Elementary Floquet Theory

The next step is to analyze the stability of the interior rest point. To do this one considers the variational matrix at 

The determinant of M is positive hence the real parts of the eigenvalues have the same sign, and stability depends on the trace (the sum of the eigenvalues of M). This is just the term in the upper left-hand corner. The rest point will be locally asymptotically stable if 

If the inequality is reversed then the rest point E. is unstable - a repeller. The Poincar -Bendixson theorem then allows one to conclude that there exists a limit cycle. Unfortunately, there may (theoretically) be several limit cycles. If all limit cycles are hyperbolic then there is at least one asymptotically stable one, for if there are multiple limit cycles the innermost one must be asymptotically stable. Moreover, since all trajectories eventually lie in a compact set, there are only a finite number of limit cycles and the outermost one must be asymptotically stable. Since the system is (real) analytic, one could also appeal to results for such systems. For example, Erie, Mayer, and Plesser [EMP] and Zhu and Smith [ZSJ show that if E is unstable then there exists at least one limit cycle that is asymptotically stable. Stability of limit cycles will be discussed in the next section. We make a brief digression to outline the principal parts of this theory, and then return to the food-chain problem. 

A standard reference for the material in this section is [CLJ. Here, the basic definitions and theorems are given but no proofs are presented. Floquet theory deals with the structure of linear systems of the form 

Theorem 4.1. LetA(t) be periodic of period T Then if 4 (f) is a fundamental matrix, so is (0 = t + T). Corresponding to any fundamental matrix (f) there exists a periodic nonsingular matrix P(t) of period T and a constant matrix B such that 

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