Suppose a certain bacteria is grown in a lab, perhaps for medical use. Left alone, the bacteria population will grow exponentially, with growth rate r. A chemical nutrient is known to speed the reproduction process of the bacteria when added. However, use of the chemical by the bacteria creates a second chemical byproduct, which hinders growth. It is also known that the level of hinderance is related to the size of the bacteria population. Namely, the larger the bacteria population is, the smaller the effect this byproduct will have. It is believed this relation is roughly exponential. Therefore, if x(t) is the bacteria concentration at time t, then the growth is given by
x′(t) = rx(t) +Au(t)x(t)−Bu(t)2e−x(t), where u(t) is the amount of the chemical being added at time t, A is the relative strength of the chemical nutrient increasing growth, and B is the strength of the byproduct. Let x0 > 0 be the given initial concentration. We will consider growth and supplementation over the normalized time interval [0, 1]. We wish to maximize x at the end of this interval while simultaneously minimizing the amount of chemical agent used. Thus, our problem can be stated
Cx(1)− ∫ 1 0
subject to x′(t) = rx(t) +Au(t)x(t)−Bu(t)2e−x(t), x(0) = x0, A,B,C ≥ 0.