Differential games in evolutionary theory: Height growth strategies of trees
The author considers a forest of N trees and looks for evolutionarily stable height growth strategies for these trees. Each tree consists of a ''productive'' part comprising the crown and root, and an ''unproductive'' part comprising the stem. The size of the productive part at time t is denoted by x(t) (unit biomass), and the height of the stem by h(t). It is assumed that the chance of survival of a tree's genetic information depends on the size of the productive part at a certain time T (time of seed production), i.e. the tree's payoff function is \(J=x(T)\). x(t) evolves over time according to the equation \[ \dot x(t)=a(t)Q(t)x(t)- dx(t) \] where Q(t) is the growth rate of the tree per unit biomass, a(t) is the fraction of growth allocated to the productive part, and d is the turnover rate of this part (senescent biomass). Both a(t) and Q(t) depend on tree height h(t) as follows: the fraction of growth allocated to the stem increases linearly with \(h: a(t)=1-c.h(t)\); and Q(t) depends on the shade cast on the subject tree by the rest of the stand, i.e. on h(t) relative to the average height of the canopy H(t): \[ Q(t)=Q_ 0+b(h(t)-H(t)). \] The objective is to find a time-evolving height pattern h(t) that is evolutionarily stable under these assumptions. For a spatially uniform, even-aged, single-species tree stand it is shown that the equilibrium strategy is of the bang-bang type: all trees grow in height at the maximum feasible rate \(\eta\) until a certain moment \(\tau =(b-Q_ 0c)/\eta bc\), and do not grow in height afterwards. The author considers also the effect of mutations in the parameters c and \(\eta\), and compares his model to other studies in the literature.
evolutionarily stable height growth strategies
scientific article; zbMATH DE number 3952569