Threshold dynamics of a reaction-advection-diffusion schistosomiasis epidemic model with seasonality and spatial heterogeneity.

Most water-borne disease models ignore the advection of water flows in order to simplify the mathematical analysis and numerical computation. However, advection can play an important role in determining the disease transmission dynamics. In this paper, we investigate the long-term dynamics of a periodic reaction-advection-diffusion schistosomiasis model and explore the joint impact of advection, seasonality and spatial heterogeneity on the transmission of the disease. We derive the basic reproduction number R 0 and show that the disease-free periodic solution is globally attractive when R 0 < 1 whereas there is a positive endemic periodic solution and the system is uniformly persistent in a special case when R 0 > 1 . Moreover, we find that R 0 is a decreasing function of the advection coefficients which offers insights into why schistosomiasis is more serious in regions with slow water flows.

Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping.

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.

Global dynamics of a diffusive competition model with habitat degradation.

In this paper, we propose a diffusive competition model with habitat degradation and homogeneous Neumann boundary conditions in a bounded domain that is partitioned into the healthy region (undisturbed habitat) and the degraded region (due to anthropogenic habitat disturbance). Species follow the Lotka-Volterra competition in the healthy region while in the degraded region species experience only exponential decay (not necessarily at the same rate). This setup is novel in that it requires no positivity assumption on the environmental heterogeneity, either absolute or on average, which would be far too restrictive for the study of the effects of habitat degradation. We rigorously show competitive exclusion and coexistence via global stability analysis. A remarkable finding is that the quality heterogeneity of landscapes can lead to the competitive exclusion of the slower species by the faster species. This result is robust as long as the degraded region has positive area, and moreover is at odds with classical results predicting the deterministic extinction of the stronger species. On the other hand, if the degraded region has intermediate negative effect on the faster competitor, species can coexist. Differing from comparable existing results, coexistence does not rely on a limit as the diffusion coefficients tend to zero or infinity. Together, these results imply that coexistence is always a possibility under this basic, yet general, configuration, providing insights into the varying impacts found through empirical study of habitat loss and fragmentation on species.