Development Of Stand Density Logistic Growth Model Based On Difference Equation

As one of the important factors affecting the growth of stands,stand density can be effectively regulated by human activities and has an active role in forest management.The self-thinning relationship of a stand is the main point of stand density management,and the maximum size-density line is often used to describe the self-thinning relationship of a stand.Currently,the common methods of maximum density line construction often rely on the high density data of realistic stands,and lack the concept of "limit" of density growth,so that the limit value of stand density growth can not be effectively estimated.Therefore,In this study,the growth model of stand density was constructed and the asymptotic parameters of the growth model were used to construct the self-thinning line at the limit level.The data in this study are the remeasurement data of four paired adjacent interval fixed sample plots of tree forests in the continuous inventory system of forest resources in Zhejiang Province from 1999-2004,2004-2009,2009-2014,and 2014-2019.The modeling was divided into three broad tree species categories: pine,fir,and broadleaf.The specific approach of this study: the stand density index,which is not constrained by the stand age structure,was constructed based on Reineke’s self-thinning rule,and the difference equation was constructed by the inverse of the stand density index in the pre-late period of the fixed sample plots,and the parameters of the difference equation were estimated by least squares based on its nonlinear difference equation.A discrete stand density growth model and a continuous stand density index growth model with nominal age as the independent variable were constructed by the difference equation.It was pointed out that the continuous stand density exponential growth model had inflection points and the same asymptotic parameters as the discrete model.When the independent variable t was integer,the continuous model and the discrete model coincided exactly.The limiting density in the study is based on the asymptotic parameter of the density growth model,the limiting level of density growth,which is generally greater than the maximum density in a realistic stand.The limit size-density line constructed from the limit density as a parameter can break the limiting effect of the maximum density value of the real forest stand on the boundary self-thinning line.While the quantile regression method can get the maximum size-density line under different quantile by changing the value of quantile,but it cannot exceed the realistic maximum density value of the stand.The estimation results of stand limiting density(ha)were calculated by combining the data of four re-measurement periods of the three tree species major categories together,which showed that: fir(314,967)> pine(310,246)> broadleaf(294,460),and the magnitude of limiting density index was negatively correlated with wood density of the tree species major categories,which was consistent with the findings of previous studies.Innovations of this study:(1)A stand density index that is not constrained by the age structure of the stand was constructed,and a density differential equation was constructed and derived using the density index data from the pre and post retest sample plots.And the logistic growth model of continuous stand density with nominal age as the independent variable was obtained by solving the density differential equation.It provides a new way to construct a continuous growth model with nominal age as the independent variable in the case of having retest data but no age data.(2)The stable equilibrium point of the difference equation or the asymptotic parameter of the continuous-type growth model was proposed as the limiting growth density of the forest stand.And this is used to construct the limit size-density line which is fundamentally different from the maximum size-density line obtained by quantile regression method,the latter is directly influenced by the collection of sample data,while the former has relatively less influence.