Acanthaster and the Coral Reef: A Theoretical Perspective

Under the title of The Acanthaster Phenomenon: a Modelling Approach, the organisers of this workshop have invited you to meet to:

Models capable of simulating large-scale hydrodynamics and Acanthaster larval dispersal are presented for the Cairns Section of the Great Barrier Reef Marine Park. The models identify asymmetries in the patterns of larval dispersal which can explain many observed features of the recent episodes of Acanthaster activity. The northern part of the Cairns Section, between the ribbon reefs and the mainland, is seen to be hydrodynamically distinctive, with the potential to be self-seeding, and therefore the source of Acanthaster activity. Implications of the emerging pattern of stochastic connectivity among reefs are explored using a simple population model. This model is based on a hypothetical assemblage of reefs, with patterns of connectivity derived from the results of the larval dispersal simulation. Experiments with the model demonstrate the potential for the long-term maintenance of populations in the northern region. By simulating the propagation of population outbreaks through the system, it is shown how such a model could be used to test certain hypotheses concerning the cause of outbreaks and to identify the most likely primary outbreak areas.

Effects of diffusion and transport of larvae through space are included with previous limit-cycle dynamics of the mesoscale of the Great Barrier Reef (GBR). For this nonlinear diffusion-reaction-transport process, Neumann boundary conditions are employed for an essentially elliptical model GBR. It is demonstrated that the newly discovered wave of outbreaks at the macroscale (the Reichelt wave) should have period in agreement with the mesoscale limit-cycle period. This, conclusion is corroborated by the data. It is suggested that the lack of large-scale synchronous outbreaks extending to the southern end of the GBR may be due to weakened or erratic properties of the larvae transport process, at least in part. Results in this paper extend previous work of the authors on modelling starfish outbreaks on the GBR.

To attempt to further identify these tentative cyclic structures, the historical data were analysed using techniques of grammar inference and the results displayed using the derivation sets. This permitted the identification of several classes which reflected the processes of an outbreak of starfish. While in some cases recovery of coral after attack seems possible, at present there is no evidence of an overall fall in starfish numbers which would presumably be a significant indicator of cessation of the outbreaks.

A numerical model of currents, based on the results of an oceanographic experiment at John Brewer Reef, is used to simulate the spawning of passive neutrally-buoyant particles for a four and a half month period in 1987. The simulation shows that the proportion of particles removed from the model grid per hour is proportional to the speed of the free-stream low-frequency current. Also, despite low-frequency current speeds of 0.20 m/s, some particles remained within 10 km of John Brewer Reef after 5 days. By allowing the simulated particles to settle after surviving 10 days in the water column, it was found that, although both the spatial and temporal variability of settlement occurred, some consistencies were observed.

The direct application of numerical hydrodynamic models to develop an understanding of biological phenomena is described. The paper reviews the gains in knowledge which have arisen from the reef-scale numerical modelling study of Great Barrier Reef hydrodynamics, developed to investigate crown-of-thorns starfish outbreaks. The models indicated an unusual and complex hydrodynamic environment, but one in which a number of patterns recur.

A scheme for modelling of trajectories of particles in Lagrangian advection/dispersion models is presented. The second-order accurate method is numerically tractable and eliminates the tendency for particles to spiral outwards on curved streamlines. The method is compared with Euler and Taylor series solutions.

Spatially explicit simulation of within-reef movement and activity of the crown-of-thorns starfish provides a tool with which to test hypotheses about the nature of outbreaks. Factors determining whether an outbreak destroys a reef or not include the initial number of starfish, their movement rates, and the degree of aggregation shown by the starfish. Behaviour of individual starfish may be less critical in determining the course of very large outbreaks compared with smaller outbreaks when a combination of reef shape, coral patchiness and food limitation causes aggregation of starfish regardless of the degree of attraction between individuals.

Notwithstanding the fact that the Acanthaster phenomenon manifests itself at the scale of the entire Great Barrier Reef, we show in this paper that a model of the behaviour of individual starfish, at a maximum resolution of an order of magnitude smaller than the size of individual starfish (i.e. at a resolution of 1 cm²) can shed light on the instabilities at the micro-, meso-, and macro-scale of the system. In particular we show that:

Most important however, we show that the state variable most often used in ecology, namely the number (density) of a population, is insufficient to determine a unique next state function (i.e. the number/density of the population at the next time step) in such locally defined models: the spatial pattern of the population is a crucial additional state.

