198
Approaching Planktonic Food Webs: Competition, Coexistence, and Chaos
After the peak in relative zooplankton C the system is overgrazed: food C
is inadequate to support grazer growth, thus making the grazer population
decline. High grazing loss rates are reflected by high algal growth rates and,
correspondingly, high algal P:C ratios. P-rich food means efficient food
carbon utilization and a high P : C ratio in zooplankton release products.
The high P:C supply ratio gives a competitive disadvantage of bacteria
relative to algae, and gives a net loss rate from the detritus pool. Both
detritus and bacteria therefore decline while the relative abundance of algal
carbon increases.
Grazing loss rates diminish in proportion to the grazer population
decline, thus decreasing algal growth rates and, correspondingly, algal P:C
ratio. The decrease in food P content leads to increased grazer carbon
egestion and to a net accumulation of detritus carbon. The low P:C ratio in
zooplankton release products is to the advantage of bacteria, which can
compete more efficiently for phosphorus when the supply rate of dissolved
organic carbon is high. On a diet composed mainly of detrital C and bacterial P, the grazers are again able to attain positive net growth and build up a
new biomass peak, thus completing the cycle.
The dependency on nonalgal food in the accumulation phase preceding a
peak in grazer biomass has the effect of decoupling secondary production
from the instantaneous primary production. Secondary production is
instead supported, at least partly, by primary production that has been
delayed though the detrital pathways of the carbon cycle. Since time delays
are generally found to have a destabilizing effect on dynamical systems
(MacDonald 1978), the present model potentially contains two sources of
dynamical instability; that of the prey-predator interaction itself and that of
the time-delayed coupling between primary and secondarY production.
Such interactions between two or more oscillating processes with incommensurable frequencies (that is, the ratio between the two frequencies is
not a rational number) have long been recognized as a common source of
chaotic behavior in dynamic systems (e.g., Guckenheimer and Holmes
1983; Thompson and Stewart 1986).
Chaos - and So What? A strictly formal assertion of deterministic chaos in a
dynamical system requires an extended set of computational measures,
such as computing the fractal dimension of the attractor (e.g., Grassberger
1990) and the Lyapunov exponents of the flow (e.g., Souza-Machado et al.
1990). Although such tests have not been performed, the appearance of
Figs. 6.25-6.28 strongly suggests that the present system has chaotic properties, and that these properties might be caused by time-delayed preypredator interactions that are the results of nonlinear storage effects in the
particulate and dissolved detrital compartments. Some indication of the
sensitivity to initial conditions in the present model is given by Fig. 6.23,
which shows two trajectories resulting from adding 1 % random noise to
the same initial conditions located on the aperiodic attractor.
Approaching Planktonic Food Webs: Competition, Coexistence, and Chaos
After the peak in relative zooplankton C the system is overgrazed: food C
is inadequate to support grazer growth, thus making the grazer population
decline. High grazing loss rates are reflected by high algal growth rates and,
correspondingly, high algal P:C ratios. P-rich food means efficient food
carbon utilization and a high P : C ratio in zooplankton release products.
The high P:C supply ratio gives a competitive disadvantage of bacteria
relative to algae, and gives a net loss rate from the detritus pool. Both
detritus and bacteria therefore decline while the relative abundance of algal
carbon increases.
Grazing loss rates diminish in proportion to the grazer population
decline, thus decreasing algal growth rates and, correspondingly, algal P:C
ratio. The decrease in food P content leads to increased grazer carbon
egestion and to a net accumulation of detritus carbon. The low P:C ratio in
zooplankton release products is to the advantage of bacteria, which can
compete more efficiently for phosphorus when the supply rate of dissolved
organic carbon is high. On a diet composed mainly of detrital C and bacterial P, the grazers are again able to attain positive net growth and build up a
new biomass peak, thus completing the cycle.
The dependency on nonalgal food in the accumulation phase preceding a
peak in grazer biomass has the effect of decoupling secondary production
from the instantaneous primary production. Secondary production is
instead supported, at least partly, by primary production that has been
delayed though the detrital pathways of the carbon cycle. Since time delays
are generally found to have a destabilizing effect on dynamical systems
(MacDonald 1978), the present model potentially contains two sources of
dynamical instability; that of the prey-predator interaction itself and that of
the time-delayed coupling between primary and secondarY production.
Such interactions between two or more oscillating processes with incommensurable frequencies (that is, the ratio between the two frequencies is
not a rational number) have long been recognized as a common source of
chaotic behavior in dynamic systems (e.g., Guckenheimer and Holmes
1983; Thompson and Stewart 1986).
Chaos - and So What? A strictly formal assertion of deterministic chaos in a
dynamical system requires an extended set of computational measures,
such as computing the fractal dimension of the attractor (e.g., Grassberger
1990) and the Lyapunov exponents of the flow (e.g., Souza-Machado et al.
1990). Although such tests have not been performed, the appearance of
Figs. 6.25-6.28 strongly suggests that the present system has chaotic properties, and that these properties might be caused by time-delayed preypredator interactions that are the results of nonlinear storage effects in the
particulate and dissolved detrital compartments. Some indication of the
sensitivity to initial conditions in the present model is given by Fig. 6.23,
which shows two trajectories resulting from adding 1 % random noise to
the same initial conditions located on the aperiodic attractor.