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.