Theoretical Population Biology 61, 1–13 (2002)
doi:10.1006/tpbi.2001.1554, available online at http://www.idealibrary.com on
Structural Instability, Multiple Stable States, and
Hysteresis in Periphyton Driven by Phosphorus
Enrichment in the Everglades
Quan Dong
Southeast Environmental Research Center, Florida International University,
OE 148 University Park, Miami, Florida 33199
Paul V. McCormick and Fred H. Sklar
Everglades Systems Research Division, South Florida Water Management District,
3301 Gun Club Road, West Palm Beach, Florida 33416-4680
and
Donald L. DeAngelis
Biological Resources Division, U.S.G.S., Department of Biology, University of Miami,
Miami, Florida 33124-0421
Received March 10, 1999
Periphyton is a key component of the Everglades ecosystems. It is a major primary producer,
providing food and habitat for a variety of organisms, contributing material to the surface soil, and
regulating water chemistry. Periphyton is sensitive to the phosphorus (P) supply and P enrichment
has caused dramatic changes in the native Everglades periphyton assemblages. Periphyton also
affects P availability by removing P from the water column and depositing a refractory portion into
sediment. A quantitative understanding of the response of periphyton assemblages to P supply and
its effects on P cycling could provide critical supports to decision making in the conservation
and restoration of the Everglades. We constructed a model to examine the interaction between
periphyton and P dynamics. The model contains two differential equations: P uptake and periphyton
growth are assumed to follow the Monod equation and are limited by a modified logistic equation.
Equilibrium and stability analyses suggest that P loading is the driving force and determines the
system behavior. The position and number of steady states and the stability also depend upon the
rate of sloughing, through which periphyton deposits refractory P into sediment. Multiple equilibria
may exist, with two stable equilibria separated by an unstable equilibrium. Due to nonlinear
interplay of periphyton and P in this model, catastrophe and hysteresis are likely to occur. © 2002
Elsevier Science (USA)
Key Words: catastrophe; hysteresis; threshold; periphyton; phosphorus; Everglades.
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0040-5809/02 $35.00
© 2002 Elsevier Science (USA)
All rights reserved.
2
INTRODUCTION
Phosphorus (P) is the principal nutrient limiting
the productivity of many freshwater ecosystems, and
P enrichment has become a major problem in these
ecosystems worldwide (DeAngelis et al., 1989; Tiessen,
1995; Vymazal, 1995). Algae respond rapidly to
increased nutrient loading, and excessive algal growth is
one of the common and undesirable consequences of P
enrichment in lakes and rivers (Carpenter et al., 1998).
Attached algal assemblages, or periphyton, are a ubiquitous and ecologically important feature of wetlands
(Goldsborough and Robinson, 1996) and might be
expected to respond in a similar manner to increased P
loading. However, relatively few studies have investigated the effects of increased P loading on wetland
processes in general and, specifically, on periphyton
abundance and growth.
The Florida Everglades is one of the few wetlands where
the ecology of the indigenous periphyton assemblage and
its response to P enrichment have been studied in detail.
The Everglades is an extensive subtropical wetland that
encompasses a large portion of southern Florida. Pristine
areas of this wetland consist of extensive stands of the
emergent macrophyte sawgrass (Cladium jamaicense)
interspersed with more sparsely vegetated areas dominated
by low stature emergent (wet prairies) and floating
macrophytes (sloughs) (Gunderson, 1994). Attached and
floating periphyton mats account for a large fraction of the
vegetative biomass and primary productivity in wet prairies
and sloughs. These mats are formed by the calcium-precipitating (i.e., calcareous) cyanobacteria Scytonema and
Schizothrix and have a three-dimensional, calcified
exoskeletonal structure (Gleason and Spackman, 1974).
During the wet season, floating mats several centimeters
thick can develop, covering most submerged surfaces and
much of the water surface, and they provide habitats and
food for a variety of animals (Browder et al., 1994;
McCormick et al., 1998).
The Everglades periphyton assemblage is extremely
sensitive to changes in P supply and provides an early
signal of eutrophication in this wetland (McCormick and
Stevenson, 1998). Increased P loading in the form of
agricultural runoff produces dramatic changes in
periphyton standing crop, productivity, and species
composition (reviewed in Browder et al., 1994 and
McCormick and Scinto, 1999). One of the more sensitive
and obvious changes caused by enrichment is the loss of
the abundant calcareous mats that characterize pristine
(low P) areas. The contribution of periphyton photosynthesis to ecosystem primary productivity declines
with increased P loading (McCormick et al., 1998).
Dong et al.
Periphyton changes caused by increased P supply may
in turn affect P dynamics within the wetland. Periphyton
functions as a major transformer and sink for P in the
Everglades (Craft and Richardson, 1993; McCormick et
al., 1998). Periphyton mats rapidly remove excess P from
the water column (Scinto and Reddy, 1997) and may be
responsible for maintaining low P availability in pristine
areas where periphyton is abundant (McCormick et al.,
1998). The importance of periphyton to P removal in
enriched areas may be considerably diminished due to
low standing crop. Calcareous mats precipitate P in the
form of calcium–phosphate complexes, which are relatively insoluble and are deposited as calcitic muds, a
common soil type in the pristine Everglades (Gleason
and Spackman, 1974). Thus, increased P loading in
association with the loss of calcareous periphyton
mats may reduce the ability of this wetland to regulate
water-column P availability.
