Lotka–Volterra models and the global vegetation pattern

Abstract

How can such a classic object of mathematical ecology as the Lotka–Volterra competition model for a community of n competing species be used for description of the global vegetation dynamics under the climate change? We assume that the present spatial distribution of global vegetation is a stable equilibrium solution of corresponding Lotka–Volterra equations. The problem is how to construct some discrete structure on a continuous set of parameters, in this context remaining in a framework of a continuous model description? The problem can be solved if a dynamical system with multiple equilibria is considered. In this case, a continuous quasi-stationary change of parameters induces a jump from one to another equilibrium, since they have different stability domain in the parametric space. On this conceptual base a special class of Lotka–Volterra equations is developed and a biologically interpreted procedure for estimating their coefficients is suggested. For this we must have maps of the annual production and biomass, a life span of each vegetation type, and a map of the current geographical distribution of different vegetation, i.e. the current global vegetation pattern (GVP). The latter is needed for the construction of the ‘elementary map’, which is an important part of the model. Another important part is the formula, which describes an annual production dependence on the temperature and precipitation (for instance, Lieth's formula can be used). Considering a one-dimensional particular case for n=2 we have the analytical formulas describing a shift of borders between two vegetation zones under the climate change. It was shown that two different types of the transition zones, namely, ‘soft’ and ‘hard’, exist in this case. If the soft zone is characterised by the continuous and smooth replacement of one type of vegetation by another, then the hard zone is a typical ‘fractal’ structure with a mosaic of different vegetation. A special calibration procedure for the Lotka–Volterra model describing the dynamics of one-dimensional GVP with n types of vegetation is suggested. It is based on the stability theorem for a system with multiple equilibria and a special averaging method applying to real geographical distributions of the annual production and the biomasses. The results are used for estimating the shift of one transition zone between taiga and steppe in the Central Siberia under the climate change (CO 2-doubling scenario).