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Averaging Principle for Backward Stochastic Differential Equations
The averaging principle for BSDEs and one-barrier RBSDEs, with Lipschitz coefficients, is investigated. An averaged BSDEs for the original BSDEs is proposed, as well as the one-barrier RBSDEs, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square.
Research on the Impact of Ecological Civilization Construction on Environmental Pollution Control in China—Based on Differential Game Theory
With environmental pollution problems becoming more and more serious, administrators need to change their original system of performance achievement evaluation and build a new performance achievement view of ecological civilization. Based on the view of achievement evaluation of ecological civilization, this research develops a continuous-time differential game of environmental pollution control between local government and enterprises by taking into consideration dynamic change in pollutants in the environment into account. By analyzing the equilibrium solution of the game, the role of the assessment mechanism of ecological civilization is explored, and countermeasures are proposed to provide a scientific basis for improving the assessment mechanism of ecological civilization in China’s environmental pollution control. The results have shown that enterprises’ equilibrium pollutants output is negatively correlated to Ecological Civilization Index. Instead, the government’s efforts to curb pollutions are positively correlated to the environmental review. With th`at, lowering enterprises’ cost of production and pollution reduction alongside imposing more severe punishment would help improve pollution management. Setting up reasonable reward-punishment system would improve local governments’ pollution management. Elsewhere, the research employs the method of numerical simulation to testify the relationship among the importance of environmental review, enterprises’ pollutant output, and government’s efforts to manage pollutions. Finally, the research also suggests necessary measures to solve environmental pollutions and introduces a numerical simulation analysis.
Pricing Corporate Bonds with Credit Risk, Liquidity Risk, and Their Correlation
This paper proposes a generalized bond pricing model, accounting for all the effects of credit risk, liquidity risk, and their correlation. We use an informed trading model to specify the bond liquidity payoff and analyze the sources of liquidity risk. We show that liquidity risk arises from reduced information accuracy and market risk tolerance, and it is market risk tolerance that links credit and liquidity. Then, we extend the traditional bond pricing model with only credit risk by incorporating liquidity risk into the framework in which the probabilities of the two risk events are estimated by a joint distribution. Using numerical examples, we analyze the role of the correlation between credit and liquidity in bond pricing, especially during a financial crisis. We document that the varying correlation between default and illiquidity explains the phenomenon of bond death spiral observed in a financial crisis. Finally, we take the US corporate bond market as an example to demonstrate our conclusions.
Implications of Advertising Lag on the Dynamic Optimal Decisions in an O2O Supply Chain
The advertising lag effect and the reference quality effect are intrinsic properties of the market that influence supply chain performance. In this paper, we use the differential game theory considering the above two effects under an O2O (online to offline) environment. We investigate how these two effects impact supply chain performance. By the extended maximum principle, the optimal analytical solutions of the decision variables in two different game scenarios are obtained, i.e., with and without advertising cooperation. We conclude that the optimal wholesale price (advertising effort level, product quality level, and retail price) increases or decreases exponentially to its steady state, which depends on whether the initial goodwill is lower than the static goodwill or not. The analysis shows that the advertising lag effect has a negative impact on the supply chain, while the reference quality effect is positive. The vertical advertising cooperation strategy is effective in facilitating the channel performance. From a comparison analysis, we obtain that the manufacturer concentrates more effort on quality improvement if the advertising lag time is larger or if consumers are more sensitive to quality, and the manufacturer pays more attention to advertising relative to quality improvements with a higher advertising cost sharing ratio.
Bounds of Degree-Based Molecular Descriptors for Generalized -sum Graphs
A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and total -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized -sum graphs as the consequences of the obtained results. At the end, -graphical presentations are also included to illustrate the results for better understanding.
About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls
This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model.