Stochastic Modeling of Bamboo Population Growth and Optimal Harvesting

Abstract

Population growth and harvest modeling is an active area of current research. There has been an effort to move from deterministic Ordinary Differential Equations (ODE) to Stochastic Differential Equations (SDE) modeling. Moreover, the latter is most realistic in describing life systems that are often perturbed by unpredictable environmental activity. Bamboo growth and harvest modeling was motivated by the “Tobacco to Bamboo” (TTB) Project where farmers in selected sections of Homabay and Migori Counties in Kenya were persuaded to plant bamboo instead of tobacco. This was met with pessimism due to the lengthy wait, at least three years, before harvesting. They also needed to know the expected income compared to the tobacco income they used to earn. This study therefore sought to explore suitable models that could be used to determine optimal expected sustainable bamboo yield. In view of this, data from the TTB project was analyzed to determine parameters including population growth rate r, carrying capacity K and population size at time t,Nt. ODEs and SDEs were used in modeling equilibrium populations and maximum sustainable yield. SDEs were solved using Itˆo calculus and associated Fokker–Planck equations. The Monte- Carlo simulation procedure was used to construct population trajectories under various model parameter values. A stochastic model with both growth rate and harvest parameters coupled with white noise and a three year delayed continuous harvest proportional to population size was developed. This was found to be most suitable since it ensures maximum mean sustainable yield without the risk of extinction as long as noise was kept at low levels. The model may not only be applied in bamboo harvesting strategies but also other renewable resources that have similar population dynamics.