Selected topics on reaction-diffusion-advection models from spatial ecology

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

On the dynamics of an epidemic patch model with mass-action transmission mechanism and asymmetric dispersal patterns

This paper examines an epidemic patch model with mass-action transmission mechanism and asymmetric connectivity matrix. Results on the global dynamics of solutions and the spatial structures of endemic solutions are obtained. In particular, we show that when the basic reproduction number $\mathcal{R}_0$ is less than one and the dispersal rate of the susceptible population $d_S$ is large, the population would eventually stabilize at the disease-free equilibrium. However, the disease may persist if $d_S$ is small, even if $\mathcal{R}_0<1$. In such a scenario, explicit conditions on the model parameters that lead to the existence of multiple endemic equilibria are identified. These results provide new insights into the dynamics of infectious diseases in multi-patch environments. Moreover, results in [27], which is for the same model but with symmetric connectivity matrix, are generalized and improved

Global dynamics of epidemic network models via construction of Lyapunov functions

In this paper, we study the global dynamics of epidemic network models with standard incidence or mass-action transmission mechanism, when the dispersal of either the susceptible or the infected people is controlled. The connectivity matrix of the model is not assumed to be symmetric. Our main technique to study the global dynamics is to construct novel Lyapunov type functions

Migration paths saturations in meta-epidemic systems

In this paper we consider a simple two-patch model in which a population affected by a disease can freely move. We assume that the capacity of the interconnected paths is limited, and thereby influencing the migration rates. Possible habitat disruptions due to human activities or natural events are accounted for. The demographic assumptions prevent the ecosystem to be wiped out, and the disease remains endemic in both populated patches at a stable equilibrium, but possibly also with an oscillatory behavior in the case of unidirectional migrations. Interestingly, if infected cannot migrate, it is possible that one patch becomes disease-free. This fact could be exploited to keep disease-free at least part of the population

Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments

This work examines the dynamics of solutions of a two-strain SIS epidemic model in patchy environments. The basic reproduction number $\mathcal{R}_0$ is introduced, and sufficient conditions are provided to guarantee the global stability of the disease-free equilibrium (DFE). In particular, the DFE is globally stable when either: (i) $\mathcal{R}_0\le \frac{1}{k}$, where $k\ge 2$ is the total number of patches, or (ii) $\mathcal{R}_0<1$ and the dispersal rate of the susceptible population is large. Moreover, the questions of competition-exclusion and coexistence of the strains are investigated when the single-strain reproduction numbers are greater than one. In this direction, under some appropriate hypotheses, it is shown that the strain whose basic reproduction number and local reproduction function are the largest always drives the other strain to extinction in the long run. Furthermore, the asymptotic dynamics of the solutions are presented when either both strain's local reproduction functions are spatially homogeneous or the population dispersal rate is uniform. In the latter case, the invasion numbers are introduced and the existence of coexistence endemic equilibrium (EE) is proved when these invasion numbers are greater than one. Numerical simulations are provided to complement the theoretical results.Comment: 35 page