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Interpreting Box Models with Transition Matrices and POPS
Barry Ruddick, Pierluigi Pantalone, Keith Thompson
Department of Oceanography, Dalhousie University
barry.ruddick@dal.ca(Abstract received 04/30/2005 for session A)
ABSTRACT
Pantalone et al (this conference) describe the use of first order Markov Chain theory to estimate the "Transition Matrix" ecological connectivity - the probability of larvae spawned at one location arriving at specific other locations in a specified time window, as required for recruitment. This matrix succinctly summarizes complex flow paths and biological behaviours in terms of transfer and retention probabilities, but can be too complicated to easily understand. We discuss the use of Peripheral Oscillation patterns (POPS, Hasselman, 1988, J Geophys Res 93D : 11015o11021) to interpret the inherent time scales and associated modal patterns of the Transition Matrix. We illustrate the technique with a simple advective box model and its transition matrix. We find three types of modes:
1. A uniformly-mixed steady state.
2. A "checkerboard" pattern that decays rapidly by mixing within boxes at each time step.
3. Complex conjugate pairs that form complementary modes, giving propagating patterns analogous to the sin and cosine parts of a travelling wave. These modes have a decay time and a period that can be interpreted in terms of advection and mixing.
Any initial state can be represented as the sum of the above modes, which decay to leave the steady mode. Thus the time evolution of the system can be understood in terms of the modes and their time scales. The technique should be useful in a variety of applications using box models, to identify the dominant time scales and associated mechanics. The effect of larval settlement onto a reef is investigated by allowing a "sticky" reef to cover part of one box. This breaks the (spatial) symmetry of the transition matrix and modifies the modal shapes and time scales. The steady state has an enhanced concentration of larvae in the box containing the reef. The complex conjugate modes no longer form orthogonal pairs, causing their patterns to form more of a standing than propagating pattern.
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2005 LAPCOD Meeting, Lerici, Italy, June 13-17, 2005