CROSS-DIFFUSION INDUCED INSTABILITY IN LVLEV–TANNER MODEL

2011 ◽  
Vol 04 (04) ◽  
pp. 431-442 ◽  
Author(s):  
WENZHEN GAN ◽  
PENG ZHU ◽  
JIE BAO
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Asymptotically Stable ◽  
Homogeneous Steady State ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Ode System ◽  
Uniform Steady State ◽  
Parameter Values

A Lvlev–Tanner model with cross-diffusion is considered. We analyze the positive uniform steady state and obtain conditions on the parameter values such that the homogeneous steady state is locally asymptotically stable both in the related ODE system and in the PDE system with self-diffusion. Once also cross-diffusion is considered in the model, the uniform steady state is shown to be unstable under some conditions. Numerical simulations are also presented.

scholarly journals Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Feifan Zhang ◽  
Wenjiao Zhou ◽  
Lei Yao ◽  
Xuanwen Wu ◽  
Huayong Zhang
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Analysis ◽  
Discrete Model ◽  
Continuous Model ◽  
Spatiotemporal Patterns ◽  
Homogeneous Steady State ◽  
Simulation Results

In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.

Solitary waves in excitable systems with cross-diffusion

2005 ◽  
Vol 461 (2064) ◽  
pp. 3711-3730 ◽  
Author(s):  
Vadim N Biktashev ◽  
Mikhail A Tsyganov
Keyword(s):  
Asymptotic Behaviour ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
System Of Equations ◽  
Soliton Interaction ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Excitable Systems

We consider a FitzHugh–Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh–Nagumo system with self-diffusion, but similar to a predator–prey model with taxis of populations on each other's gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.

scholarly journals Numerical Simulations of a Delay Model for Immune System-Tumor Interaction

2018 ◽  
Vol 23 (1) ◽  
pp. 19 ◽  
Author(s):  
Mohamed A. Hajji ◽  
Qasem Al-Mdallal
Keyword(s):  
Steady State ◽  
Immune System ◽  
Numerical Simulations ◽  
Relative Rate ◽  
Delay Model ◽  
Effector Cells ◽  
Rate Of Increase ◽  
System Tumor ◽  
Delay Differential ◽  
Parameter Values

In this paper we consider a system of delay differential equations as a model for the dynamics of tumor-immune system interaction. We carry out a stability analysis of the proposed model. In particular, we show that the system can have up to two steady states: the tumor free steady state, which always exist, and the tumor persistent steady state, which exists only when the relative rate of increase of the tumor cells exceeds the ratio between the natural proliferation rate and the relative death rate of the effector cells. We also determine an upper bound for the delay, such that stability is preserved. Numerical simulations of the system under different parameter values are performed.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

scholarly journals Mathematical Analysis of Malaria-Schistosomiasis Coinfection Model

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
E. A. Bakare ◽  
C. R. Nwozo
Keyword(s):  
Numerical Simulations ◽  
Endemic Equilibrium ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Mathematical Techniques ◽  
Parameter Values ◽  
The Impact ◽  
Number Of Individuals ◽  
Disease Free

We formulated and analysed a mathematical model to explore the cointeraction between malaria and schistosomiasis. Qualitative and comprehensive mathematical techniques have been applied to analyse the model. The local stability of the disease-free and endemic equilibrium was analysed, respectively. However, the main theorem shows that if RMS<1, then the disease-free equilibrium is locally asymptotically stable and the phase will vanish out of the host and if RMS>1, a unique endemic equilibrium is also locally asymptotically stable and the disease persists at the endemic steady state. The impact of schistosomiasis and its treatment on malaria dynamics is also investigated. Numerical simulations using a set of reasonable parameter values show that the two epidemics coexist whenever their reproduction numbers exceed unity. Further, results of the full malaria-schistosomiasis model also suggest that an increase in the number of individuals infected with schistosomiasis in the presence of treatment results in a decrease in malaria cases. Sensitivity analysis was further carried out to investigate the influence of the model parameters on the transmission and spread of malaria-schistosomiasis coinfection. Numerical simulations were carried out to confirm our theoretical findings.

Turing Patterns in the Lengyel–Epstein System with Superdiffusion

2017 ◽  
Vol 27 (08) ◽  
pp. 1730026 ◽  
Author(s):  
Biao Liu ◽  
Ranchao Wu ◽  
Naveed Iqbal ◽  
Liping Chen
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Complex Dynamics ◽  
Multiple Scale ◽  
Turing Instability ◽  
Amplitude Equations ◽  
Homogeneous Steady State ◽  
Weakly Nonlinear ◽  
Homogeneous Solutions ◽  
Normal Diffusion

Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

scholarly journals Dynamical Analysis of a Plateau Pika with Cross-Diffusion under Contraception Control

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiaoyan Wang ◽  
Junyuan Yang ◽  
Fengqin Zhang
Keyword(s):  
Steady State ◽  
Local Stability ◽  
Degree Theory ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Plateau Pika ◽  
Globally Asymptotically Stable ◽  
Cross Diffusion ◽  
Contraception Control ◽  
Diffusion Rates

A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed whend21is small enough. However, whend21is large enough, the model shows Turing bifurcation ifB2 -4AC > 0. Furthermore, it is proved that if,R > R0, βK > dand cross-diffusion rates are zero, the positive coexistence steady state is globally asymptotically stable. A nonconstant positive solution bifurcates from the coexistent steady state by the Leray-Schauder degree theory. Numerical simulations are carried out to illustrate the main results.

Bifurcation of Reaction Cross-Diffusion Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750049 ◽  
Author(s):  
Rong Zou ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Neumann Boundary ◽  
Reaction Cross ◽  
Homogeneous Steady State ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Existence And Multiplicity ◽  
Steady State Solutions

This paper is devoted to a reaction cross-diffusion system under Neumann boundary conditions. Firstly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov–Schmidt reduction. Next, the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail. In particular, the Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are determined by using center manifold reduction and normal form theory. Finally, some of the main results are illustrated by an application to a predator–prey model with Allee effect and one-dimensional spatial domain [Formula: see text].

Reexamination of the Serendipity Theorem from the stability viewpoint

2019 ◽  
Vol 85 (1) ◽  
pp. 43-70
Author(s):  
Akira Momota ◽  
Tomoya Sakagami ◽  
Akihisa Shibata
Keyword(s):  
Growth Rate ◽  
Steady State ◽  
Population Growth ◽  
Numerical Simulations ◽  
Policy Implication ◽  
Population Growth Rate ◽  
Golden Rule ◽  
The Social ◽  
The Stability ◽  
Parameter Values

AbstractThis paper reexamines the Serendipity Theorem of Samuelson (1975) from the stability viewpoint, and shows that, for the Cobb–Douglas preference and CES technology, the most-golden golden-rule lifetime state being stable depends on parameter values. In some situations, the Serendipity Theorem fails to hold despite the fact that steady-state welfare is maximized at the population growth rate, since the steady state is unstable. Through numerical simulations, a more general case of CES preference and CES technology is also examined, and we discuss the realistic relevance of our results. We present the policy implication of our result, that is, in some cases, the steady state with the highest utility is unstable, and thus a policy that aims to achieve the social optima by manipulating the population growth rate may lead to worse outcomes.

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