scholarly journals The influence of autotoxicity on the dynamics of vegetation spots

2020 ◽  
Author(s):  
Annalisa Iuorio ◽  
Frits Veerman
Keyword(s):  
Dynamical System ◽  
Reaction Diffusion ◽  
Homoclinic Orbits ◽  
Spatial Dimension ◽  
Travelling Wave ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Scaling Analysis ◽  
Water Models ◽  
Vegetation Water

AbstractPlant autotoxicity has proved to play an essential role in the behaviour of local vegetation. We analyse a reaction-diffusion-ODE model describing the interactions between vegetation, water, and autotoxicity. The presence of autotoxicity is seen to induce movement and deformation of spot patterns in some parameter regimes, a phenomenon which does not occur in classical biomass-water models. We aim to analytically quantify this novel feature, by studying travelling wave solutions in one spatial dimension. We use geometric singular perturbation theory to prove the existence of symmetric, stationary and non-symmetric, travelling pulse solutions, by constructing appropriate homoclinic orbits in the associated 5-dimensional dynamical system. In the singularly perturbed context, we perform an extensive scaling analysis of the dynamical system, identifying multiple asymptotic scaling regimes where (travelling) pulses may or may not be constructed. We discuss the agreement and discrepancy between the analytical results and numerical simulations. Our findings indicate how the inclusion of an additional ODE may significantly influence the properties of classical biomass-water models of Klausmeier/Gray–Scott type.

On the Application of Geometric Singular Perturbation Theory to Some Classical Two Point Boundary Value Problems

1998 ◽  
Vol 08 (02) ◽  
pp. 189-209 ◽  
Author(s):  
Michael Hayes ◽  
Tasso J. Kaper ◽  
Nancy Kopell ◽  
Kinya Ono
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation

In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly-perturbed boundary value problems. The central theme is that the criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: ∊y″+αy′+βy=0, semilinear: ∊y″+αy′+f(y)=0, and quasilinear: ∊y″+g(y) y′+f(y)=0, on the interval [0,1], where t∈[0,1], ′=d/dt, 0<∊≪1, and general boundary conditions y(0)=A, y(1)=B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

1991 ◽  
Vol 334 (1633) ◽  
pp. 1-24 ◽  
Keyword(s):  
Travelling Waves ◽  
Chemical Species ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Travelling Wave ◽  
Diffusion Waves ◽  
Long Time ◽  
Diffusion Rates ◽  
Cubic Autocatalysis

We study the isothermal autocatalytic system , A + n B → ( n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v * n ,where v * n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v * 2 , this rate of decay is only algebraic for faster travelling waves with v > v * 2 . This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.

scholarly journals A Laplace transform finite difference scheme for the Fisher-KPP equation

2021 ◽  
Vol 15 ◽  
pp. 174830262199958
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

scholarly journals Canard solutions in neural mass models: consequences on critical regimes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Elif Köksal Ersöz ◽  
Fabrice Wendling
Keyword(s):  
Temporal Dynamics ◽  
Spatial Scales ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Critical Transitions ◽  
Intracerebral Eeg ◽  
Neural Mass ◽  
Mesoscopic Level ◽  
Eeg Recordings ◽  
Physiological Background

AbstractMathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

scholarly journals Allee dynamics: growth, extinction and range expansion

2017 ◽  
Author(s):  
I Bose ◽  
M Pal ◽  
C Karmakar
Keyword(s):  
Population Density ◽  
Range Expansion ◽  
Population Biology ◽  
Allee Effect ◽  
Additive Noise ◽  
Finite Population ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
One Dimension ◽  
Negative Growth

AbstractIn population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this Letter, we study a reaction-diffusion (RD) model of population growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortality rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e., extinction. The early signatures of the transition to extinction at a bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of travelling wave solutions of the population density is demonstrated. The parameter regimes in which the travelling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.

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