Trends in the percentage cover of macroalgae are generally used as an important indicator of the health of coral reefs. While considerable data exist on patterns of macroalgal cover, the underlying patch dynamics of macroalgae are poorly understood. It is important to appreciate the dynamical processes of algae because they determine the frequency and duration of competitive interaction with other taxa, such as corals, and thereby contribute to our understanding of competitive interactions. We used transition matrices to quantify and describe the patch dynamics of 2 common macroalgae in the Caribbean, Dictyota pulchella and Lobophora variegata, over a 9 mo period. Size-based matrices were constructed for both species from forereef habitats subjected to contrasting levels of wave exposure. The dynamics of D. pulchella are highly sensitive to the physical environment, whilst those of L. variegata are far less sensitive. The patch dynamics of D. pulchella showed a higher temporal variation than L. variegata, but only on the exposed forereef. D. pulchella virtually disappeared in January at both the exposed and sheltered sites, whereas L. variegata exhibited a continuing increase in patch size irrespective of exposure. D. pulchella formed larger, more dynamic patches under high exposure. This implies that contact interactions with coral colonies and space preemption for settling planulae will occur more frequently than in sheltered environments. Our results demonstrate the need to investigate patch dynamics of macroalgae at the individual species level, whilst taking into account environmental conditions.

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For the 3-D linear oscillator with damping and disturbed by multiplicative white noise, we numerically compute the unique invariant density of the associated system obtained by projection onto the unit sphere. We show how varying feedback gains and noise intensities affect the corresponding density and consequently the stability properties.