Mathematical Structure for Forest Bynamics with Interaction between Trees and Soils

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We are concerned with the dynamics of forest system. Considering the system as a hybrid system of trees and soils with some interactions, we introduced a mathematical model which describes the kinetics of system. The model is expressed by a system of nonlinear partial differential equations. In this research program we have studied analytically and numerically the model mainly in a case where the soil level is constant and stable. In such a case we constructed a dynamical system determined from the model equations, constructed a Iyapunov function for the dynamical system, showed that every trajectory converges asymptotically to a stationary solution and clarified that the structure of equilibria, i.e., stationary solutions of model equations, changes drastically depending on the various parameters in the equations which represent the state of the forest. Furthermore we have found out that this model gives us a clear boundary which may seem to correspond to the forest boundary which the forest possesses inherently: This means that observing the best boundary we can inversely identify all the parameters in the model to know the mathematical structure of the forest system. This suggests, furthermore, that it is possible in the future to know from observations of the bust boundary the degree of health or the power of restitution from destruction for the forest itself. As for the case where the soil level is non constant and unstable, we have only performed numerical simulations. We found out some interesting quasi-periodic trajectories. But the mathematical analysis remains to be studied in the future.