Stationary distribution of a stochastic vegetation–water system with reaction–diffusion

<abstract><p>In this paper, a reaction-diffusion vegetation-water system with time-varying delay, impulse and L$ {\rm \acute{e}} $vy jump is proposed. The existence and uniqueness of the positive solution are proved. Meanwhile, mainly through the principle of comparison, we obtain the sufficient conditions for finite-time stability which reflect the effect of time delay, diffusion, impulse, and noise. Besides, considering the planting, irrigation and other measures, we introduce control variable into the vegetation-water system. In order to save the costs of strategies, the optimal control is analyzed by using the minimum principle. Finally, numerical simulations are shown to illustrate the effectiveness of our theoretical results.</p></abstract>

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The influence of autotoxicity on the dynamics of vegetation spots

10.1101/2020.07.29.226522 ◽

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.

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Normal approximations to discrete unimodal distributions

Journal of Applied Probability ◽

10.1017/s0021900200115931 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1013-1018

Author(s):

B. G. Quinn ◽

H. L. MacGillivray

Keyword(s):

Stationary Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Hypergeometric Distribution ◽

Death Process ◽

Normal Approximations ◽

Discrete Random Variables ◽

Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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Singular limit of reaction–diffusion systems and modified motion by mean curvature

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050200046x ◽

2002 ◽

Vol 132 (04) ◽

pp. 951

Keyword(s):

Mean Curvature ◽

Reaction Diffusion ◽

Singular Limit ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Motion By Mean Curvature

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Surface reactions observed by photoemission electron microscopy (PEEM)

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100121831 ◽

1992 ◽

Vol 50 (1) ◽

pp. 286-287

Author(s):

H.H. Rotermund

Keyword(s):

Electron Microscopy ◽

Electron Microscope ◽

Work Function ◽

Chemical Reactions ◽

Reaction Diffusion ◽

Dynamic Processes ◽

Clean Surface ◽

Crystal Surfaces ◽

Single Crystal Surfaces ◽

Photoemission Electron

Chemical reactions at a surface will in most cases show a measurable influence on the work function of the clean surface. This change of the work function δφ can be used to image the local distributions of the investigated reaction,.if one of the reacting partners is adsorbed at the surface in form of islands of sufficient size (Δ>0.2μm). These can than be visualized via a photoemission electron microscope (PEEM). Changes of φ as low as 2 meV give already a change in the total intensity of a PEEM picture. To achieve reasonable contrast for an image several 10 meV of δφ are needed. Dynamic processes as surface diffusion of CO or O on single crystal surfaces as well as reaction / diffusion fronts have been observed in real time and space.

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