Variable-Yield Models

The purpose of this chapter is to give a complete global analysis of the variable-yield model. Essentially, we confirm that the variable-yield models make the same predictions - concerning the growth of a single population, and concerning the outcome of competition between two... 

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. 

The predictions of the variable-yield model (3.1) and the corresponding constant-yield model (7.1) are identical. Typical solutions of each model approach the corresponding equilibrium in a monotone fashion (see Proposition 5.3). 

In one respect, the variable-yield model has been a disappointment in the sense that it was hoped that the transient behavior of its solutions would better fit the transient behavior seen in experiments with certain algae [CNIJ. The experiments, described in [CM], involved the growth of a Chlamydomonas reinhardii population on a nitrogen substrate. Following a step increase in the dilution rate, damped oscillations were observed in cell numbers. Cunningham and Nisbet [CNl] note that the singlepopulation variable-yield model could not reproduce these oscillations without the introduction of time delays into the equations. See also the monograph [NG]. 

Similarly, our analysis of the variable-yield model in Chapter 8 is limited to two competing populations because we rely on the techniques of monotone dynamical systems theory. One would expect the main result of Chapter 8 to remain valid regardless of the number of competitors, just as it did for the simpler constant-yield model treated in Chapters 1 and 2. Perhaps the LaSalle corollary of Chapter 2 can be used to carry out such an extension, using arguments similar to those used in [AM] (described in Chapter 2). As noted in [NG], a structured model in which... 

G2] J. P. Grover (1992), Constant- and variable-yield models of population growth Responses to environmental variability and implications for competition, Journal of Theoretical Biology 158 409-28. 

SW3] H. Smith and P. Waltman (1994), Competition for a single limiting resource in continuous culture The variable-yield model, SIAM Journal on Applied Mathematics 54 1113-31. 

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