Appendices
The following sections contain material that should not be necessary to
read in order to follow the general line of reasoning in the preceding
chapters, but which may still be of importance to those that are specially
interested in some particular topic.
Al Coexistence in a Gradient ofN:P Supply Ratios
Ifwe consider the equilibrium situation for species i alone (that is, III = D),
we have from Eqs. (3.18), (3.19) that uptake and growth must balance (i.e.,
V R . I = IIIQR.I) for both Nand P. If growth is limited by element R, this means
that all state variables and process rates for this element will attain their
equilibrium values for this particular loss rate, which can be written as
IIR.I = D, SR = S'R.i' QR.I = Q'R.i' aR.1 = a'R.I' etc. On the other, hand, if gr?wth is
not li~ited by element R, we must have IIR.i > D, SR > S R.i' QR.i > Q R.i' and
a R ,< a R ,·
If we "identify the abstract resources, labeled I and II in Fig. 3.8, as
concentrations of dissolved inorganic P and N (Sp and SN)' the isoclines for
species i in Fig. 3.8 will be formed by the intersection of the two lines
Sp =S'P.i and SN =SN.i' If we maintain the arrangement of isoclines for species 1
and 2 as in Fig. 3.8, a necessary condition for coexistence will be that the
isoclines intersect; that is, species 1 is the superior competitor for P
(SP.I< Sp), and species 2 is the superior competitor for N (SN.I > SN.2)' Since
the maximal growth rates are assumed to be identical (Ji'l = 11"2 = jI'), the
condition SR.I < S'R.2 becomes equivalent to K R.I < K R.2' where K RJ [(~g R) 1"1] is
the Monod half-saturation parameter for species i when limited by element
R. By substituting the definition of K R • i [Eq. (3.11)], the conditions for
intersecting isoclines can be written as:
a'
Q'
-Ll <......£..1
a~.1 Q~.l
(A1.l)
The following sections contain material that should not be necessary to
read in order to follow the general line of reasoning in the preceding
chapters, but which may still be of importance to those that are specially
interested in some particular topic.
Al Coexistence in a Gradient ofN:P Supply Ratios
Ifwe consider the equilibrium situation for species i alone (that is, III = D),
we have from Eqs. (3.18), (3.19) that uptake and growth must balance (i.e.,
V R . I = IIIQR.I) for both Nand P. If growth is limited by element R, this means
that all state variables and process rates for this element will attain their
equilibrium values for this particular loss rate, which can be written as
IIR.I = D, SR = S'R.i' QR.I = Q'R.i' aR.1 = a'R.I' etc. On the other, hand, if gr?wth is
not li~ited by element R, we must have IIR.i > D, SR > S R.i' QR.i > Q R.i' and
a R ,< a R ,·
If we "identify the abstract resources, labeled I and II in Fig. 3.8, as
concentrations of dissolved inorganic P and N (Sp and SN)' the isoclines for
species i in Fig. 3.8 will be formed by the intersection of the two lines
Sp =S'P.i and SN =SN.i' If we maintain the arrangement of isoclines for species 1
and 2 as in Fig. 3.8, a necessary condition for coexistence will be that the
isoclines intersect; that is, species 1 is the superior competitor for P
(SP.I< Sp), and species 2 is the superior competitor for N (SN.I > SN.2)' Since
the maximal growth rates are assumed to be identical (Ji'l = 11"2 = jI'), the
condition SR.I < S'R.2 becomes equivalent to K R.I < K R.2' where K RJ [(~g R) 1"1] is
the Monod half-saturation parameter for species i when limited by element
R. By substituting the definition of K R • i [Eq. (3.11)], the conditions for
intersecting isoclines can be written as:
a'
Q'
-Ll <......£..1
a~.1 Q~.l
(A1.l)