408 John Woods
the ambient climate is quite different. We start the integrations on the lines where
the annual heat budget is in balance, and the ecosystem is in balance at its attractor
(defined in the next section). By starting at different times on the same track, we
can ensure that the water column passes through the target location at different
times of year, and so build up the Eulerian-equivalent annual cycle at that location.
Before we can trust that simulation of annual cycle of the ecosystem at a target
location, we need to consider the stability of a drifting VPE. That will be the subject ofthe next section.
18.12 Stability
There is a substantialliterature on the conditions under which models incorporating non-linear equations may yield simulations that exhibit oscillations or chaos.
Systems that are chaoticalIy unstable produce quite different simulations for tiny
differences in the initial conditions used to start the integration of the model.
Another characteristic ofunstable solutions is the existence of an Attractor (sometimes called a Strange Attractor), which comprises alI the locations occupied by the
simulation in phase space, which has axes based on two orthogonal variables. The
existence of an attractor is good news in the sense that the simulation, while unstabIe, is constrained to particular configurations. But it is bad news, in the sense that
it limits predictability. The existence of chaos in simulations from simple models
has stimulated the search for similar fluctuations in observations of the natural system being simulated. It has been found in some cases (Gleick 1998; Lorenz 1993).
It has been shown that very simple models of predator-prey interactions exhibit
chaotic instability for some ranges of parameter values (May 1976). That is also
true for simple NPZD (Nitrogen, Phytoplankton, Zooplankton and Detritus) models ofthe plankton ecosystem (Asciotti et al 1993). This suggests that the plankton
ecosystem may itself be chaotically unstable. The available observations are not
sufficient to confirm or reject that hypothesis. So our knowledge of the stability of
the plankton system depends critically on the properties of mathematical models.
Simple models have unstable solutions, but what about full-blown simulations?
This is an important issue for the value of plankton ecosystem modelling in
addressing practical problems; for example, in fisheries, toxic algal blooms, epidemics and climate changes. If the simulations are chaotically unstable they will
have limited (if any) value in making predictions ofphenomena of concern to society. It is a matter of some urgency to establish the stability of complex simulations
of the plankton ecosystem, including Virtual Plankton Ecosystems.
Eulerian simulations do seem to exhibit signs of chaos (Popova et al 1997).
However that result may be an artefact ofthe model's population-based equations,
which are strongly non-linear, not least in the closure term (Caswell and Neubert
1998). Furthermore Eulerian integration is known to cause instability in engineering models. Fluid dynamicists find that, because the Lagrangian integration follows characteristics, it avoids the instability created by Eulerian integration
the ambient climate is quite different. We start the integrations on the lines where
the annual heat budget is in balance, and the ecosystem is in balance at its attractor
(defined in the next section). By starting at different times on the same track, we
can ensure that the water column passes through the target location at different
times of year, and so build up the Eulerian-equivalent annual cycle at that location.
Before we can trust that simulation of annual cycle of the ecosystem at a target
location, we need to consider the stability of a drifting VPE. That will be the subject ofthe next section.
18.12 Stability
There is a substantialliterature on the conditions under which models incorporating non-linear equations may yield simulations that exhibit oscillations or chaos.
Systems that are chaoticalIy unstable produce quite different simulations for tiny
differences in the initial conditions used to start the integration of the model.
Another characteristic ofunstable solutions is the existence of an Attractor (sometimes called a Strange Attractor), which comprises alI the locations occupied by the
simulation in phase space, which has axes based on two orthogonal variables. The
existence of an attractor is good news in the sense that the simulation, while unstabIe, is constrained to particular configurations. But it is bad news, in the sense that
it limits predictability. The existence of chaos in simulations from simple models
has stimulated the search for similar fluctuations in observations of the natural system being simulated. It has been found in some cases (Gleick 1998; Lorenz 1993).
It has been shown that very simple models of predator-prey interactions exhibit
chaotic instability for some ranges of parameter values (May 1976). That is also
true for simple NPZD (Nitrogen, Phytoplankton, Zooplankton and Detritus) models ofthe plankton ecosystem (Asciotti et al 1993). This suggests that the plankton
ecosystem may itself be chaotically unstable. The available observations are not
sufficient to confirm or reject that hypothesis. So our knowledge of the stability of
the plankton system depends critically on the properties of mathematical models.
Simple models have unstable solutions, but what about full-blown simulations?
This is an important issue for the value of plankton ecosystem modelling in
addressing practical problems; for example, in fisheries, toxic algal blooms, epidemics and climate changes. If the simulations are chaotically unstable they will
have limited (if any) value in making predictions ofphenomena of concern to society. It is a matter of some urgency to establish the stability of complex simulations
of the plankton ecosystem, including Virtual Plankton Ecosystems.
Eulerian simulations do seem to exhibit signs of chaos (Popova et al 1997).
However that result may be an artefact ofthe model's population-based equations,
which are strongly non-linear, not least in the closure term (Caswell and Neubert
1998). Furthermore Eulerian integration is known to cause instability in engineering models. Fluid dynamicists find that, because the Lagrangian integration follows characteristics, it avoids the instability created by Eulerian integration