The Single-Population Growth Model

The chapter proceeds as follows. In the next section the variable-yield model of single-population growth is derived and analyzed. In Section 3, the competition model is formulated and its equilibrium solutions identified. The conservation principle is introduced in Section 4 in order to reduce the dimension of the system of equations by one local stability properties of the equilibrium solutions are also determined. The global behavior of solutions of the reduced system is treated in Section 5, and the global behavior of solutions of the original competitive system is discussed in Section 6. The chapter concludes with a discussion of the main results. 

Throughout this book we have made use of the conservation principle, which is the defining property of a chemostat. In the previous discussions this principle followed from the form of the equations, but in this formulation we assume the principle and use it to construct the equation for Q. 

The principle simply states that if everything is expressed in nutrient equivalents then the sum of the variables should behave as a chemostat without consumption. Nutrient is neither created nor lost, but rather is merely converted from a free to a stored state. For the situation just described, the sum at time t of free and stored nutrient, called E, should satisfy 

In previous chapters we have usually made use of the consequence of this - namely, that 

Assuming that x(t) remains positive, one has the following equation for the cell quota  

---