Many models have been built to simulate P dynamics
in wetlands. These models often use a constant (settling)
rate to describe P deposition (e.g., Walker, 1995). Estimating the settling rate for such models is often difficult.
Uncertainties are probably great but very difficult to
quantify and measure due to their numerous uncontrollable sources (see Qian, 1997a, 1997b, and Walker,
1997). Recently, many data have been collected about the
relationship between periphyton and P dynamics. The
current information about periphyton responses to P
supply suggests that the P settling may vary greatly as a
consequence of biological processes. Nevertheless, it is
not clear if and how the responses alter the settling rate.
It is desirable and necessary to quantitatively understand
how biotic processes in periphyton affect P fluxes in
wetlands in general and in the Everglades in particular.
In addition, scientific hypotheses concerning the quantitative relationship between periphyton and P supply
have significant implications in the formulation of policies and administrative rules. For example, Florida law
requires the establishment of a P criterion that can be
used to make rules ‘‘to prevent an ecological imbalance’’
and to restore areas impacted by high P loading.
Periphyton has been identified as a biological indicator
of the P threshold (Lean et al., 1992).
In this study, we used a modeling approach to examine
the interaction between periphyton and P supply. We
constructed a suite of models to serve as a quantitative
framework that can be used to (a) generate hypotheses
about the threshold and the retention capacity of P,
(b) evaluate importance of various ecological processes
and parameters, such as sloughing and mortality of
periphyton, and (c) identify critical information missing
in current ecological understanding. The models are
3
Catastrophe and Hysteresis in Periphyton
designed to address two questions: (a) how do periphyton communities respond to P supply, and (b) how does
the response of periphyton communities in turn affect P
dynamics. The present paper is largely restricted to
description of one particular model with limited structural complexity. With this model, we used equilibrium
analysis, stability analysis, and bifurcation analysis to
reveal general features of interactions between periphyton
and P and the system dynamics.
MODEL STRUCTURE
Our model contains two dependent variables: (1) the
amount of dissolved, available P per unit volume in
the water, P, and (2) the amount of P bound in
living periphyton per unit volume, A. A system of two
equations describes the dynamics of periphyton and P:
dA/dt=F(A, P),
(1)
dP/dt=G(A, P).
(2)
and
The forms of F(A, P) and G(A, P) specify the way
periphyton and P behave and interact. These forms are
based on empirical information. Observations from field,
laboratory, and mesocosm studies suggest that P
enrichment in a previously oligotrophic marsh could
have two major effects on periphyton (Steward and
Ornes, 1975; Swift and Nicholas, 1987; Gleason and
Spackman, 1974; Flora et al., 1988; McCormick and
O’Dell, 1996; McCormick et al., 1996, 1998; Richardson
et al., 1996; McCormick and Scinto, 1999; McCormick
and Stevenson, 1998). First, on periphytometers (artificial substrata), the biomass accumulation increases with
P additions on the short term (a couple of weeks), in field
dosing experiments. This indicates that the growth rate
per unit biomass of periphyton increases with P, particularly in the low concentration range of water column P.
However, the biomass accumulation decelerates and
soon stops in several weeks. Second, increases in P supply
exert a detrimental effect on calcareous periphyton mats.
At high P concentrations, the mat matrices lose their
architectural integrity. Mats collapse and dissolve.
Without structural support of calcareous mats, the
standing crop of periphyton remains low. We incorporated these two effects into a simple model. A plausible
representation of the net growth rate per unit biomass of
periphyton is
F(A, P)/A=umax (P/(ku +P))(1 − A/Amax (P)) − m, (3)
where, umax is the maximum growth rate of periphyton
(g/g · day); Amax (P) is the maximum standing crop of
periphyton, incorporating self-limiting effects (g/m 3); ku
is the half-saturation coefficient for growth rate of
periphyton as a function of P(g/m 3); and m is the
mortality (g/g · day).
In Eq. (3), the term umax P/(ku +P) is the classic
Monod function (Monod, 1950) for periphyton growth
and P uptake, and it has been used widely to represent
the P limitation on algal growth in freshwater systems in
the absence of toxic and prohibitive effects (Grover,
1990, 1991; Borchardt et al., 1994). This type of model
has been used previously in the empirical studies of
interactions between periphyton and P in South Florida
wetlands (Scinto and Reddy, 1997; Hwang et al., 1998).
The Monod function assumes that the growth rate of
periphyton increases rapidly with P at low concentrations of P, while the increase per unit increment of
P diminishes and becomes insignificant at high P
concentrations.
The next factor in Eq. (3), (1 − A/Amax (P)) is in
the form of a modified logistic function. The logistic
equation also has been used widely in algal studies
(Rodriguez, 1987; Momo, 1995). In this model, the factor
represents a biomass-dependent, self-limiting effect on
growth, resulting from a combination of competition for
substrates, light, or nutrients other than P. The limiting
effect of P and depletion of P are explicitly included in the
equations. It is assumed here that the architecturally
supportable maximum of standing crop, Amax , is a
decreasing function of P, Amax (P). This can be represented by the nonlinear curve shown in Fig. 1. Field and
experimental observations (Flora et al., 1988; Richardson
et al., 1996; McCormick et al., 1998; McCormick et al.,
2001; McCormick and Stevenson, 1998; McCormick and
Scinto, 1999) indicate that phosphorus effects on the
periphyton mat matrices are slight at low P concentrations,
but become substantial as concentrations increase
above background levels. For simplicity, the Amax (P)
curve can be approximately represented in a piecewise
linear form
˛
a1 , (P < P1 ),
a1 +(P − P1 )(a1 − a2 )/(P1 − P2 ),
Amax (P)=
(P1 < P < P2 ),
a2 , (P, P2 ).
