Stationary Points of the Nutrients. Algae. and Herbivores Model
245
indefinitely, unless it by some means is perturbed away from the stationary
point. The stationary points are found by solving the set of equations
f(i) = 0; if the equations are nonlinear, there can be many of them or, in
some cases, there might be no stationary points at all.
The set of equations determining the stationary points of Eqs. (5.1)-(5.3)
are found by setting the time derivatives to zero:
DP L -«(1+D)P-()gZ=O
(A6.1)
(p- «(1+ D»)C-/Z= 0
(g - (0 + D»)Z= O·
(A6.2)
(A6.3)
Three of the coefficients in Eqs. (A6.1)-(A6.3) are nonlinear functions
(f.J, J, g) of the state variables (P, C, Z), such that the steady-state equations
will be nonlinear. Among the five stationary points that can be solutions of
Eqs. (A6.1)-(A6.3), we will start by examining two apparently trivial ones,
which are nevertheless important in determining the dynamics of the
system Eqs. (5.1)-(5.3).
The Washout Point. This stationary point corresponds to the lifeless
situation where both the biomasses of algae and grazers are zero (C = 0 and
Z = 0). Due to the simplifying assumption of neglecting dissolved P, the
concentration of algal P will still be nonzero:
P=~P.
(A6.4)
(1+D L
This apparent contradiction between zero algal C and nonzero algal P is
resolved if the state variable P is interpreted as the amount of P potentially
available for algal growth. Formally, the algal P quota will be infinite at this
stationary point, but the algal growth rate will still be finite and equal to the
asymptotic maximum (f.J = Ii) because of the hyperbolic relationship
between cell quota and growth rate [Eq. (3.2)].
The Grazer Extinction Point. This stationary point corresponds to the
ungrazed situation where grazer biomass is zero (Z = 0), while algal
biomass is nonzero (C *" 0). From Eq. (A6.2), the conditions Z = 0 and C*"O
together imply that f.J = (j + D, or that algal growth rate is exactly balanced
by losses from flushing and sedimentation. The algal P concentration (P)
will still be determined by P supply rate, flushing, and sedimentation as in
Eq. (A6.4), so that the algal biomass at this stationary point is found by
substituting f.J = (j + D in the Droop equation [Eq. (3.2)], solving for the
steady-state cell quota Q, and combining this result with the requirement
that C= PIQ:
245
indefinitely, unless it by some means is perturbed away from the stationary
point. The stationary points are found by solving the set of equations
f(i) = 0; if the equations are nonlinear, there can be many of them or, in
some cases, there might be no stationary points at all.
The set of equations determining the stationary points of Eqs. (5.1)-(5.3)
are found by setting the time derivatives to zero:
DP L -«(1+D)P-()gZ=O
(A6.1)
(p- «(1+ D»)C-/Z= 0
(g - (0 + D»)Z= O·
(A6.2)
(A6.3)
Three of the coefficients in Eqs. (A6.1)-(A6.3) are nonlinear functions
(f.J, J, g) of the state variables (P, C, Z), such that the steady-state equations
will be nonlinear. Among the five stationary points that can be solutions of
Eqs. (A6.1)-(A6.3), we will start by examining two apparently trivial ones,
which are nevertheless important in determining the dynamics of the
system Eqs. (5.1)-(5.3).
The Washout Point. This stationary point corresponds to the lifeless
situation where both the biomasses of algae and grazers are zero (C = 0 and
Z = 0). Due to the simplifying assumption of neglecting dissolved P, the
concentration of algal P will still be nonzero:
P=~P.
(A6.4)
(1+D L
This apparent contradiction between zero algal C and nonzero algal P is
resolved if the state variable P is interpreted as the amount of P potentially
available for algal growth. Formally, the algal P quota will be infinite at this
stationary point, but the algal growth rate will still be finite and equal to the
asymptotic maximum (f.J = Ii) because of the hyperbolic relationship
between cell quota and growth rate [Eq. (3.2)].
The Grazer Extinction Point. This stationary point corresponds to the
ungrazed situation where grazer biomass is zero (Z = 0), while algal
biomass is nonzero (C *" 0). From Eq. (A6.2), the conditions Z = 0 and C*"O
together imply that f.J = (j + D, or that algal growth rate is exactly balanced
by losses from flushing and sedimentation. The algal P concentration (P)
will still be determined by P supply rate, flushing, and sedimentation as in
Eq. (A6.4), so that the algal biomass at this stationary point is found by
substituting f.J = (j + D in the Droop equation [Eq. (3.2)], solving for the
steady-state cell quota Q, and combining this result with the requirement
that C= PIQ: