Predator-prey fluctuations

Example 14.2. Foxes and rabbits. Predator-prey fluctuations are well-understood periodic phenomena in nature as well as the basis of board games such as Struggle or Foxes and Rabbits [9]. In terms of these Grass grows as available area permits, rabbits feast on grass and multiply when doing so, foxes feast on rabbits and multiply when doing so, and hunters in constant number shoot foxes. 

A reaction may be periodic if its network provides for restoration of a reactant or intermediate that has been depleted, while conversion of main reactants to products continues. Periodic behavior often results from competition of two or more contending mechanisms. Predator-prey fluctuations in ecology (Lotka-Volterra mechanisms) provide an easily visualized example. The Belousov-Zhabotinsky reaction—catalyzed oxidation of malonic acid by bromate—involves a similar competition between two pathways. 

Examples include multiple steady states in isothermal CSTRs, predator-prey fluctuations, the Belousov-Zhabotinsky reaction, and a test for stability of quasi-stationary states in reactions with a self-accelerating intermediate steps. 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

The system rotates irreversibly in a direction determined by the sign of /. An example of such a system is the well-known Lotka-Volterra prey-predator interaction given as an exercise (exc. 18.9). We can also apply this inequality to derive a sufficient condition for the stability of a steady state. If all fluctuations fipP > 0 then the steady state is stable. But here it is more expedient to use the Lyapunov theory of stability to which we turn now. 

The results can be displayed by a plot of [Y] as a function of [X], A mathematical space with time-dependent variables plotted on the axis is called a phase space. If [Y] is plotted as a function of [X] for a constant value of [A], there a closed curve that is retraced over and over again as time passes, exhibiting periodic behavior. However, there are different curves for different initial states, but the oscillations predicted by the mechanism resemble the actual fluctuations in predator and prey populations in actual ecosystems. 

---