부산대학교 도서관

The authors study a model of vegetation and water interaction in arid or semi-arid environments. To account for the lags and nonlocal effects observed in nature they modify a reaction-diffusion-type model of C.~A. Klausmeier [Science {\bf 284} (1999). no. 5421, 1826--1828, \doi{10.1126/science.284.5421.1826}] by introducing integral terms: $$ \align \frac{\partial w(\Vec{x}, t)}{\partial t} &= d\Delta w(\Vec{x}, t) +a - w(\Vec{x}, t) - w(\Vec{x}, t)p(\Vec{x}, t)\int_\Omega \phi_1(\|\Vec{x}-\Vec{x}\|)p(\Vec{y}, t) d\Vec{y},\\ \frac{\partial p(\Vec{x}, t)}{\partial t} &= \Delta p(\Vec{x}, t) - m p(\Vec{x}, t)\\ &+\int_\Omega \int_{-\infty}^0\phi_2(\|\Vec{x}-\Vec{x}\|, t-s)p(\Vec{y}, s)w(\Vec{y}, s)p(\Vec{x}, s) ds d\Vec{y}, \endalign $$ where $w$ is water density, $p$ is biomass density. The system is de-dimensionalized to put forward the ratio $d$ of the diffusion rate between water and vegetation. In the words of the authors: \par From the summary: ``It is shown that the temporal kernel function does not affect Turing bifurcation. For better understanding the influences of lag effect and nonlocal competition on the vegetation pattern formation, we choose some special kernel functions and obtain some insightful results: (i) Time delay does not trigger the vegetation pattern formation, but can postpone the evolution of vegetation. In addition, in the absence of diffusion, time delay can induce the occurrence of stability switches, while in the presence of diffusion, spatially nonhomogeneous time-periodic solutions may emerge, but there are no stability switches; (ii) The spatial nonlocal interaction may trigger the pattern onset for small diffusion ratio of water and vegetation, and can change the number and size of isolated vegetation patches for a large diffusion ratio. (iii) The interaction between time delay and spatial nonlocal competition may induce the emergence of traveling wave patterns, so that the vegetation remains periodic in space, but is oscillating in time. These results demonstrate that precipitation can significantly affect the growth and spatial distribution of vegetation.'' \par The authors perform local stability analysis of equilibria to analyze the bifurcations and present numerical simulations as well.