Global dynamics of a competition model with non-local dispersal I: The shadow system

Equations with non-local dispersal have been widely used as models in biology. In this paper we focus on logistic models with non-local dispersal, for both single and two competing species. We show the global convergence of the unique positive steady state for the single equation and derive various properties of the positive steady state associated with the dispersal rate. We investigate the effects of dispersal rates and inter-specific competition coefficients in a shadow system for a two-species competition model and completely determine the global dynamics of the system. Our results illustrate that the effect of dispersal in spatially heterogeneous environments can be quite different from that in homogeneous environments.     

Transient and asymptotic dynamics of a prey–predator system with diffusion

In this paper, we study a prey–predator system associated with the classical Lotka–Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞ . Next, we take the shadow system to approximate the original system as D → ∞ . The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞ . Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D₀ > 0 such that if , the time period of the ODE, satisfies , then the solution to the original system with D ≥ D₀ cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence estimates of the dynamics of a hyperbolic system with variable coefficients

A damped hyperbolic equation with a dissipative nonlinearity posed in the energy space is considered. The differential operator involved is not the Laplace operator but rather the operator ? div(a_?(x) ? ( · )) that has its coefficient depending on a parameter ?. We analyze the behavior of the global attractors as the parameter ? tends to 0. It is assumed that the coefficient a_?(x) has a well determined behavior with the parameter, and the idea is to relate the distance of the global attractors with the magnitude a_? ? a₀, measured in an appropriate norm. Copyright © 2013 John Wiley & Sons, Ltd.     

Global dynamics of the Boussinesq‐Burgers system with large initial data

In this paper, we study the global existence and asymptotic behavior of the Boussinesq‐Burgers system subject to the Dirichlet boundary conditions. Based on the L^p(p > 2) estimates of the solution, which are different from the standard L²‐based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

Water resources planning based on complex system dynamics: A case study of Tianjin city

A complex system dynamic (SD) model focusing on water resources, termed as TianjinSD, is developed for the integrated and scientific management of the water resources of Tianjin, which contains information feedback that governs interactions in the system and is capable of synthesizing component-level knowledge into system behavior simulation at an integrated level, thus presenting reasonable predictive results for policy-making on water resources allocation and management. As for the Tianjin city, interactions among 96 components for 12 years are explored and four planning alternatives are chosen, one of which is based on the conventional mode assuming that the existing pattern of human activities will be prevailed, while the others are alternative planning designs based on the interaction of local authorities and planning researchers. Optimal mode is therefore obtained according to different scenarios when compared the simulation results for evaluation of different decisions and dynamic consequences.