Optimal reproductive efforts and the timing of reproduction of annual plants in randomly varying environments

A simple model of the allocation of biomass in annual plants to vegetative and reproductive growth is presented. The model is shown to generate testable hypotheses with respect to several characteristic features of annual plants growing in various environmental situations. Among the features examined are the timing of reproduction, the size of the mature vegetative body, and the annual yield of seeds. A "bang-bang" growth strategy is optimal in stochastic environments if natural selection acts to maximize the expected annual yield of seeds. Given this strategy plants follow the "most rapid approach path" to an optimal vegetative body, and they initiate reproductive growth once this mature body is reached. The size of the mature body depends on the environment, and the study expounds this dependency. If natural selection acts to maximize the expected annual rate of growth, which is equivalent to maximizing the expected value of the logarithm of the annual yield of seeds, a "bang-bang" growth plan is not likely to result in random environments. The optimal plan is graded. It is comprised of rather long and continuous period of allocating biomass to reproductive and vegetative growth simultaneously. This plan degenerates into a "bang-bang" plan in deterministic environments. Though the logarithmic plan is inferior in terms of expected annual yield, it is superior in terms of lower variance and susceptibility to probable extinction.