Modified Lotka-Volterra Model
Now a days, most modern physical chemistry extbooks discuss nonlinear chemical kinetics. Although it is common practice to use the Lotka-Volterra Model (LVM) to introduce the concept of chemical oscillations, it is recognized however that the LVM is not an entirely optimal example of oscillations. On the one hand, it lacks a bifurcation relation, its oscillations only show marginal stability, and the LVM has a fixed point at infinity, which is nonphysical. On the other hand, a linear stability analysis is accessible to most undergraduates, making a simple two variable model with stable oscillation and with a relatively simple stability analysis desirable.
In a past issue of this journal, we introduced the Higgins Model (HM) as an alternative to the LVM. The HM is a more physical two variable model in enzyme kinetics and a better example of chemical oscillations. In spite of the HM's physical foundation, its linear stability analysis is quite challenging and time-consuming for most undergraduates. To alleviate the cumbersome algebra associated with the HM, we introduce a Modified Lotka-Volterra Model (MLVM) as an alternative modelto study chemical oscillations at the undergraduate level.
Even though a given Lotka-Volterra-type model has no chemical system associated with it, its linear stability analysis is simple compared to the analyses of any of the two variable chemical models available. Also, these types of models allow us to introduce several essential concepts in nonlinear chemical kinetics. First, we typically discuss the LVM in the classroom to introduce linear stability analysis. Second, we assign the analytical study of the MLVM as a problem set. This model shows multiple physical steady state solutions, stable oscillations and a simple bifurcation relation. Also, all the linear stability analysis associated with this model is extremely accessible to undergraduates. Third, we study the MLVM numerically in a lab session. The numerical analysis can be carried out using any available software package capable of numerically integrating a system of differential equations. Finally, a lab report is required, which should include several examples of the different bifurcations as well as the predictions derived from the analytical analysis. As a result, the students have a better sense of how the analytical and numerical analyses, carried out in chemical kinetics, complement each other. With this knowledge, the students are ready to study complex but more chemical models, where, in many cases, only numerical integration is possible.
Although changes to the LVM are not new, none of the previous modified versions of the LVM have been used to introduce chemical oscillations at the undergraduate level. Of the several modified models, we consider the changes to the LVM first suggested by Holling, which define the MLVM studied in this paper. In Section 2, we use linear stability analysis to obtain a bifurcation relation among other results. This analysis yields dynamic information that is studied numerically in Section 3.
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