182
Approaching Planktonic Food Webs: Competition. Coexistence. and Chaos
Mapping the N:P Loading Gradient. If we choose a constant dilution rate
(D = 0.01 dai\ as in the foregoing simulations), we can construct a gradient ofN:P loading ratios by varying the input concentrations ofN and P (N I
and PI)' The competitive abilities of the two species at different external N:P
supply ratios can then be studied by simulated reciprocal invasion experiments, as in Section 6.2. Each set of invasion experiments was started with a
simulation run designed to establish one of the species as the resident.
Species 1 was established as resident at an N:P loading ratio sufficiently high
to ensure P-limitation in the stationary state (N I : PI = 15; cf. Fig. 3.10), while
species 2 was established as resident at a low N:P loading ratio (N I : PI = 5; cf.
Fig. 3.10) to ensure N-limitation. As in the previous Section, initial conditions for the initial runs were randomly chosen, but with all state variables
for the invading species set to zero. Each invasion experiment in the same
set used an initial condition based on the stationary state with only the resident species present, with the addition of a small inoculum of the invading
species (equal to 1% of resident biomass). The outcome of the invasion was
classified as exclusion of species i if fmal species i biomass was <1 % of total
phytoplankton biomass, and as coexistence ifboth species had> 1 % of final
biomass.
By performing a set of invasion experiments at a fixed P loading rate, and
varying the N loading rate, the intervals of N supply rates leading to exclusion or coexistence can be efficiently found by a binary search procedure
similar to the one used in Section 6.2. Repeating this process for a range of
P loading rates mapped out the regions of exclusion and coexistence in the
two-dimensional gradient of Nand P loading rates. By slowly increasing
the P loading from a low value, or by decreasing it from an initial high
value, the two major dynamic modes of the system (the stable focus and the
limit cycle) can be tracked individually, as in previous sections.
Using either species 1 or species 2 as resident gave practically the same
results in the simulated invasion experiments: an abrupt shift from competitive dominance by one species to the other taking place at critical N:P
loading ratio equal to the N:P ratio ofthe grazer (NLIP L = 0JOp), leading to
the exclusion of species 2 at high N:P ratios (NLIP L > 0JOp) and species 1 at
low N:P ratios (NLIP L < 0JOp). The positive feedback generated by differential nutrient recycling apparently makes N:P supply ratios diverge to the
extent that coexistence becomes impossible, even for small deviations from
the optimal N:P supply ratio relative to grazer requirements (that is, NLIP L
= OJ Op). Even if some kind of evolutionary design tradeoff could make
species able to coexist at certain N : P supply ratios by conforming to the
conditions illustrated in Fig. 3.9 (which by itself might seem quite unlikely),
such a situation of competitive balance would be very sensitive to the
effects of differential nutrient utilization by zooplankton grazers. This
makes it less likely that the explanation to the paradox of the plankton
should be found in the theory of competition for two or more essential
resources.
Approaching Planktonic Food Webs: Competition. Coexistence. and Chaos
Mapping the N:P Loading Gradient. If we choose a constant dilution rate
(D = 0.01 dai\ as in the foregoing simulations), we can construct a gradient ofN:P loading ratios by varying the input concentrations ofN and P (N I
and PI)' The competitive abilities of the two species at different external N:P
supply ratios can then be studied by simulated reciprocal invasion experiments, as in Section 6.2. Each set of invasion experiments was started with a
simulation run designed to establish one of the species as the resident.
Species 1 was established as resident at an N:P loading ratio sufficiently high
to ensure P-limitation in the stationary state (N I : PI = 15; cf. Fig. 3.10), while
species 2 was established as resident at a low N:P loading ratio (N I : PI = 5; cf.
Fig. 3.10) to ensure N-limitation. As in the previous Section, initial conditions for the initial runs were randomly chosen, but with all state variables
for the invading species set to zero. Each invasion experiment in the same
set used an initial condition based on the stationary state with only the resident species present, with the addition of a small inoculum of the invading
species (equal to 1% of resident biomass). The outcome of the invasion was
classified as exclusion of species i if fmal species i biomass was <1 % of total
phytoplankton biomass, and as coexistence ifboth species had> 1 % of final
biomass.
By performing a set of invasion experiments at a fixed P loading rate, and
varying the N loading rate, the intervals of N supply rates leading to exclusion or coexistence can be efficiently found by a binary search procedure
similar to the one used in Section 6.2. Repeating this process for a range of
P loading rates mapped out the regions of exclusion and coexistence in the
two-dimensional gradient of Nand P loading rates. By slowly increasing
the P loading from a low value, or by decreasing it from an initial high
value, the two major dynamic modes of the system (the stable focus and the
limit cycle) can be tracked individually, as in previous sections.
Using either species 1 or species 2 as resident gave practically the same
results in the simulated invasion experiments: an abrupt shift from competitive dominance by one species to the other taking place at critical N:P
loading ratio equal to the N:P ratio ofthe grazer (NLIP L = 0JOp), leading to
the exclusion of species 2 at high N:P ratios (NLIP L > 0JOp) and species 1 at
low N:P ratios (NLIP L < 0JOp). The positive feedback generated by differential nutrient recycling apparently makes N:P supply ratios diverge to the
extent that coexistence becomes impossible, even for small deviations from
the optimal N:P supply ratio relative to grazer requirements (that is, NLIP L
= OJ Op). Even if some kind of evolutionary design tradeoff could make
species able to coexist at certain N : P supply ratios by conforming to the
conditions illustrated in Fig. 3.9 (which by itself might seem quite unlikely),
such a situation of competitive balance would be very sensitive to the
effects of differential nutrient utilization by zooplankton grazers. This
makes it less likely that the explanation to the paradox of the plankton
should be found in the theory of competition for two or more essential
resources.