At different stages of life, the growth rate of a tree is not the same. This presentation is to: build a mathematical model according to the regular pattern which has been observed and then solve the problem and optimize the established model. Problem Proposition A newly planted tree grows slowly, but gradually the tree grows tall and will grow at a faster speed. But when it grows to a certain height, the growth rate will gradually become stable and then slowly go down. This patter is universal. Problem Analysis
If we assume that the growth rate of a tree is proportional with its current height, it obviously does not meet the two ends, particularly the latter part of the growth process, because the tree won’t grow faster and faster boundlessly But if we assume that the growth rate of the tree is proportional to the difference between the maximum height and the current height, it is obviously not in conformity with the middle section of the growth process. We made a compromise assuming that the growth rate is proportional with both its current height and difference between the maximum height and the current height.
Assumptions Assume that there is a maximum height of a tree can grow to, when this height is reached the tree will stop growing higher. Assume that the growth rate of a tree is only related to its current height and the difference between the maximum height and its current height, its not influenced by other environmental factors. Symbol Descriptors Assume the maximum height of the tree is H (m) and at time t (year) the height is h(m). The proportional coefficient of tree growth rate and the current height and the difference between the maximum height and its current height is k.
Model building and solving According to the analysis of the problem and the assumptions made above, we obtain the following equation: Where the proportional constant k >0 . This is a first-order differential equation with separable variables. To solve the equation, we separate the variables and get Therefore, the general solution of the equation is Where is a positive constant. Take H=1, we get different curve of h(t) with different value of C and k.
Conclusions. We can conclude that when K is constant, the slope of Growth of Tree’s Curve will approach 0 as C get bigger. However the slope of the curve gets its peak at middle period where the x-coordinate is 1. 8 when C=5. The slope is largest at middle period which means that the trees grow fastest at that period. This model shows a similar pattern compared with the growth of human-beings. In old age, people grow slowly, even stop growing. When we are babies, the growth rate is also very low but higher than that of the old people.