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Discrete and Continuous Dynamical Systems - S

August 2020 , Volume 13 , Issue 8

Issue dedicated to SFBT 2017

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Special issue dedicated to SFBT 2017
Alain Miranville
2020, 13(8): i-i doi: 10.3934/dcdss.2020179 +[Abstract](1150) +[HTML](389) +[PDF](67.56KB)
Prey-predator model with nonlocal and global consumption in the prey dynamics
Malay Banerjee, Nayana Mukherjee and Vitaly Volpert
2020, 13(8): 2109-2120 doi: 10.3934/dcdss.2020180 +[Abstract](2030) +[HTML](294) +[PDF](790.92KB)

A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation
Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco and René Thomas
2020, 13(8): 2121-2134 doi: 10.3934/dcdss.2020181 +[Abstract](2034) +[HTML](320) +[PDF](1597.75KB)

We start by coupling negative 2-circuits, which are characteristic of the presence of a regulation loop in a dynamical system. This loop can be modelled with coupled differential equations represented in a first approach by a conservative differential system. Then, an example of regulation loop with a dissipative component will be given in human physiology by the vegetative system regulating the cardio-respiratory rhythms.

The stability analysis of brain lactate kinetics
Jean-Pierre Françoise and Hongjun Ji
2020, 13(8): 2135-2143 doi: 10.3934/dcdss.2020182 +[Abstract](1384) +[HTML](251) +[PDF](321.84KB)

Our aim in this article is to study properties of a generalized dynamical system modeling brain lactate kinetics, with \begin{document}$ N $\end{document} neuron compartments and \begin{document}$ A $\end{document} astrocyte compartments. In particular, we prove the uniqueness of the stationary point and its asymptotic stability. Furthermore, we check that the system is positive and cooperative.

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case
Lucile Mégret and Jacques Demongeot
2020, 13(8): 2145-2163 doi: 10.3934/dcdss.2020183 +[Abstract](1535) +[HTML](269) +[PDF](380.91KB)

We generalize the results on the existence of an over-stable solution of singularly perturbed differential equations to the equations of the form \begin{document}$ \varepsilon\ddot{x}-F(x,t,\dot{x},k(t), \varepsilon) = 0 $\end{document}. In this equation, the time dependence prevents from returning to the well known case of an equation of the form \begin{document}$ \varepsilon dy/dx = F(x,y,a, \varepsilon) $\end{document} where \begin{document}$ a $\end{document} is a parameter. This can have important physiological applications. Indeed, the coupling between the cardiac and the respiratory activity can be modeled with two coupled van der Pol equations. But this coupling vanishes during the sleep or the anesthesia. Thus, in a perspective of an application to optimal awake, we are led to consider a non autonomous differential equation.

What mathematical models can or cannot do in glioma description and understanding
Angélique Perrillat-Mercerot, Alain Miranville, Nicolas Bourmeyster, Carole Guillevin, Mathieu Naudin and Rémy Guillevin
2020, 13(8): 2165-2193 doi: 10.3934/dcdss.2020184 +[Abstract](1814) +[HTML](399) +[PDF](2928.67KB)

Our aim in this article is not to provide a review of the existing literature but to make a critical analysis on glioma behavior mathematical modeling. We present here mathematical modeling history, interests, modalities and limitations on the study of glioma growth. We also make a point on model consideration according to glioma comportments. We finally introduce medical imaging coupled with in silico models as the next step in glioma research. We do not claim completeness of the bibliography but we tried to cover a large amount of mathematical considerations for glioma behavior illustrated with selected representative papers.

Paradoxes of vulnerability to predation in biological dynamics and mediate versus immediate causality
Evariste Sanchez-Palencia and Philippe Lherminier
2020, 13(8): 2195-2209 doi: 10.3934/dcdss.2020185 +[Abstract](1244) +[HTML](255) +[PDF](532.15KB)

The causality scheme of an (essentially non symmetric) predator-prey system involves automaticaly advantages and disadvantages highly dependent on time. We study systems with one predator and one or two preys furnishing issues which involve mediate and inmediate causality (naturally associated with the attractor and the previous transient). The issues are highly dependent on the parameter accounting for the vulnerability of the preys. When the vulnerability is small, an increase of it implies a (demographic) disadvantage for the preys, but, when it is large (involving periodic cycles) an increase turnes out in an advantage because of the rarefaction of predators (this is associated with average populations on the periodic cycles). When two preys with different vulnerability are present, the most vulnerable may desappear (i. e. the attractor does not contain such prey). This phenomenon only occurs when the less vulnerable prey is nevertheless able to support the predator; otherwise, this one keeps eating anyway the other preys. The mechanism of such patterns are better described in terms of attractors and stability than in terms of advantages versus disadvantages (which are drastically dependent on the viewpoints of the three species).

Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation
Kelong Cheng, Cheng Wang, Steven M. Wise and Zixia Yuan
2020, 13(8): 2211-2229 doi: 10.3934/dcdss.2020186 +[Abstract](1719) +[HTML](385) +[PDF](377.68KB)

The existence and uniqueness of Gevrey regularity solutions for the functionalized Cahn-Hilliard (FCH) and Cahn-Hilliard-Willmore (CHW) equations are established. The energy dissipation law yields a uniform-in-time \begin{document}$ H^2 $\end{document} bound of the solution, and the polynomial patterns of the nonlinear terms enable one to derive a local-in-time solution with Gevrey regularity. A careful calculation reveals that the existence time interval length depends on the \begin{document}$ H^3 $\end{document} norm of the initial data. A further detailed estimate for the original PDE system indicates a uniform-in-time \begin{document}$ H^3 $\end{document} bound. Consequently, a global-in-time solution becomes available with Gevrey regularity.

