174,442 research outputs found
Hopf Bifurcation in an Oscillatory-Excitatory Reaction-Diffusion Model with Spatial heterogeneity
We focus on the qualitative analysis of a reaction-diffusion with spatial
heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo
system in which the excitability parameter is space dependent. This
heterogeneity allows to exhibit concomitant stationary and oscillatory
phenomena. We prove the existence of an Hopf bifurcation and determine an
equation of the center-manifold in which the solution asymptotically evolves.
Numerical simulations illustrate the phenomenon
Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment
In this paper, we are concerned with the cooperative system in which âtuâÎu=ÎŒu+α(x,t)vâa(x,t)up and âtvâÎv=ÎŒv+ÎČ(x,t)uâb(x,t)vq in ΩĂ(0,â); (âÎœu,âÎœv)=(0,0) on âΩĂ(0,â); and (u(x,0),v(x,0))=(u0(x),v0(x))>(0,0) in Ω, where p,q>1, ΩâRN(Nâ„2) is a bounded smooth domain, α,ÎČ>0 and a,bâ„0 are smooth functions that are T-periodic in t, and ÎŒ is a varying parameter. The unknown functions u(x,t) and v(x,t) represent the densities of two cooperative species. We study the long-time behavior of (u,v) in the case that a and b vanish on some subdomains of ΩĂ[0,T]. Our results show that, compared to the nondegenerate case where a,b>0 on ΩĂ[0,T], such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system.Pablo Ălvarez-Caudevilla was partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258. Yihong Du was partially supported by the Australian Research Council. Rui Peng was partially supported by NSF of China (11271167, 11171319), the Program for New Century Excellent Talents in University, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Natural Science Fund for Distinguished Young Scholars of Jiangsu Province
Influence of a road on a population in an ecological niche facing climate change
We introduce a model designed to account for the influence of a line with
fast diffusion-such as a road or another transport network-on the dynamics of a
population in an ecological niche. This model consists of a system of coupled
reaction-diffusion equations set on domains with different dimensions (line /
plane). We first show that the presence of the line is always deleterious and
can even lead the population to extinction. Next, we consider the case where
the niche is subject to a displacement, representing the effect of a climate
change or of seasonal variation of resources. We find that in such case the
presence of the line with fast diffusion can help the population to persist. We
also study several qualitative properties of this system. The analysis is based
on a notion of generalized principal eigenvalue developed by the authors in
[5]
Speed of pattern appearance in reaction-diffusion models: Implications in the pattern formation of limb bud mesenchyme cells
It has been postulated that fibroblast growth factor (FGF) treatment of cultured limb bud mesenchyme cells reinforces the lateral inhibitory effect, but the cells also show accelerated pattern appearance. In the present study, we analyze how a small change in a specific parameter affects the speed of pattern appearance in a Turing reaction-diffusion system using linear stability analysis. It is shown that the sign of the change in appearance speed is qualitatively decided if the system is under the diffusion-driven instability condition, and this is confirmed by numerical simulations. Numerical simulations also show that a small change in parameter value induced easily detectable differences in the appearance speed of patterns. Analysis of the Gierer-Meinhardt model revealed that a change in a single parameter can explain two effects of FGF on limb mesenchyme cellsâreinforcement of lateral inhibition and earlier appearance of pattern. These qualitative properties and easy detectability make this feature a promising tool to elucidate the underlying mechanisms of biological pattern formationwhere the quantitative parameters are difficult to obtain
Pattern selection models: From normal to anomalous diffusion
âPattern formation and selection is an important topic in many physical, chemical, and biological fields. In 1952, Alan Turing showed that a system of chemical substances could produce spatially stable patterns by the interplay of diffusion and reactions. Since then, pattern formations have been widely studied via the reaction-diffusion models. So far, patterns in the single-component system with normal diffusion have been well understood. Motivated by the experimental observations, more recent attention has been focused on the reaction-diffusion systems with anomalous diffusion as well as coupled multi-component systems. The objectives of this dissertation are to study the effects of superdiffusion on pattern formations and to compare them with the effects of normal diffusion in one-, and multi-component reaction-diffusion systems. Our studies show that the model parameters, including diffusion coefficients, ratio of diffusion powers, and coupling strength between components play an important role on the pattern formation. Both theoretical analysis and numerical simulations are carried out to understand the pattern formation in different parameter regimes. Starting with the linear stability analysis, the theoretical studies predict the space of Turing instability. To further study pattern selection in this space, weakly nonlinear analysis is carried out to obtain the regimes for different patterns. On the other hand, numerical simulations are carried out to fully investigate the interplay of diffusion and nonlinear reactions on pattern formations. To this end, the reaction-diffusion systems are solved by the Fourier pseudo-spectral method. Numerical results show that superdiffusion may substantially change the patterns in a reaction-diffusion system. Different superdiffusive exponents of the activator and inhibitor could cause both qualitative and quantitative changes in emergent spatial patterns. Comparing to single-component systems, the patterns observed in multi-component systems are more complexâ--Abstract, page iv
Qualitative Analysis of Reaction-Diffusion Systems in Neuroscience context
The aim of this article is to provide some insights in the qualitative
analysis of some Nonlinear Reaction-Diffusion systems arising in Neuroscience
context. We first intoduce a nonhomogeneous FitzHugh-Nagumo (nhFHN) featuring
both excitability and oscillatory properties. Then, we discuss the qualitative
analysis of a toy model related to nhFHN. In particular, we focus on the
convergence of solutions of the toy model toward different solutions (fixed
point, periodic) and show the existence of a cascade of Hopf Bifurcations.