(4)
(see Fig. 1)
At any given point on the Amax (P) curve, the derivative, d(Amax (P))/dP, measures the effect of changes in P
on the maximum supportable standing crop of periphyton assemblage at a particular P concentration. Note that
4
Dong et al.
A, Amax , P, m, and v \ 0,
and
Pin , umax , ku , a1 , a2 , P1 , and P2 > 0.
When the uptake of P by periphyton is taken into
account, the equations describing changes in periphyton
and in water P through time are,
dA/dt=A[umax (1 − A/Amax (P))(P/(ku +P)) − m],
(6)
and
dP/dt=v(Pin − P)
− A[umax (1 − A/Amax (P))(P/(ku +P)) − m]
− fmA.
FIG. 1. A hypothetical curve depicting the effect of P on the
maximum supportable standing crop of periphyton. The horizontal
axis is water-column P concentration. The vertical axis indicates the
maximum supportable standing crop of periphyton in the unit of P. A
piecewise linear representation is used to approximate the smooth
curve. Roughly, there are three regions along the P axis, divided by
points P1 and P2 . The region between P1 and P2 is the ‘‘transitional
region’’ for periphyton, in which the periphyton response to P is most
sensitive and conspicuous architectural changes occur. In the other two
regions, periphyton standing crop is relatively stable to changes in P.
Amax (P) is a monotonically decreasing curve (Fig. 1), i.e.,
d(Amax (P))/dP is either negative or zero. Thus,
−d(Amax (P))/dP measures the strength of the P effect.
As we show below, this measure is an important parameter. Finally, in Eq. (3) the mortality, m, represents the
loss rate per unit biomass of periphyton through sloughing. During sloughing, dead periphyton tissues deposit
into sediment. We assume that m is a constant.
We assume a constant flow of P into the system and a
loss rate due to washout that is dependent on watercolumn P concentration. In the absence of periphyton,
the equation for P can be written,
dP/dt=(Pin − P) v,
(5)
where Pin is the P concentration in water that flows into
the system from external sources (g/m 3); and v is the
daily water flux through the system, measured as a proportion of the total water volume of the system. v=vŒ/w.
vΠis the velocity of water through the system (m 3/day).
w is the total volume of the system (m 3).
In Eq. (5), vPin is the per space unit P loading flux from
external sources (g/day/m 3). All of the state variables
and parameters in the above equations are nonnegative
and most parameters are positive:
(7)
In Eq. (7), we have made the assumption that a constant portion of the dead tissue material, f, is recalcitrant
and does not decompose before going into the sediment.
The remainder of the dead periphyton biomass, which
dies at the rate mA, decomposes and releases its P back to
the water. The deposition of refractory fraction of tissue
P into sediment after death removes P from the pools of
periphyton and water. The importance of periphyton on
P cycling lies in the deposition. The term, fmA, is the rate
of P transport from water to sediment by the periphyton.
It can be regarded as the system’s biological assimilative
capacity of P.
EQUILIBRIA AND STABILITY
Equilibria and their stability are two useful attributes,
delineating the long-term behavior of a system. We
evaluated equilibria and their stability analytically and
graphically, and examined how different processes can
affect them. We were interested only in those biologically
reasonable equilibria for which A and P are positive
(A > 0, and P > 0).
1. Steady State Equilibria
We evaluate the equilibria first. The equilibria,
(A g, P g), were determined by setting the right-hand sides
of Eqs. (6) and (7) to zero: dA/dt=0 and dP/dt=0.
Solving for the A in terms of P in both equations, we
first noted that the equilibria are easily found as the
intersections of the dA/dt zero isocline
A=Amax (P)(1 − m(1+ku /P)/umax )
(8)
5
Catastrophe and Hysteresis in Periphyton
with the line
A=v(Pin − P)/fm.
(9)
Figures 2, 3, and 4 show possible configurations of the
zero isoclines, dA/dt=0 and dP/dt=0. We will discuss
different configurations below. In general, the intersection points of these two isoclines represent equilibria.
The straight line defined by the Eq. (9) will always intersect the equilibria. The line defined by Eq. (9) intersects
the P axis at the point Pin , therefore illustrating the relation between Pin and the equilibrium point. Equation (9)
suggests and the straight line in Figs. 2, 3, and 4 shows
that P concentration of the in-flow, Pin , is the dominant
factor determining both the number and position of
equilibria. Nevertheless, the number and position of
equilibria also depend on several other parameters, particularly, the slope of Eq. (9), −v/fm, and the steepest
downward slope of the dA/dt=0 curve, min(dA/dP).
The pattern of A g and P g and their responses to Pin can
be examined under the two opposite conditions:
min(dA g/dP) \ −v/fm,
and
min(dA g/dP) < −v/fm.
First, consider the case that min(dA g/dP) \ −v/fm.
