A Memory-Lattice Model for Vegetation Dynamics and Pattern Formation in Resource-strained Environments.

Abstract:

Vegetation over a variety of terrains with adverse environmental conditions is reported to abandon spatial homogeneity, and exhibit a number of curious self-organizing patterns instead. Over the years, numerous studies have been put forth to explain the origins of such behaviour and the mechanisms involved in the formation of these observed structures. Several of these use a two-species reaction-diffusion model that describes the patterns formed as a consequence of an interplay between the species of vegetation and the necessary resources in the region, usually considered arid. Some, through a detailed analysis of a few widely accepted models, have suggested interpreting these as warning signs of an impending desertification. However, the current understanding and insights that these models provide us with seem to be partial and not entirely satisfactory. We investigate the phenomenon through a memory lattice model and attempt to explain the resulting criticalities by invoking ideas from non-linear dynamics as well as non-equilibrium statistical mechanics. A novel feature of our proposal is that, while the dynamics are controlled by the diffusion rate and survival probability of the species, the effects of resource depletion are incorporated as a memory element that additionally decides both the survival and the diffusion of vegetation at a given site on the landscape. Besides exhibiting some of the well-known vegetation patterns, our model also shows critical behaviour possibly leading to desertification under certain conditions. Moreover, our simulations also suggest that such critical transitions may be averted by a suitable range of initial density values for vegetation.

Attached with this, is one example of a lattice-state simulation with certain model parameter values. A small number of lattice sites were randomly seeded and numerically evolved using our model. Circles/ring-like structures, arcs, spirals, periodic banded patterns, fragmented maze-like structures and finally isolated patches/spots are all seen as a part of a single evolution! Isotropic and homogeneous landscape/lattice conditions were considered for the simulation in consideration.