Elemental Composition and Allocation Constraints
237
where the last inequality comes from the condition Q ~ Q'. Likewise, we
can use the non-negativity of the food P:C ratio (Q ~ 0) to obtain an
additional constraint
(A3.9)
The two inequalities [Eqs. (A3.8), (A3.9)], and the necessary conditions
o ~ Sc ~ s·c ~ 1 and rp ~ 0, define a feasible region containing all combinations
of carbon assimilation efficiency Sc and P excretion rate rp that are compatible with Eq. (A3.7) when food is phosphorus deficient (0 ~ Q ~ Q\ In Fig.
A3.1 all trajectories confined within the feasible region, starting at the intersection of the lines corresponding to Sc = s·c and Q = Q', and ending on the
line where Q = 0, will describe a valid strategy for maintaining balanced
growth when food P content decreases from Q' to O.
A particularly simple set of alternative food utilization strategies can be
formulated if we can assume that there is a common upper limit to the
assimilation efficiencies of both elements (for example determined by the
maximum gut residence time), such that and if we assume that the optimal
food composition is equal to the P:C ratio of the grazer; that is, Q'= 0. Under
these conditions, the requirements of balanced growth [Eq. (A3.7)] on Pdeficient food (Q' ~ fJ) can be written as
ecI-rc=e'(Q/O)I-rp'
(A3.10)
At the optimal food composition, where Q = Band Sp = Sc = s·, Eq. (A3.10)
implies that rp = r C' Although such strategies can be contained within the
feasible region of Fig. A3.1, it seems unnecessarily wasteful if P excretion
increases such that rp > r c> when food becomes P deficient (Q < 0). If we
assume that r p ~ r c> then the family of power functions
(A3.ll)
are able to describe an interesting range of possible P utilization strategies.
Fig. A3.2 shows that as the exponent n decreases towards 0, the P excretion
rate approaches the limiting case where rp = rc for all Q ~ B. At the other
extreme, when the exponent n increases towards infinity, the P excretion
rate approaches the limiting case where rp = 0 for all Q < 0. The exponent n
might therefore be interpreted as the animal's ability to economize with P,
by reclaiming catabolites for anabolic purposes. The limiting cases n --+ 0
and n --+ 00 would then correspond to no and complete reutiIization of
catabolites.
If we substitute Eq. (A3.1l) into the growth-balancing equation [Eq.
(A3.10)], the relationship between food P content and C assimilation
efficiency can be written as
(A3.12)
237
where the last inequality comes from the condition Q ~ Q'. Likewise, we
can use the non-negativity of the food P:C ratio (Q ~ 0) to obtain an
additional constraint
(A3.9)
The two inequalities [Eqs. (A3.8), (A3.9)], and the necessary conditions
o ~ Sc ~ s·c ~ 1 and rp ~ 0, define a feasible region containing all combinations
of carbon assimilation efficiency Sc and P excretion rate rp that are compatible with Eq. (A3.7) when food is phosphorus deficient (0 ~ Q ~ Q\ In Fig.
A3.1 all trajectories confined within the feasible region, starting at the intersection of the lines corresponding to Sc = s·c and Q = Q', and ending on the
line where Q = 0, will describe a valid strategy for maintaining balanced
growth when food P content decreases from Q' to O.
A particularly simple set of alternative food utilization strategies can be
formulated if we can assume that there is a common upper limit to the
assimilation efficiencies of both elements (for example determined by the
maximum gut residence time), such that and if we assume that the optimal
food composition is equal to the P:C ratio of the grazer; that is, Q'= 0. Under
these conditions, the requirements of balanced growth [Eq. (A3.7)] on Pdeficient food (Q' ~ fJ) can be written as
ecI-rc=e'(Q/O)I-rp'
(A3.10)
At the optimal food composition, where Q = Band Sp = Sc = s·, Eq. (A3.10)
implies that rp = r C' Although such strategies can be contained within the
feasible region of Fig. A3.1, it seems unnecessarily wasteful if P excretion
increases such that rp > r c> when food becomes P deficient (Q < 0). If we
assume that r p ~ r c> then the family of power functions
(A3.ll)
are able to describe an interesting range of possible P utilization strategies.
Fig. A3.2 shows that as the exponent n decreases towards 0, the P excretion
rate approaches the limiting case where rp = rc for all Q ~ B. At the other
extreme, when the exponent n increases towards infinity, the P excretion
rate approaches the limiting case where rp = 0 for all Q < 0. The exponent n
might therefore be interpreted as the animal's ability to economize with P,
by reclaiming catabolites for anabolic purposes. The limiting cases n --+ 0
and n --+ 00 would then correspond to no and complete reutiIization of
catabolites.
If we substitute Eq. (A3.1l) into the growth-balancing equation [Eq.
(A3.10)], the relationship between food P content and C assimilation
efficiency can be written as
(A3.12)