Only one positive equilibrium point exists, at the intersection of the two isoclines (Figs. 2a and 2b). Roughly four
regions can be seen, in each of which A g and P g respond
differently, with respect to the Pin . These four regions are
divided by three levels of P; P0 (P0 =mku /(umax − m)), Pa ,
and Pb (Fig. 2a). When Pin is extremely low, Pin < P0 , the
periphyton, A, cannot maintain positive growth, and thus
disappears. If Pin decreases from a higher level, negative
growth may occur. If the Pin level is between P0 and Pa
(P0 < Pin < Pa ), A g will increase rapidly with Pin , while P g
does not show much change. The Monod growth function
plays a large role in shaping the equilibrium pattern in the
low Pin environment. In the third region, Pa < Pin < Pb , the
algal biomass and water column P respond sensitively to
changes in Pin . A dramatic but continuous transition of the
steady state equilibrium occurs. When Pin is high, Pin > Pb ,
P g is high and A g is low. A g is insensitive to Pin (Fig. 2). P g
is almost linearly increasing with Pin in this range.
It is interesting to see the influence of P effect on
periphyton mats, d(Amax (P))/dP, on the relationship of
A g and P g. By taking the derivative of A in Eq. (8) with
respect to P, we obtain
dA/dP=(1 − m(1+ku /P)/umax ) d(Amax (P))/dP
+Amax (P) mku /(umax P 2).
FIG. 2. Equilibrium and its stability, and the effect of changes in
Pin on the equilibria and stability. The solid curves are isoclines, where
dP/dt=0 and dA/dt=0. Their intersection point is the nonnegative
equilibrium. The straight broken line representing the equation,
A=v(Pin − P)/fm, intersects at equilibrium. Sloughing, measured by
fm, is one of the major determinant of the positions and stability of
equilibria. In this case, the value of fm is relatively low, thus one equilibrium exists. Arrows show the moving direction of system around that
point. Inward arrows indicate a tendency to return to the equilibrium,
and thus a stable equilibrium. Outward arrows indicate unstable equilibrium. Instability may occur only when Pin is between Pa and Pb . In
(a) the slope of Amax (P) curve is moderate, and the equilibrium is stable.
In (b) the slope of Amax (P) curve is steep and the equilibrium is
unstable. See text for more details.
This implies that the downward slope of dA/dP becomes
steeper as d(Amax (P))/dP becomes more negative.
The sloughing rate, fm, is another important parameter. Equation (9) (and Fig. 2) shows how fm affect the
system attributes. Note that the difference between Pin
6
Dong et al.
FIG. 4. Equilibria and their stability. The solid curves are isoclines,
where dP/dt=0 and dA/dt=0. Their intersection points are nonnegative equilibria. The straight broken line representing the equation,
A=v(Pin − P)/fm, lies across the equilibria. In this case, the value of
fm is relatively high. The steepest slope of Amax (P) curve is steeper than
the slope of the broken line, −v/fm. Thus, multiple equilibria may
exist. Stable equilibria are surrounded by inward arrows from all directions. Outward arrows from an equilibrium to any direction indicate
instability. The equilibria E1 and E3 are both stable nodes, while the
equilibrium E2 is a saddle point.
dA/dt=A[umax (P/(ku +P))(1 − A/Amax (P))],
dP/dt=v(Pin − P)
− A[umax (P/(ku +P))(1 − A/Amax (P))].
FIG. 3. Equilibrium and its stability; outcome from the model that
does not include the effect of sloughing. The solid lines are isoclines,
where dP/dt=0 or dA/dt=0. The intersection point of two nonlinear
isoclines is the nonnegative equilibrium. At equilibrium, P g=Pin . One
equilibrium exists for a given Pin . Inward arrows indicate a tendency to
return to the equilibrium, and thus a stable equilibrium. Outward
arrows indicate unstable equilibrium. Instability may occur only when
Pin is between P1 and P2 . (a) The equilibrium in the transitional region is
stable, if the slope of Amax (P) curve is moderate, and (b) the equilibrium
is unstable when the slope is steep.
and P g measures the removal of P by the system at steady
state. The difference between P g and Pin decreases, as
−v/fm decreases with fm. Periphyton removes a
smaller portion of P from the water column. The special
case, m=0, means that sloughing does not occur in the
model. Many earlier models that simulated interactions
between algae and nutrients ignored P sedimentation
through sloughing (e.g., Edelstein-Keshet, 1988), and
thus were constructed in this way. In this case, the model
is equivalent to
The equilibrium solution of this model is P g=Pin , and
A g=Amax (Pin ). Thus, first the P concentration will reach
Pin at steady state, without sloughing. The periphyton
biomass does not affect the equilibrium value of P.
Second, P limitation on growth is unimportant in
determining the steady states. Periphyton biomass is high
in the low Pin region (Pin < P1 , in Figs. 3a and 3b). Thus,
the position of equilibrium (A g, P g) and its response to
Pin differ significantly from the case in which fm > 0, in
the low Pin regions (Figs. 3a and 3b). In the high Pin
region, periphyton biomass is low (P2 < Pin , in Figs. 3a
and 3b). The effect of changes in Pin , on periphyton is
small within both low and high P regions. With intermediate levels of Pin , P1 < Pin < P2 , the significant transition
occurs. The periphyton biomass responds sensitively to a
change in Pin and declines approximately with a magnitude of (a1 − a2 )(P1 − P2 ) for a unit increase in Pin . There
are only three regions with respect to the driving force Pin
(divided by the points P1 and P2 ). Third, the water flow is
also insignificant in determining the equilibrium. The
comparison between results of models with fm > 0, and
with fm=0 reveals the contribution of sloughing to
system behavior and demonstrates the significance of fm
7
Catastrophe and Hysteresis in Periphyton
and its explicit representation in model. It is interesting
to note that the width between Pa and Pb could be
narrower than that between P1 and P2 (Figs. 2a, and 1,
or, 3a). Thus the transition of A g may appear more
dramatic than Amax (P), due to the interactive feedback of
biomass change.
