An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift.

In this paper, we consider a predator-prey model describing intraguild predation with memory-based diffusion. We first investigate the stability of two semi-trivial constant steady states and find that they are unstable if the positive steady state exists and can not be stable at the same time. Then we study the stability of the positive constant steady state and find that system has no Turing bifurcation when the predator is assumed to forage the prey and multiple stability switches can arise through Hopf bifurcation with joint effects of the memory-based diffusion and delay. Numerical simulations also show that the competition between two consumers has also important influence on complex spatiotemporal patterns formation of system.

In this paper, we propose a reaction-diffusion-advection SIS epidemic model with linear external source to study the effects of open advective environments on the persistence and extinction of infectious diseases. Threshold-type results on the global dynamics in terms of the basic reproduction number $ \mathcal{R}_{0} $ are established. It is found that the introduction of open advective environments leads to different monotonicity and asymptotic properties of the basic reproduction number $ \mathcal{R}_0 $ with respect to the diffusion rate $ d_I $ and advection speed $ q $. Our analytical results suggest that increasing the advection speed or decreasing the diffusion rate of infected individuals helps to eradicate the diseases in open advective environments.

In this paper, we investigate the global dynamics of a Leslie-Gower predator-prey system in advective homogeneous environments. We discuss the existence and uniqueness of positive steady-state solutions. We study the large time behavior of solutions and establish threshold conditions for persistence and extinction of two species when they live in open advective environments. Numerical simulations indicate that the introduction of advection leads to the evolution of spatial distribution patterns of species and specially it may induce spatial separation of the prey and predator under some conditions.