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Register a new userSelected topics on reaction-diffusion-advection models from spatial ecology
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We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented.
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In this paper, we prove some qualitative properties of pushed fronts for the periodic reaction-diffusion-equation with general monostable nonlinearities. Especially, we prove the exponential behavior of pushed fronts when they are approaching their unstable state which has been left open so far. Through this property, we also prove the stability of pushed fronts.
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.
This paper aims to explore the temporal-spatial spreading and asymptotic behaviors of West Nile virus by a reaction-advection-diffusion system with free boundaries, especially considering the impact of advection term on the extinction and persistence of West Nile virus. We define the spatial-temporal risk index $R^{F}_{0}(t)$ with the advection rate and the general basic disease reproduction number $R^D_0$ to get the vanishing-spreading dichotomy regimes of West Nile virus. We show that there exists a threshold value $mu^{*}$ of the advection rate, and obtain the threshold results of it. When the spreading occurs, we investigate the asymptotic dynamical behaviors of the solution in the long run and first give a sharper estimate that the asymptotic spreading speed of the leftward front is less than the rightward front for $0<mu<mu^*$. At last, we give some numerical simulations to identify the significant effects of the advection.
This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.
Finite difference/element/volume methods of discretising PDEs impose a subgrid scale interpolation on the dynamics. In contrast, the holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is fully informed by the underlying microscale dynamics. We establish a new proof that in principle there exists an exact closure of the dynamics of a general class of reaction-advection-diffusion PDEs, and show how our approach constructs new systematic approximations to the in-principle closure starting from a simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers PDE, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to be able to systematically construct automatically good, macroscale discretisations to a wide range of PDEs, including wave PDEs.
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