250
Appendices
so that P will be an increasing function of Z, with P = (}C" at Z = Z,. The
singularity at Z = Z' implies that P ~ Q?when Z ~ Z", or, by reversing the
argument, that grazer biomass increases asymptotically to Z' as P ~ Q? As
Z remains finite, Eq. (A6.22) implies that P ~ Q? as P L ~ Q? The domain of
this stationary point in terms of input P concentration will thus be P L ~ PIlL.
Across the boundary of this domain, the trajectory of the stationary point
with C-limited grazer growth will be continuous with the trajectory of the
stationary point of simultaneous C- and P-limitation.
A7 Local Stability Analysis of the Nutrients, Algae, and
Herbivores Model
Consider a general non-linear dynamic system x = f(x), with a stationary
point i such that f(i) = o. If the system is perturbed a small distance lix.
from the stationary point, such that x = i + lix., the effect of this perturbation can be studied by considering the locally linearized system ax = Alix.,
where A is the Jacobian matrix of the non-linear system, evaluated at the
stationary point (for further details, see Luenberger 1979). The Jacobian is
defined as the matrix A = OfJOx r of partial derivatives with respect to the
state variables; that is, an element alJ of A is the partial derivative of equation i by state variablej (alJ = Of/Ox J ).
.
The behavior of the linear approximation to a small perturbation will be
determined by the signs of the eigenvalues of the Jacobian matrix. If the
real parts of all the eigenvalues of the Jacobian are negative, then small
perturbations will decay exponentially with time such that the system will
eventually return to the stationary point. A system with this behavior is
said to be locally, asymptotically stable. If any eigenvalue has a positive real
part, then any small perturbation from the equilibrium can grow exponentially with time, leading eventually to the system being driven away from
the stationary point. If the functions f(x) are analytical everywhere in the
state space (i.e., have continuous derivatives with respect to x), then the
existence of a single, locally stable stationary point implies that this point
will be a global equilibrium, where the system will end up in from any
arbitrary initial condition. In more general cases, with multiple stationary
points having different local stability properties, the relationship between
local and global stability can be very complex (Guckenheimer and Holmes
1983).
The eigenvalues A,,. .• ,l. of a general n x n matrix A are found by solving
the characteristic equation IAI -A I = 0, which is equivalent to solving an
n'th order polynomial equation in A.. Fortunately, for the case of local stability analysis, some properties of the roots of a general polynomial equation can be investigated without explicitly solving the equation (which is
Appendices
so that P will be an increasing function of Z, with P = (}C" at Z = Z,. The
singularity at Z = Z' implies that P ~ Q?when Z ~ Z", or, by reversing the
argument, that grazer biomass increases asymptotically to Z' as P ~ Q? As
Z remains finite, Eq. (A6.22) implies that P ~ Q? as P L ~ Q? The domain of
this stationary point in terms of input P concentration will thus be P L ~ PIlL.
Across the boundary of this domain, the trajectory of the stationary point
with C-limited grazer growth will be continuous with the trajectory of the
stationary point of simultaneous C- and P-limitation.
A7 Local Stability Analysis of the Nutrients, Algae, and
Herbivores Model
Consider a general non-linear dynamic system x = f(x), with a stationary
point i such that f(i) = o. If the system is perturbed a small distance lix.
from the stationary point, such that x = i + lix., the effect of this perturbation can be studied by considering the locally linearized system ax = Alix.,
where A is the Jacobian matrix of the non-linear system, evaluated at the
stationary point (for further details, see Luenberger 1979). The Jacobian is
defined as the matrix A = OfJOx r of partial derivatives with respect to the
state variables; that is, an element alJ of A is the partial derivative of equation i by state variablej (alJ = Of/Ox J ).
.
The behavior of the linear approximation to a small perturbation will be
determined by the signs of the eigenvalues of the Jacobian matrix. If the
real parts of all the eigenvalues of the Jacobian are negative, then small
perturbations will decay exponentially with time such that the system will
eventually return to the stationary point. A system with this behavior is
said to be locally, asymptotically stable. If any eigenvalue has a positive real
part, then any small perturbation from the equilibrium can grow exponentially with time, leading eventually to the system being driven away from
the stationary point. If the functions f(x) are analytical everywhere in the
state space (i.e., have continuous derivatives with respect to x), then the
existence of a single, locally stable stationary point implies that this point
will be a global equilibrium, where the system will end up in from any
arbitrary initial condition. In more general cases, with multiple stationary
points having different local stability properties, the relationship between
local and global stability can be very complex (Guckenheimer and Holmes
1983).
The eigenvalues A,,. .• ,l. of a general n x n matrix A are found by solving
the characteristic equation IAI -A I = 0, which is equivalent to solving an
n'th order polynomial equation in A.. Fortunately, for the case of local stability analysis, some properties of the roots of a general polynomial equation can be investigated without explicitly solving the equation (which is