The ―Determinant Of The Jacobian Matrix‖ As APurely Determining Factor Of A Vegetation PatternFormation Under Turing Instability

摘要

The mechanism for growth, spread and vegetation pattern formation is largely unknown and poorly understood. To improve understanding of this mechanism, a mathematical model consisting of two nonlinear partial differential equations for soil water balance (N) and plant biomass density variable (P) to investigate the dynamics of forest growth and vegetation pattern formation was developed. The methods used include Michaelis-Menten Kinetics for the rate of nutrients uptake by a cell or organism for growth and Continuous-Time Markov (CTM) method as a standardized methodology that describes plant metabolism responses to multiple resource inputs. This CTM technique was used to obtain a simple plant growth component by synthesizing the four resources (light, water and nutrients together with temperature). To linearize the nonlinear model formulated in order to explain the dynamics of the growth, spread and vegetation pattern formation of the forest, the Taylor Series Expansion method was applied. 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. The results revealed that, stability conditions needed for pattern formation is possible provided that . Thus, the homogeneous plant equilibrium decreases with decreasing rainfall until plant become extinct. Based on this condition, the trace and determinant criteria for stability were obtained as and respectively. Again, as increases or decreases, also increases or decreases respectively irrespective of the values of the other parameters. This suggests that which is a surrogate for a dimensionless infiltration capacity prohibits pattern formation at high levels. Hence, one may therefore not expect vegetation patterns to exist in situations of high fertility level and rich water condition. However, this is not the case.