Existence and Stability of Stationary States of a Reaction–Diffusion-Advection Model for Two Competing Species

2020 ◽  
Vol 30 (05) ◽  
pp. 2050065
Author(s):  
Li Ma ◽  
De Tang
Keyword(s):  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Similar Species ◽  
Resident Species ◽  
Competition System ◽  
Special Cases ◽  
Existence And Stability ◽  
Maximal Growth Rate ◽  
Steady State Solutions ◽  
Monotone Dynamical Systems

It is well known that the research of two species in the Lotka–Volterra competition system could create very interesting dynamics. In our paper, we investigate the global dynamical behavior of a classic Lotka–Volterra competition system by studying the steady states and corresponding stability by mainly employing the methods of monotone dynamical systems theory, Lyapunov–Schmidt reduction and spectral theory and so on. It illustrates that the dynamical behavior substantially relies on certain variable of the maximal growth rate. Furthermore, we obtain that one of the semi-trivial steady state solutions is a global attractor in some special cases. In biology, these results show that both of the species do not coexist and the mutant forces the extinction of resident species under some condition for two similar species system.

scholarly journals Dynamics of an SIRS epidemic model with cross-diffusion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yaru Hu ◽  
Jinfeng Wang
Keyword(s):  
Dynamical Behavior ◽  
Disease Model ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Threshold Dynamics ◽  
Cross Diffusion ◽  
Asymptotic Profile ◽  
Reaction Diffusion Model ◽  
Special Cases ◽  
Globally Stable

<p style='text-indent:20px;'>The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>, it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_{0} $\end{document}</tex-math></inline-formula> even in the case of chemotaxis effects: if <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_{0}&lt;1 $\end{document}</tex-math></inline-formula>, the unique disease free equilibrium is globally stable; if <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{0}&gt;1 $\end{document}</tex-math></inline-formula>, the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.</p>

scholarly journals Numerical results on existence and stability of steady state solutions for the reaction-diffusion and Klein–Gordon equations

2014 ◽  
Vol 7 (6) ◽  
pp. 723-742 ◽  
Author(s):  
Miles Aron ◽  
Peter Bowers ◽  
Nicole Byer ◽  
Robert Decker ◽  
Aslihan Demirkaya ◽  
...  
Keyword(s):  
Steady State ◽  
Reaction Diffusion ◽  
Numerical Results ◽  
Existence And Stability ◽  
Klein Gordon ◽  
Steady State Solutions

Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis

2018 ◽  
Vol 28 (11) ◽  
pp. 2275-2312 ◽  
Author(s):  
Sainan Wu ◽  
Jinfeng Wang ◽  
Junping Shi
Keyword(s):  
Reaction Diffusion ◽  
Sensitivity Coefficient ◽  
Spatial Dimension ◽  
Epidemic Spreading ◽  
Susceptible Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability ◽  
Steady State Solutions

We propose a new reaction–diffusion predator–prey model system with predator-taxis in which the preys could move in the opposite direction of predator gradient. A similar situation also occurs when susceptible population avoids the infected ones in epidemic spreading. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any predator-taxis sensitivity coefficient are proved. It is also shown that such predator-taxis does not qualitatively affect the existence and stability of coexistence steady state solutions in many cases. For diffusive predator–prey system with diffusion-induced instability, it is shown that the presence of predator-taxis may annihilate the spatial patterns.

scholarly journals CAS WAVELETS METHOD TO SOLVE REACTION-DIFFUSION SYSTEM AND COMPARE IT WITH (G.F.E) METHOD

2020 ◽  
Vol 17 (35) ◽  
pp. 1110-1123
Author(s):  
Badran Jasim SALIM ◽  
Oday Ahmed JASIM
Keyword(s):  
Steady State ◽  
Finite Elements ◽  
Reaction Diffusion ◽  
Partial Differential ◽  
Reaction Diffusion Systems ◽  
Time Frequency ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Cas Wavelets ◽  
Steady State Solutions

Wavelet analysis plays a prominent role in various fields of scientific disciplines. Mainly, wavelets are very successfully used in signal analysis for waveform representation and segmentation, time-frequency analysis, and fast algorithms in the propagation equations and reaction. This research aimed to guide researchers to use Cos and Sin (CAS) to approximate the solution of the partial differential equation system. This method has been successfully applied to solve a coupled system of nonlinear Reaction-diffusion systems. It has been shown CAS wavelet method is quite capable and suited for finding exact solutions once the consistency of the method gives wider applicability where the main idea is to transform complex nonlinear partial differential equations into algebraic equation systems, which are easy to handle and find a numerical solution for them. By comparing the numerical solutions of the CAS and Galerkin finite elements methods, the answer of nonlinear Reaction-diffusion systems using the CAS wavelets for all tˆ and x values is accurate, reliable, robust, promising, and quickly arrives at the exact solution. When parameters 𝜀1 𝑎𝑛𝑑 𝜀2 are growing and with L decreasing, then the CAS method converges to steady-state solutions quickly (the less L, the more accurate the solution). It is converging towards steady-state solutions faster than and loses steps over time. Moreover, the results also show that the solution of the CAS wavelets is more reliable and faster compared to the Galerkin finite elements (G.F.E).

Practical persistence in ecological models via comparison methods

1996 ◽  
Vol 126 (2) ◽  
pp. 247-272 ◽  
Author(s):  
Robert Stephen Cantrell ◽  
Chris Cosner
Keyword(s):  
Reaction Diffusion ◽  
Basic Question ◽  
Positive Equilibrium ◽  
Uniform Persistence ◽  
Interacting Species ◽  
Attracting Set ◽  
Alternative Approach ◽  
Interacting Populations ◽  
Existence And Stability ◽  
Stable Periodic

A basic question in mathematical ecology is that of deciding whether or not a model for the population dynamics of interacting species predicts their long-term coexistence. A sufficient condition for coexistence is the presence of a globally attracting positive equilibrium, but that condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibrium, it may be difficult to establish its existence and stability, especially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or permanence, where coexistence is inferred from the existence of a globally attracting positive set. The advantage of that approach is that often uniform persistence can be shown much more easily than the existence of a globally attracting equilibrium. The disadvantage is that most techniques for establishing uniform persistence do not provide any information on the size or location of the attracting set. That is a serious drawback from the applied viewpoint, because if the positive attracting set contains points that represent less than one individual of some species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to seek asymptotic lower bounds on the populations or densities in the model, via comparison with simpler equations whose dynamics are better known. If such bounds can be obtained and approximately computed, then the prediction ofpersistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be established for some classes of reaction–diffusion models for interacting populations. Somewhat surprisingly, themodels need not be autonomous or have any specific monotonicity properties.

scholarly journals NEW EXACT SOLUTIONS OF SPATIALLY AND TEMPORALLY VARYING REACTION-DIFFUSION EQUATIONS

2008 ◽  
Vol 06 (04) ◽  
pp. 371-381 ◽  
Author(s):  
NALINI JOSHI ◽  
TEGAN MORRISON
Keyword(s):  
Reaction Diffusion ◽  
Elliptic Functions ◽  
Point Of View ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Source Terms ◽  
Symmetry Approach ◽  
New Exact Solutions ◽  
Special Cases ◽  
New Feature

This paper considers reaction-diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painlevé transcendents. We consider reaction-diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painlevé equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that are not integrable are also considered. Solutions are constructed using a Lie symmetry approach.

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