Second, consider the opposite case, where min(dA g/dP)
< −v/fm. Multiple equilibria may now exist. There are
still roughly four regions with respect to the Pin , in which
the positions and the number of equilibria differ (Fig. 4).
The pattern of A g and P g is similar in the first, second,
and fourth regions, as before in the case in which
min(dA g/dP) \ −v/fm. One equilibrium exists. A g
either stays at zero, rises, or is insensitive as Pin increases,
in the first, second, or fourth regions, respectively. In the
third region, where Pin is between Pa and Pb , the isoclines
may cross more than once. Up to three equilibria with
positive A g and P g could exist, one at high A g and low
P g (E1 ), one at low A g and high P g (E3 ), and one at
intermediate A g and P g (E2 ) (Fig. 4). As discussed in
more detail below, the equilibrium at intermediate A g
and P g is unstable. Thus, transition from one equilibrium
to another is an abrupt change, in contrast to the continuous change in the case, min(dA g/dP) \ −v/fm. The
abrupt change can occur even when the response of mat
integrity to P is gradual and continuous, due to the
interplay of the periphyton response to P and the consequent changes in the P removal by periphyton.
With high A g and relatively low Pin , periphyton removes a
relatively large portion of P, and thus maintains P g at low
levels. With low A g and relatively high Pin , periphyton only
removes a small portion of P from the water column. P g
remains high. Thus, a positive feedback operates in certain
circumstances in the interaction between periphyton and P.
The architectural response of the periphyton mat seems to
be an important mechanism that shapes the equilibrium
pattern.
Overall, multiple steady states may occur. P loading
(Pin v) is driving the system. The sloughing process that
removes P from the system at the rate fmA and the P effect
on periphyton mat architecture (Amax (P)) are major processes determining the system states. Periphyton affects
wetland P dynamics mainly through P uptake from the
water and P deposition in sediment after death. Thus,
growth, death, decomposition, and deposition are all critical processes affecting the capacity of P retention and
assimilation in periphyton-dominated marshes.
2. Stability
We examined the local stability of equilibria and the
determinants of the stability with the Jacobian of the
system of equations; i.e., the first-order equations linearized
about the equilibrium points (e.g., Edelstein-Keshet, 1988).
Such an analysis helped to identify potential stabilizing and
destabilizing processes.
The real parts of eigenvalues (l) of the Jacobian are
measures of stability.
l1, 2 =(b ± `b 2 − 4c)/2;
where we define,
b=(dF/dA+dG/dP)A*, P* ,
and,
c=[(dF/dA)(dG/dP)− (dF/dP)(dG/dA)]A*, P* .
If one or both of the eigenvalues are positive, the system is
unstable (Lyapunov instability). The sufficient condition
for Lyapunov instability of the equilibrium point is either,
b > 0, or, c < 0. The rate at which the system returns to
equilibrium is termed ‘‘relative stability.’’ A system with
negative real eigenvalues has a relative stability in proportion to the absolute values of eigenvalues. In a loose sense,
the largest l increases as b increases and as c decreases.
Calculation of b, c, and l suggests that the general
condition for an equilibrium to be unstable is either
b=m − umax P g/(ku +P g) − v − (dF/dP)A*, P* > 0,
(10)
or
c=fm(dF/dP)A*, P* − v(m − umax P/(ku +P)) < 0.
(11)
For A g to be positive, it must be true that m − umax P/
(ku +P) < 0. Thus, Eqs. (10) and (11) imply that unstable
equilibria could occur only when (dF/dP)A*, P* is negative.
(dF/dP)A*, P* is the derivative of Eq. (6) with respect to P at
the equilibrium point
(dF/dP)A*, P* =Aumax [(ku /(ku +P) 2)(1 − A/Amax (P))
+(A(P/(ku +P))/Amax (P) 2)
× d(Amax (P))/dP].
This suggests, the instability occurs mainly in the transitional region of Pin with a steep slope of Amax (P) (Figs. 2b, 3b,
and 4). A more negatives d(Amax (P))/dP with a larger fm
tends to destabilize the system.
8
Dong et al.
Inequality (10) also shows that b (and thus the
real parts of the eigenvalues and the instability of the
system) increases with an increase in −d(Amax (P))/dP.
d(Amax (P))/dP is the slope of the tangent lines of Amax (P)
curve in Fig. 1. It stands for the strength of the deteriorating
effect of P on periphyton mats, and is a critical
parameter to stability. If Amax (P) were an increasing
function of P, so that d(Amax (P))/dP > 0, then the equilibrium would always be stable. However, field and
experimental observations suggest, and we are assuming,
the opposite. The inequalities (10) and (11) suggest that
an increase of v, water flux, is stabilizing. Among other
parameters, umax and ku tend to play opposite roles in
stabilization. Inequality (11) shows that the equilibrium
is stable in both the high and low Pin regions. Since
(dF/dA)A*, P* =m − umax P/(ku +P) < 0,
we can rewrite Eq. (11) as
(dA/dP)A*, P* < −v/fm.
