Generic dynamics of a simple plankton population model with a non-integer exponent of closure

Low-dimensional plankton models are used to help understand measurements of plankton in the world's oceans. The full dynamics of these models and the effects of varying the functional forms are not completely understood. Moreover, the effects of small-scale physical influences are only recently becoming apparent. In particular, turbulence may play a pivotal role in the strategies adopted by predators of zooplankton, and thus may alter the so-called closure term, which models predation on zooplankton when the predators themselves are not being explicitly simulated. We investigate the use of a closure term with a non-integer exponent, allowing determination of the dynamics as the closure term varies continuously between the commonly used linear and quadratic forms. We determine which characteristics of the dynamics are generic, in that they occur for any exponent of closure, and which are purely a consequence of the usual integer exponents. A three-way transcritical bifurcation of three steady states is the generic situation, occurring for all except the purely linear closure term. Hopf bifurcations, consequent limit cycles, and chaotic attractors appear to be generic across all exponents of closure. Oscillations, and hence chaos, had been hypothesised to be eliminated with the use of quadratic closure.