258
Appendices
c,c2 -co = (rC03 +(0+ DXC03 -e~»u,
(A7.42)
+«0'+ D)+ ~XeCO,C02 +(0' +D)COJ)
The term ~ - e~ > ~ - ~ = p -P" in Eq. (A7.42) will be nonnegative as
long as p ~ P", which is satisfied since it is a condition for existence of this
stationary point. In other words: as all Routh-Hurwitz criteria are satisfied,
all eigenvalues will have negative real parts, which again implies that the
stationary point with C-limited grazer growth will be locally stable.
A8 The Persistence Boundary
In Appendix 7 it was shown that the existence of a locally stable stationary
point with zero grazer biomass, called the grazer extinction point, will be
possible only when the grazers are unable to maintain positive net growth
rate [g < (8 + D)) on algae growing at equilibrium with their dilution and
sedimentation losses [p = (0'+ D»).
If we assume that Q < 8 and C ~ C' at the grazer extinction point, then the
grazer ingestion rate will be saturated (I = n and the grazer growth rate will
thus be determined by algal P content alone. From the Droop equation [Eq.
(3.2)], the phosphorus cell quota of algae growing atp= (0'+ D) will be
Q _
JI
Q'
(A8.I)
-p,'-(O'+D) ,
giving a grazer growth rate
_ ,Q_,
Il'
Q'=
p,'g"
g-g 6 -g p,'-(O'+D) 6 p,'-(O'+D)'
(A8.2)
where g' = &/' - rand g" = g'Q'I8. The inequality g < (8 + D), determining
the local stability of the grazer extinction point, can then be written as
(0+ DX(p,' -a)-D» p,' g".
(A8.3)
The implications of this quadratic inequality in D become somewhat
clearer on looking at a simple illustration (Fig. A8.I). When viewed as functions of D, the left-hand side of Eq. (AS.3) is a concave parabola which intersects the D axis at D = p' - 00 and D = - 6, and which is equal to 8(p' - 00)
when D = 0, while the right-hand side is a straight line, parallel to the D axis,
at distance p'g". The local stability condition [Eq. (A8.3)] for the grazer
extinction point is satisfied for the closed interval on the D axis where the
parabola lies above the straight line. The bounds of this interval are determined by the roots of the quadratic equation(8 + D)«J.L' - 0) - D) = p'g",
which can, after some manipulation, be expressed as
Appendices
c,c2 -co = (rC03 +(0+ DXC03 -e~»u,
(A7.42)
+«0'+ D)+ ~XeCO,C02 +(0' +D)COJ)
The term ~ - e~ > ~ - ~ = p -P" in Eq. (A7.42) will be nonnegative as
long as p ~ P", which is satisfied since it is a condition for existence of this
stationary point. In other words: as all Routh-Hurwitz criteria are satisfied,
all eigenvalues will have negative real parts, which again implies that the
stationary point with C-limited grazer growth will be locally stable.
A8 The Persistence Boundary
In Appendix 7 it was shown that the existence of a locally stable stationary
point with zero grazer biomass, called the grazer extinction point, will be
possible only when the grazers are unable to maintain positive net growth
rate [g < (8 + D)) on algae growing at equilibrium with their dilution and
sedimentation losses [p = (0'+ D»).
If we assume that Q < 8 and C ~ C' at the grazer extinction point, then the
grazer ingestion rate will be saturated (I = n and the grazer growth rate will
thus be determined by algal P content alone. From the Droop equation [Eq.
(3.2)], the phosphorus cell quota of algae growing atp= (0'+ D) will be
Q _
JI
Q'
(A8.I)
-p,'-(O'+D) ,
giving a grazer growth rate
_ ,Q_,
Il'
Q'=
p,'g"
g-g 6 -g p,'-(O'+D) 6 p,'-(O'+D)'
(A8.2)
where g' = &/' - rand g" = g'Q'I8. The inequality g < (8 + D), determining
the local stability of the grazer extinction point, can then be written as
(0+ DX(p,' -a)-D» p,' g".
(A8.3)
The implications of this quadratic inequality in D become somewhat
clearer on looking at a simple illustration (Fig. A8.I). When viewed as functions of D, the left-hand side of Eq. (AS.3) is a concave parabola which intersects the D axis at D = p' - 00 and D = - 6, and which is equal to 8(p' - 00)
when D = 0, while the right-hand side is a straight line, parallel to the D axis,
at distance p'g". The local stability condition [Eq. (A8.3)] for the grazer
extinction point is satisfied for the closed interval on the D axis where the
parabola lies above the straight line. The bounds of this interval are determined by the roots of the quadratic equation(8 + D)«J.L' - 0) - D) = p'g",
which can, after some manipulation, be expressed as