Mathematical Structure for Forest Bynamics with Interaction between Trees and Soils

我们关注的是森林系统的动态。将该系统视为具有一定相互作用的树木与土壤的杂交系统，引入了描述系统动力学的数学模型。该模型用非线性偏微分方程组表示。在本研究项目中，我们主要在土壤水平恒定和稳定的情况下对该模型进行了分析和数值研究。在这种情况下，我们构造了一个由模型方程确定的动力系统，构造了动力系统的Iyapunov函数，证明了每条轨迹都渐近地收敛于一个平稳解，并阐明了平衡结构，即模型方程的平稳解，随着代表森林状态的方程中的各种参数的变化而急剧变化。此外，我们发现该模型给出了一个清晰的边界，似乎与森林固有的森林边界相对应：这意味着通过观察最佳边界，我们可以反识别模型中的所有参数，从而了解森林系统的数学结构。这进一步表明，将来有可能通过对胸廓边界的观察来了解森林本身的健康程度或从破坏中恢复的能力。对于土壤层非恒定不稳定的情况，我们只进行了数值模拟。我们发现了一些有趣的准周期轨迹。但数学分析仍需进一步研究。

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.