Finally, we connect this analysis to the nhFHN system
Invasion moving boundary problem for a biofilm reactor model
The work presents the analysis of the free boundary value problem related to
the invasion model of new species in biofilm reactors. In the framework of
continuum approach to mathematical modelling of biofilm growth, the problem
consists of a system of nonlinear hyperbolic partial differential equations
governing the microbial species growth and a system of semi-linear elliptic
partial differential equations describing the substrate trends. The model is
completed with a system of elliptic partial differential equations governing
the diffusion and reaction of planktonic cells, which are able to switch their
mode of growth from planktonic to sessile when specific environmental
conditions are found. Two systems of nonlinear differential equations for the
substrate and planktonic cells mass balance within the bulk liquid are also
considered. The free boundary evolution is governed by a differential equation
that accounts for detachment. The qualitative analysis is performed and a
uniqueness and existence result is discussed. Furthermore, two special models
of biological and engineering interest are discussed numerically. The invasion
of Anammox bacteria in a constituted biofilm inhabiting the deammonification
units of the wastewater treatment plants is simulated. Numerical simulations
are run to evaluate the influence of the colonization process on biofilm
structure and activity.Comment: 20 pages, 11 figures, original pape
Influence of a road on a population in an ecological niche facing climate change
We introduce a model designed to account for the influence of a line with fast diffusionâsuch as a road or another transport networkâon the dynamics of a population in an ecological niche.This model consists of a system of coupled reaction-diffusion equations set on domains with different dimensions (line / plane). We first show that, in a stationary climate, the presence of the line is always deleterious and can even lead the population to extinction. Next, we consider the case where the niche is subject to a displacement, representing the effect of a climate change. We find that in such case the line with fast diffusion can help the population to persist. We also study several qualitative properties of this system. The analysis is based on a notion of generalized principal eigenvalue developed and studied by the authors (2019)
Putting reaction-diffusion systems into port-Hamiltonian framework
Reaction-diffusion systems model the evolution of the constituents distributed in space under the influence of chemical reactions and diffusion [6], [10]. These systems arise naturally in chemistry [5], but can also be used to model dynamical processes beyond the realm of chemistry such as biology, ecology, geology, and physics. In this paper, by adopting the viewpoint of port-controlled Hamiltonian systems [7] we cast reaction-diffusion systems into the portHamiltonian framework. Aside from offering conceptually a clear geometric interpretation formalized by a Stokes-Dirac structure [8], a port-Hamiltonian perspective allows to treat these dissipative systems as interconnected and thus makes their analysis, both quantitative and qualitative, more accessible from a modern dynamical systems and control theory point of view. This modeling approach permits us to draw immediately some conclusions regarding passivity and stability of reaction-diffusion systems. It is well-known that adding diffusion to the reaction system can generate behaviors absent in the ode case. This primarily pertains to the problem of diffusion-driven instability which constitutes the basis of Turingâs mechanism for pattern formation [11], [5]. Here the treatment of reaction-diffusion systems as dissipative distributed portHamiltonian systems could prove to be instrumental in supply of the results on absorbing sets, the existence of the maximal attractor and stability analysis. Furthermore, by adopting a discrete differential geometrybased approach [9] and discretizing the reaction-diffusion system in port-Hamiltonian form, apart from preserving a geometric structure, a compartmental model analogous to the standard one [1], [2] is obtaine
Bistable reaction equations with doubly nonlinear diffusion
Reaction-diffusion equations appear in biology and chemistry, and combine
linear diffusion with different kind of reaction terms. Some of them are
remarkable from the mathematical point of view, since they admit families of
travelling waves that describe the asymptotic behaviour of a larger class of
solutions of the problem posed in the real line. We
investigate here the existence of waves with constant propagation speed, when
the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In
the present setting we consider bistable reaction terms, which present
interesting differences w.r.t. the Fisher-KPP framework recently studied in
\cite{AA-JLV:art}. We find different families of travelling waves that are
employed to describe the wave propagation of more general solutions and to
study the stability/instability of the steady states, even when we extend the
study to several space dimensions. A similar study is performed in the critical
case that we call "pseudo-linear", i.e., when the operator is still nonlinear
but has homogeneity one. With respect to the classical model and the
"pseudo-linear" case, the travelling waves of the "slow" diffusion setting
exhibit free boundaries. \\ Finally, as a complement of \cite{AA-JLV:art}, we
study the asymptotic behaviour of more general solutions in the presence of a
"heterozygote superior" reaction function and doubly nonlinear diffusion
("slow" and "pseudo-linear").Comment: 42 pages, 11 figures. Accepted version on Discrete Contin. Dyn. Sys
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