(12)
Equation (12) means that the equilibrium is unstable if
the slope of the straight line is less steep than the slope of
periphyton isocline at the intersection in Fig. 4. This
suggests that the equilibrium in the middle is unstable
when three steady state equilibria exist. Figure 4 reveals
the mathematical nature of the three equilibria. Points E1
and E3 are both stable nodes, while E2 is a saddle point.
Thus, the plane is divided into two basins of attraction,
one containing all trajectories that move toward the
point E1 and one containing all trajectories that move
towards E3 (also see Fig. 5). The high value A g has a high
removal capacity of P and maintains low P, and vice
versa. This operates as a self-reinforcing mechanism,
keeping A g high and P g low, or A g low and P g high. This
mechanism creates the local stability, but global instability.
With the special case, m=0, A g=Amax (P g), and
g
P =Pin , the instability condition is simplified. First, c=
vumax P/(ku +P) > 0. Thus, only functions and parameters composing the expression for b determine the
stability of the equilibrium, and
b=(−d(Amax (P))/dP|P=Pin − 1) umax Pin /(ku +Pin ) − v.
We have b < 0 if
−d(Amax (P))/dP|P=Pin < v(ku +Pin )/umax Pin +1.
(13)
FIG. 5. Bifurcation. Panels (a) and (b) show how the steady state
periphyton and P respond to Pin changes, respectively. Broken arrows
indicate the direction of change as a consequence of the increase of Pin
and solid arrows indicate the direction of change with the reduction of
Pin . A gradual increase in Pin tends to result in a stable equilibrium with
increasing values of A g until Pin goes beyond Pp . Then a dramatic drop
of A g occurs, because the mat structure cannot support the high
biomass. The state plane suddenly becomes a basin of attraction for the
stable equilibrium at low level of A g and high level of P g. When Pin
decreases from a high level, the system remains at low A g, even after the
level of Pin decreases below Pp . One must decrease Pin below Pr to
restore a high A g equilibrium. The state plane then becomes a basin of
attraction of the equilibrium of low P g and relatively high A g.
Inequality (13) is the stability criterion for the single
equilibrium, when fm=0.
In summary, in this model system the in-flow P
concentration, Pin , is the major determinant of the
Catastrophe and Hysteresis in Periphyton
equilibrium. The strength of the detrimental effect of P
increase on periphyton mat is a major determinant of the
stability. The sloughing rate, fm, is another important
determinant of the equilibrium and its stability.
PERTURBATION, HYSTERESIS, AND
NONEQUILIBRIUM BEHAVIOR
The P input, Pin , is a driving force in this type of
system. A question of much management interest for
ecosystems such as the Everglades, parts of which have
been enriched by increased loads of P in recent decades,
is what will happen if (a) loading continues and expands
into the areas that are currently not impacted, and
(b) loading decreases in areas that have already been
enriched. These two scenarios were explored by increasing or decreasing Pin in the model.
Figure 5 explicitly shows the outcome of these two
scenarios. First, when Pin increases from a low level
gradually, a single stable equilibrium exists. At this phase
of equilibrium state, P serves as a limiting factor and A g
increases with Pin (Fig. 5a). A large portion of P is taken up
by periphyton from water column and then deposited into
the sediment. Thus, P g shows a little change and remains at
a very low level (Fig. 5b). As A g reaches the peak value, a
further small increase of Pin beyond a threshold level, Pp ,
would lead the equilibrium to an abrupt switch, from a
stable state with high A g and low P g to another stable state
at low A g and high P g. This drop in A g represents the dissolution of periphyton mats. P is no longer limiting. Such a
large change in the equilibrium, as a response to a small
difference in Pin around the threshold, is a result of a structural instability or a ‘‘fold catastrophe’’ (see Zeeman,
1977). The abrupt change occurs even when the response
of mat integrity to P is gradual and continuous, due
to the interplay of the periphyton response to P and the
consequent changes in the P removal by periphyton. Pp is a
threshold for the equilibrium at high A g and low P g to
exist. We may call this threshold the ‘‘protection
threshold.’’ The equilibrium with high A g and low P g
represents a thick periphyton mat in an oligotrophic environment, which is a feature of the slough communities
dominated by native periphyton in the Everglades.
Now, suppose that efforts are made to reduce P
loading from a high level gradually. As Pin is reduced, the
system remains at low A g and high P g, even after the level
of Pin decreases below the protection threshold. In fact,
Pin may have to be decreased to a much lower level, Pr , to
restore the high A g, low P g equilibrium (Fig. 5). Pa indicates a threshold for the system to recover from a crash in
A g due to high Pin . Below this Pin level, the entire state
9
plane becomes a basin of attraction for high A g, and the
system returns to that attractor (Fig. 5, also see Fig. 6). We
may call Pr the ‘‘restoration threshold.’’ If the protection
threshold differs from the restoration threshold, hysteresis
exists. Hysteresis means that the system going through a
change in a control variable (e.g., the change of Pin from a
to b in Fig. 5) does not return to its initial state when the
control variable is returned back to its initial point (e.g., the
change in Pin from b back to a in Fig. 5). Instead, the
control variable must be returned beyond its starting point
before recovery can occur. Hysteresis occurs in this system
because of a strong, nonlinear interplay of periphyton and
water column P. Periphyton removes P from the water. The
removal capacity depends on the standing crop. When the
P loading rate is low, periphyton standing crop and the
capacity for periphyton to remove P are high. Under these
conditions, it may be possible for the marsh to assimilate
increased P loads and maintain low P availability. In contrast, if P inputs are reduced from previously high loading
rates, conditions under which periphyton standing crop
and removal capacity are low, then the capacity for P
removal may remain low. Therefore, even under a scenario
of reduced P loading, excess P may remain in the water
column and inhibit the formation of calcareous mats that
support high standing crops. The occurrence of this type of
hysteresis is likely to be an important feature of this system.
