A higher-order predator-prey interaction with application to observed starfish waves and cycles

Higher-order community (i.e. ‘social’) interactions are notoriously difficult to model, mathematically. But they have been rigorously established in population data. Here, it is demonstrated that replacing the usual cubics, quartics, etc., by 2nd degree homogeneous functions of the population sizes can lead to tractable differential equation models with biological meaning. A generation of the much studied ( m = 2 (2 coral/1 starfish)-model of Antonelli and Kazarinoff which now allows mth order (coral/coral)-interactions, m ≥ 2, m an integer, studies the effect on the real number parameter called the Floquet multiplier or asymptotic orbital stability, β 2, of the starfish aggregation induced limit cycle of small amplitude. It is shown numerically that as m increases, the absolute value | β 2|, first increases to a maximum at m = 4, and then more gradually decreases, other parameters being held fixed. Its value at m = 8 is substantially less than its value at m = 2. (See Tables 1 and 2 and Fig. 1). Thus, the presence of ‘social’ interactions with low values of m enhances stability, while values of 8 or more destroy stability. Recent findings of J. Pandolfi and others indicate the presence of higher-order ‘social’ interactions in species of Acropora corals, the most abundant on the Great Barrier Reef, and the most frequently attacked by the starfish, Acanthaster planci. This may help explain the observed stability of both the starfish/coral cycle at the mesoscale and the ‘Reichelt-wave’ at the scale of the whole Great Barrier Reef. But the model implies that acroporan corals can only be mildly ‘social’.