Chaos and Phase-Locking in Predator-Prey Models in Relation to the Functional Response

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Abstract
The dynamics of a simple predator-prey model: .ovrhdot.X = .phi.x(1-x)-.gamma.g(x)z (prey) .ovrhdot.y = (x-y)/.tau. (lagged prey) .ovrhdot.z = .gamma.g(y)z-.upsilon.z (predator) with periodic forcing (.phi.) on the prey''s reproductive rate and a functional response, g(x), is investigated in relation to parameters and the functional response. The system is sampled at the forcing period and plotted against a parameter for each of four functional responses: Linear, Type 1, Type 2 and Type3. Sampling at the forcing period allows one to see when the system is phase-locked in some ratio with the forcing cycle. The analysis reveals very complicated and unexpected switching between different phase-locking ratios alternating with regions of quasiperiodic and chaotic behavior within each functional response as a parameter is varied. Of the four functional responses tested, the Type 2 response produces the most complex behavior.