Modified bidomain model with passive periodic heterogeneities
Yves Coudière, Anđela Davidović and Clair Poignard
2020, 13(8): 2231-2258 doi: 10.3934/dcdss.2020126 +[Abstract](1843) +[HTML](369) +[PDF](879.63KB)

In this paper we study how mesoscopic heterogeneities affect electrical signal propagation in cardiac tissue. The standard model used in cardiac electrophysiology is a bidomain model - a system of degenerate parabolic PDEs, coupled with a set of ODEs, representing the ionic behviour of the cardiac cells. We assume that the heterogeneities in the tissue are periodically distributed diffusive regions, that are significantly larger than a cardiac cell. These regions represent the fibrotic tissue, collagen or fat, that is electrically passive. We give a mathematical setting of the model. Using semigroup theory we prove that such model has a uniformly bounded solution. Finally, we use two–scale convergence to find the limit problem that represents the average behviour of the electrical signal in this setting.

Fractional Cauchy problems and applications
Mohammed AL Horani, Mauro Fabrizio, Angelo Favini and Hiroki Tanabe
2020, 13(8): 2259-2270 doi: 10.3934/dcdss.2020187 +[Abstract](1568) +[HTML](262) +[PDF](354.85KB)

We are devoted to fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications to partial differential equations are given to illustrate the abstract fractional degenerate differential problems.

Well-posedness of a one-dimensional nonlinear kinematic hardening model
David Melching and Ulisse Stefanelli
2020, 13(8): 2271-2284 doi: 10.3934/dcdss.2020188 +[Abstract](1701) +[HTML](301) +[PDF](313.37KB)

We investigate the quasistatic evolution of a one-dimensional elastoplastic body at small strains. The model includes general nonlinear kinematic hardening but no nonlocal compactifying term. Correspondingly, the free energy of the medium is local but nonquadratic. We prove that the quasistatic evolution problem admits a unique strong solution.

Polarization dynamics in a resonant optical medium with initial coherence between degenerate states
Katherine A. Newhall, Gregor Kovačič and Ildar Gabitov
2020, 13(8): 2285-2301 doi: 10.3934/dcdss.2020189 +[Abstract](1416) +[HTML](254) +[PDF](594.54KB)

We investigate the polarization switching phenomenon for a one-soliton solution to the \begin{document}$ \Lambda $\end{document}-configuration Maxwell-Bloch equations in the case of initial coherence in the material, corresponding to the forbidden transition between the two lower energy levels. We find two polarization states that are stationary. They are not the purely right- and left-circularly polarized solitons, as in the case of zero initial coherence, but rather two mixed, elliptically polarized, states. These polarization states, of which only one is asymptotically stable, depend on both the initial population levels of the lower states and the coherence value. We also find the existence of superluminal soliton propagation, but through numerical simulations show this solution to be unstable, and therefore likely not realizable physically.

Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems
Yejuan Wang and Tomás Caraballo
2020, 13(8): 2303-2326 doi: 10.3934/dcdss.2020092 +[Abstract](1996) +[HTML](472) +[PDF](388.85KB)

In this paper, we first prove that the property of being a gradient-like general dynamical system and the existence of a Morse decomposition are equivalent. Next, the stability of gradient-like general dynamical systems is analyzed. In particular, we show that a gradient-like general dynamical system is stable under perturbations, and that Morse sets are upper semi-continuous with respect to perturbations. Moreover, we prove that any solution of perturbed general dynamical systems should be close to some Morse set of the unperturbed gradient-like general dynamical system. We do not assume local compactness for the metric phase space \begin{document}$ X $\end{document}, unlike previous results in the literature. Finally, we extend the Morse decomposition theory of single-valued nonautonomous dynamical systems to the multi-valued case, without imposing any compactness of the parameter spaces.

Explicit investment setting in a Kaldor macroeconomic model with macro shock
Zhenzhen Wang, Zhenghui Li, Shuanglian Chen and Zhehao Huang
2020, 13(8): 2327-2346 doi: 10.3934/dcdss.2020093 +[Abstract](2139) +[HTML](529) +[PDF](766.45KB)

As the inevitable attributes of macro shocks on macroeconomic system, in this paper, we develop a Kaldor macroeconomic model with shock. The shock is due to the investment uncertainty. We then provide an approach for macroeconomic control by calibrating the evolvement of the shocked Kaldor macroeconomic model with some expected benchmark process. The calibration is realized through the setting for investment. The benchmark process is usually the reflection of decisions or policies. An optimal investment setting associated with a five-dimensional nonlinear system of ordinary differential equations is presented. Through a logical modification for the boundary conditions, the nonlinear system is simplified to be linear and a completely explicit formula for the optimal investment setting is achieved. The rationality of the modification is supported by some stability condition. To cope with the systematic risk caused by the macro shock, we define a dynamic Value-at-Risk(VaR) as the risk measure capturing the risk level of the shocked Kaldor macroeconomic model and introduce a risk constraint into the programming of calibration. Then a constrained investment setting is presented. Finally, we carry out an application of the theoretical results by calibrating the evolvement of the shocked Kaldor macroeconomic model with the business cycle generated from the classical Kaldor model through the investment setting.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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