordinary Differential Equations ◽  
Computer Programming ◽  
Characteristic Root ◽  
Analog Computer ◽  
State Variables ◽  
State Variable ◽  
Forcing Function ◽  
Derivatives Of

State variable techniques are reviewed and applied to analog computer programming. The concise rep resentation for ordinary differential equations made possible by this technique is used to formulate a gen eral program for all such equations. It is shown that an analog computer program based on state variables will not have redundant integrators. The fact that the use of state variables facilitates the choice of variables internal to an analog com puter program is illustrated by two techniques, namely, (1) a technique for avoiding derivatives of the forcing function in programming a large class of ordinary differential equations, and (2) a technique for simulating certain systems in such a way that the effect of each characteristic root is placed in evi dence.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Nonholonomic Constraints

2016 ◽  
Vol 17 (1) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Nonholonomic Constraints ◽  
State Variables ◽  
Well Conditioning ◽  
Fifth Order

A method is presented for formulating and numerically integrating ordinary differential equations of motion for nonholonomically constrained multibody systems. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation, yielding ordinary differential equations of motion. Orthogonal-dependent coordinates and velocities are used to enforce constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for updating local coordinates on configuration and velocity constraint manifolds, transparent to the user and at minimal computational cost. The formulation is developed for multibody systems with nonlinear holonomic constraints and nonholonomic constraints that are linear in velocity coordinates and nonlinear in configuration coordinates. A computational algorithm for implementing the approach is presented and used in the solution of three examples: one planar and two spatial. Numerical results using a fifth-order Runge–Kutta–Fehlberg explicit integrator verify that accurate results are obtained, satisfying all the three forms of kinematic constraint, to within error tolerances that are embedded in the formulation.

scholarly journals Systems biology informed deep learning for inferring parameters and hidden dynamics

2019 ◽  
Author(s):  
Alireza Yazdani ◽  
Lu Lu ◽  
Maziar Raissi ◽  
George Em Karniadakis
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Systems Biology ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
System Level ◽  
Model Parameters ◽  
Unknown Parameters ◽  
The Neural Networks ◽  
Noisy Measurements

AbstractMathematical models of biological reactions at the system-level lead to a set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a new systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, external forcing, and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems.Author summaryThe dynamics of systems biological processes are usually modeled using ordinary differential equations (ODEs), which introduce various unknown parameters that need to be estimated efficiently from noisy measurements of concentration for a few species only. In this work, we present a new “systems-informed neural network” to infer the dynamics of experimentally unobserved species as well as the unknown parameters in the system of equations. By incorporating the system of ODEs into the neural networks, we effectively add constraints to the optimization algorithm, which makes the method robust to noisy and sparse measurements.

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