부산대학교 도서관

The authors study a model of vegetation and water interaction in aridor semi-arid environments. To account for the lags and nonlocal effectsobserved in nature they modify a reaction-diffusion-type model ofC.~A. Klausmeier [Science {\bf 284} (1999). no. 5421, 1826--1828,\doi{10.1126/science.284.5421.1826}] by introducing integral terms:$$\align \frac{\partial w(\Vec{x}, t)}{\partial t} &= d\Delta w(\Vec{x}, t) +a- w(\Vec{x}, t) - w(\Vec{x}, t)p(\Vec{x}, t)\int_\Omega\phi_1(\|\Vec{x}-\Vec{x}\|)p(\Vec{y}, t) d\Vec{y},\\ \frac{\partial p(\Vec{x},t)}{\partial t} &= \Delta p(\Vec{x}, t) - mp(\Vec{x}, t)\\ &+\int_\Omega \int_{-\infty}^0\phi_2(\|\Vec{x}-\Vec{x}\|,t-s)p(\Vec{y}, s)w(\Vec{y}, s)p(\Vec{x}, s) ds d\Vec{y},\endalign$$where $w$ is water density, $p$ is biomass density. The system isde-dimensionalized to put forward the ratio $d$ of the diffusion ratebetween water and vegetation. In the words of the authors: \parFrom the summary: ``It isshown that the temporal kernel function does not affect Turing bifurcation. Forbetterunderstanding the influences of lag effect and nonlocal competition on thevegetationpattern formation, we choose some special kernel functions and obtain someinsightful results: (i) Time delay does not trigger the vegetation patternformation, but canpostpone the evolution of vegetation. In addition, in the absence of diffusion,timedelay can induce the occurrence of stability switches, while in the presence ofdiffusion, spatially nonhomogeneous time-periodic solutions may emerge, butthere are nostability switches; (ii) The spatial nonlocal interaction may trigger thepattern onsetfor small diffusion ratio of water and vegetation, and can change the number andsizeof isolated vegetation patches for a large diffusion ratio. (iii) Theinteractionbetweentime delay and spatial nonlocal competition may induce the emergence oftravelingwave patterns, so that the vegetation remains periodic in space, but isoscillating intime. These results demonstrate that precipitation can significantly affect thegrowthand spatial distribution of vegetation.''\parThe authors perform local stabilityanalysis of equilibria to analyze the bifurcations and presentnumerical simulations as well.