Environmental perturbations and stochastic events
often push the system into nonequilibrium states. When
multiple stable states exist, it is interesting to examine
how new nonequilibrium states caused by perturbation
determine the asymptotic behaviors. We ran simulations
with the model to investigate the nonequilibrium behaviors. The parameter values used in the simulation are
based on empirical studies, and thus they are realistic. As
earlier analyses indicated, with low and high Pin , the
system approaches to one attractor of high A and low A,
respectively. With an intermediate Pin level, there are two
attractors. A small difference in the initial condition may
lead to very different stable states (Fig. 6). This implies
that below a certain level, perturbations may not affect
the periphyton and P much, as they tend to approach one
asymptotic state. A small increase in the perturbation at
a certain magnitude may push a large change and the
system approaches to another asymptotic state.
In this model, we assumed that the self-limiting effect
operates through growth. The self-limiting effects may
also operate via mortality instead of growth. We
modified the model with this assumption and conducted
equilibrium and stability analyses. The analysis of the
new model also showed possible catastrophe and hysteresis. Therefore, general patterns presented above seem
to be robust, regardless of some details in the model.
10
FIG. 6. Multiple attractors and nonequilibrium simulation. Simulation trajectories are shown starting from various initial conditions.
For this value of Pin , the plane is largely divided into two basins of
attraction. One contains all trajectories that move toward the equilibrium point E1 with high A g and low P g, and one contains all trajectories
that move towards the equilibrium E3 with low A g and high P g. Two
stable equilibria are likely to occur and both equilibria are locally
stable. The parameter values are: m=0.05, f=0.5, Umax =2, ku =25,
v=0.4, a1 =1400, a2 =400, P1 =10, P2 =30.
DISCUSSION
We constructed a two-variable model to simulate the
interaction between periphyton and P dynamics. Despite
the simplicity of the model structure, our model generated some new insights and identified some important
ecological processes and missing links. For example, a
major prediction of our model is the likely occurrence of
multiple stable states, and structural instability. One of
the characteristics of this kind of system is that there can
be a large change in the state of the system resulting from
a small change in the control variable, which in this case
is the P loading. This result was supported by two lines of
evidence. First, field observations have demonstrated
that changes in periphyton biomass and taxonomic
composition occur rather abruptly along P gradients in
the marsh. Second, mesocosm dosing experiments
showed that the changes in periphyton are associated
with rather modest increases in water-column P concentrations (McCormick and O’Dell, 1996; Richardson
et al., 1996; McCormick et al., 1998; McCormick and
Scinto, 1999).
Our analyses also show the potential for hysteresis in
this type of system. The likely existence of hysteresis
implies that two thresholds may exist, one for protection
and one for restoration. Studies of lake restoration have
shown this could occur. In some lakes, algal biomass and
Dong et al.
trophic status do not always recover in a desired direction, and the symptoms of eutrophication remain after a
significant reduction of P loading (Sas, 1989; Tilzer et al.,
1991).
We predicted that increased loading rates do not result
in proportional increases in water column P, at least at
low P loading levels. This has been supported by the
studies in the Everglades (McCormick et al., 1998) and
other ecosystems (e.g., Pomeroy, 1960). McCormick et
al. (1998) found that added P is scavenged and retained
efficiently in the periphyton due to the P-limited nature
of the Everglades marsh, and they concluded that
periphyton may represent the major sink for this excess P
in open-water habitats, which occupy large areas of the
marsh interior. We believe this model captures some
realism and generality in the Everglades and similar
ecosystems.
Structural instability, multiple stable states, and
hysteresis in this system are mainly a consequence of
nonlinear interactions between periphyton and P,
and particularly two opposite effects of P on periphyton.
First, periphyton growth is limited by P at low P levels.
Second, the physical structure of periphyton mats breaks
and the maximum standing crop declines at high P level.
This type of structural instability and hysteresis has been
found in several algal and plant studies (Kempf et al., 1984;
Gatto and Rinaldi, 1987; Momo, 1995; Scheffer et al., 1997;
see Loehle 1989 for review of catastrophe in various
ecological systems). These models are similar to ours in that
they are process-oriented, contain the Monod growth
response to nutrients and the logistic density limitation, and
often include two or more opposite effects occurring in a
relationship of two variables. This kind of model seems
more realistic and offers more mechanistic insights than
Rosenzweig’s model (1971), which leads to the hypothesis
of enrichment paradox. We found that even simple modifications to the basic form of the Monod or logistic equations are usually sufficient to produce a fold catastrophe
and hysteresis. Structural instability and hysteresis might be
common features in ecological systems. Further, these and
our study suggest that structural instability and hysteresis
characterizes responses of natural systems to anthropogenic
influence, due to the nonlinear nature of the responses. Structural instability and hysteresis also can be sources of variability in system states and presents a challenge to empirical
studies. Data analyses that are based on simple, static,
linear statistical models or the assumption of a single stable
equilibrium state usually fail to deal with this kind of
variability and instability.
