Mathematical modelling of forest growth, spread and vegetation pattern formation

Abstract

The mechanism for growth, spread and vegetation pattern formation is largely unknown and poorly understood. To improve understanding of this mechanism, two mathematical models each consisting of three nonlinear partial differential equations for surface water balance (W), soil water balance (N) and plant biomass density variable (P) to investigate the dynamics of forest growth and vegetation pattern formation were developed. The models have a parameter that accounts for the influence of the interactions among multiple resources such as light, water, temperature and nutrients on the growth, spread and vegetation pattern formation. The methods used include Michaelis-Menten Kinetics for the rate of nutrients uptake by a cell or organism for growth; Continuous-Time Markov (CTM) method as a standardised methodology that describes plant metabolism responses to multiple resource inputs and the Taylor Series Expansion method used to linearise the nonlinear models formulated in order to explain the dynamics of the growth, spread and vegetation pattern formation of the forest. The linear stability analysis of homogeneous steady-state solutions provided a reliable predictor of the onset and nature of pattern formation in the reaction-diffusion systems.Thus, the homogeneous plant equilibrium decreases with decreasing rainfall until plant become extinct. Finally, numerical simulations of system of partial differential equation models were carried out based on different fertility levels under different water conditions. The simulation results show that, regardless of the parameter space, and the level of iii precipitation, the shift of the vegetation cover from uniform to gaps, labyrinths, spots, and into bare soil or almost bare soil is possible. The proposed model derived in the study could be applied to any vegetation type. The model could be used to further analyse the conditions for the development of dynamic patterns and their occurrence in different biological systems.