Both a population dynamic model and simplified calculations of the numbers of fish predators that would be required to control heavy recruitment of the coral-eating crown-of-thorns starfish (Acanthaster planci L.) predict threshold densities of fish predators below which starfish population outbreaks may occur. This prediction has been tested through comparable surveys of putative fish predators in the Red Sea, where no major outbreaks of A. planci were found or are known to have occurred, and on the Great Barrier Reef, where two major series of outbreaks have occurred during the last 25 years. In the Red Sea densities of presumed fish predators were found to be well above the predicted threshold. By contrast on the Great Barrier Reef mean densities both of proposed fish predators of juvenile and subadult A. planci, and of know fish predators of adult A. planci, were below the predicted threshold. Moreover the densities of fish predators on four mid-shelf reefs that are believed to have escaped major outbreaks of A. planci were found to be significantly higher than those on otherwise similar but impacted reefs. These data are compatible with the hypothesis that current and recent outbreaks of the crown-of-thorns starfish on the Great Barrier Reef have been facilitated by the presence of only relatively low numbers of fish predators, the numbers of which may have been decreased as a result of the intensification of fishing pressure that has occurred from the 1960s onwards.

A model of predation on subadult Acanthaster planci is developed within a metapopulation framework in which adult starfish and predators are patchily distributed, but reefs are linked by larval dispersal. The model demonstrates that it is possible for starfish to be maintained at low levels by predators with a type II functional response on some patches, provided larval mixing is incomplete and resource limitation exists on at least one reef. Because of the long pre-reproductive post settlement stage in Acanthaster, predation on this stage is found to be of particular importance in then population dynamics of the starfish.

Local Finsler geometry is used to obtain results on the passive Volterra-Hamilton theory of ecological production. The new Finsler equations are used to model defensive strategies for two species of reef-building corals under attack by starfish, Acanthaster planci, on the Great Barrier Reef. These equations are obtained by perturbation of the underlying (formal) cost functional for reef-building so that terms involving species diversity ratios (i.e. polyp number ratios) are included. Such species diversity perturbations result in raising the critical point values at which instability and bifurcation occur, thereby stabilizing the system and its production. Hence, spatial shapes and distributions of hard corals, which result in more efficient growth for the community, may provide a non-chemical defense against A. planci.

Current models to explain ‘primary outbreaks’ of Acanthaster planci assume recruitment of larvae to the natal reef. In the light of recent studies and the current awareness of the importance of larval advection in the recruitment of coral reef organisms, two new models are offered; particular attention is drawn to the possible importance of persistent breeding populations of A. planci as sources of larvae initiating sequences of secondary outbreaks. Reef habitats harbouring stable A. planci populations are characterised by hydrodynamic systems retentive to larvae, and poor coral prey availability, in contrast to the strong flushing and rich coral cover of open-water reefs where outbreaks occur. Other features of stable A. planci populations are a significant contribution to diet by alternatives to hard corals; and reduced predator pressure. A simple model of the important processes influencing starfish population stability is developed. The ability to alternate between endemic and epidemic life-styles in different habitats is a common feature of ‘outbreaking’ organisms.

A stochastic extension of the (deterministic) model of the starfish-coral predator-prey population dynamics introduced by Seymour (1989) is defined and analysed. In particular, it is shown to admit a stationary probability density which is a function of a basic “efficiency of predation” parameter. Properties of this density are used as a basis for discussion of the potential role of (small scale) stochastic effects both in the ecology of interactions of starfish and coral, and with respect to possible selection pressures pertinent to the evolution of A. planci into a predator prone to outbreak (i.e. a near-optimal predator in the terminology of Seymour, 1989).

The Zakai form of nonlinear prediction theory is used to estimate year-to-year state changes in crown-of-thorns starfish populations of the Great Barrier Reef. Taking the previously defined coral-state diffusion as observation process and the starfish-state diffusion as signal process, a least squares polynomial regression curve recently derived for simultaneously collected starfish/coral state data is used to set up the prediction preliminaries. Numerical results are not inconsistant with starfish outbreak values of the last 5 years and yield an expected high value for 1987. However, because of large error in the data it is doubtful that a more refined mesh for the Mihlstein approximations used to evaluate the Ito stochastic integrals involved in Zakai theory would improve the accuracy of predictions.

Stable limit cycles at the scale of individual reefs and hydrodynamic connections between reefs for larval transport are together sufficient to account qualitatively for the large scale dynamics of the phenomenon. Several different factors may generate the stable limit cycles, and these need not be mutually exclusive, but rather reinforcing. An important factor in the limit cycle dynamics in other areas is fish predator pressure on the starfish; increased pressure can suppress limit cycle behaviour. There is some evidence that the same factor may have been coercive on the GBR. Fish predator pressure may have been reduced through the intensification of fishing in the 1960s, leading to the two waves of outbreaks experienced on the GBR since then. There is a possibility that successive waves of outbreaks could degrade the system.