As important, potential features of this system, structural instability and hysteresis may have significant
implications for the Everglades restoration. Structural
instability may entail difficulties in preserving the
11
Catastrophe and Hysteresis in Periphyton
Everglades marshes, and necessitate cautious conservation strategies and ‘‘threshold’’ studies. The potential for
hysteresis implies that two thresholds could exist, one for
protection and one for restoration. This study predicts
that the threshold for recovery is likely to be lower than
the threshold for protection, with respect to P loading.
Some ongoing P dosing experiments with periphyton in
mesocosms will soon terminate. Continuous monitoring
of these mesocosms may provide data to test and
quantify this prediction.
Structural instability, multiple equilibria, and hysteresis also imply a likely falsification of a hypothesis
suggested by Hopkinson et al. (1997), which is important
for rule-making in the restoration and conservation of
the Everglades. The hypothesis states that small increases
in P inputs over a long time will have the same end result
as larger increases in P inputs over a short time. This is
not necessarily true. Within certain ranges, a steady P
input may lead to a stable equilibrium with a high standing crop and a fluctuating P input may lead to a switch
between equilibria of high and low standing crops.
We found that the mortality, m, and the refractory
fraction of dead tissue, f, are critical demographic
parameters, and the detrimental effect of P on periphyton, Amax (P), is a critical biological function. They
determine the number and positions of equilibria, the
stability of equilibria, and dynamic behaviors of the
system. Unfortunately, estimates, in situ or in vitro, of
many these rates, such as mortality and sloughing
(overall consequence of death and deposition), are rarely
made (Reynolds, 1984; McCormick and Stevenson,
1991). The architectural response of periphyton to P is
poorly understood. These parameters and processes
deserve more attention in periphyton studies and need to
be explicitly represented in model.
The mortality and refractory fraction of dead tissue
determine the P flux from water through periphyton to
sediment. This flux is an important measure of the assimilative capacity of marshes. Our study suggests that the
efficiency of P removal is high at low P concentrations
and low with high P loading in periphyton-dominated
slough areas. The P flux to sediment could even decline
when P concentration increases beyond a certain level.
This proposition is supported by observations of Craft
and Richardson (1993), but deviates from several widely
accepted theories and model assumptions. First, this
means the P flux is not directly proportional to watercolumn P concentrations, which is assumed by many
water quality models that ignore biological processes
(e.g., Walker, 1995). Second, this also means that the
periphyton systems do not follow the theory that P
recycling is high in low P environments (see DeAngelis
et al., 1989). According to this well-accepted theory, a high
level of recycling and reuse of nutrients in oligotrophic
environment are evolutionarily adaptive. The marshes in
the Everglades appear to offer an example of an exception to this generalization. The adaptive argument needs
to be further examined. In the Everglades, P enrichment
leads green algae to replace abundant filamentous cyanobacteria. In other freshwater systems, P enrichment
usually leads to blooms of filamentous cyanobacteria
and the replacement of green algae (Carpenter et al.,
1998). These deviations suggest a need to improve
theories in this area and call for more studies.
Periphyton is an important component of the food
web and ecosystem of Everglades marshes. Our periphyton model is a part of our food web model and ecosystem
model in our meta-frame modeling approach. The simple
model described here ignored many factors and processes
that may influence the dynamics of periphyton. The
mathematical analysis of this model and the hypotheses
of structural instability and multiple stable states help us
to identify the important components in model development. For example, we need to look at the processes
that cause the collapse of mat architecture. Periphyton
community shifts from calcareous to noncalcareous
when periphyton standing crop changes as a response to
P enrichment. Periphyton community succession also
could play a significant role in P cycling in the Everglades
wetland. P cycling between water column and periphyton
mats may differ significantly from P cycling within
mats. The mucilage production and microbial activity
may be critical processes determining mat architecture. Currently, we are increasing the model complexity, including the number of community types, other
environmental variables and the nonlinearity and discontinuity in the functional relationship. The purpose is
to capture more realism and to improve our understanding of the interaction among those important processes
and variables in the Everglades marshes. We are
developing a suite of models progressively in a metaframe modeling approach to answer different questions,
along a spectrum of temporal and spatial scales and
degree of details and specifications. In this approach,
simulations will be performed to link ecological processes
at different temporal and spatial scales, reconstruct
system dynamics, and examine the scenarios of management or engineering approaches more specifically.
ACKNOWLEDGMENTS
This paper benefited from comments by S. Dailey, T. Fontaine,
E. Gaiser, P. Geddes, A. Hastings, K. Havens, G. Huxel, J. Richards,
L. Scinto, and an anonymous reviewer on various drafts of
12
the manuscript. The sawgrass leaf blades, mosquito songs, heat,
and thunderstorms in the Everglades marshes also sharpened and
stimulated our thoughts. Financial support was provided by the South
Florida Water Management District, the U.S. Department of the
Interior South Florida Ecosystem Restoration Program ‘‘Critical
Ecosystems Studies Initiative’’ (administrated through the National
Park Service), and the U.S. Geological Survey, Florida Caribbean
Science Center.
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