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ELSEVIER
EcologicalModelling97 (1997) 59-73
The long-term bioeconomic impacts of grazing on plant
succession in a rangeland ecosystem
Kevin Cooper a,., Ray Huffaker b
a Department of Pure and Applied Mathematics, Washington State University, Pullman WA 99164, USA
b Department of Agricultural Economics, Washington State University, Pullman WA 99164, USA
Received3 May 1995;accepted8 August 1996
Abstract
The on-site environmental impacts of for-profit livestock grazing on private rangeland are conceptualized as an
interdependent pair of interrelated-species models defined over different time scales. Slow-manifold theory links the fast
(annual) dynamics of an optimization-based grazing-decision submodel (formulating the predator-prey relationship between
livestock and vegetation), with the slow (decade) dynamics of a species-competition submodel (specifying grazing-induced
succession from perennial grasses to less environmentally-desirable annual species). A stable manifold (partitioning phase
space into basins-of-attraction to equilibria representing plant states of differing social desirability) is analytically approximated, and the approximation is analyzed for its mathematical accuracy under various bioeconomic conditions. The
approximated stable manifold represents a 'successional threshold' measuring the resilience of the rangeland ecosystem in
recovering from historic overgrazing. The successional threshold provides a means of evaluating the environmental efficacy
of agricultural programs which would promote recovery of private rangeland by offering financial incentives to induce
for-profit livestock enterprises to reduce grazing. © 1997 Elsevier Science B.V.
Keywords: Plant succession;Livestockgrazing economics;Successionalthreshold
1. Introduction
The ecological structure of seral grassland communities is significantly determined by competition
among constituent plant species (Evans and Young,
1972). In the intermountain region of the United
States, and in the absence of grazing livestock,
rangeland is dominated by highly competitive perennial grasses (e.g., bluestem, grama and bunch grasses)
as understory species to sagebrush. However, his* Correspondingauthor.Tel.: + 1-509-3354308.
toric overgrazing by livestock on preferred perennial
grasses has reduced these grasses' vigor, and thus
their ability to withstand the invasion of highly
competitive alien annual grasses, introduced inadvertently by immigrant settlers. Currently, millions of
acres in the intermountain region are dominated by
alien annual grasses, principally cheatgrass ( B r o m u s
tectorura L.) (Evans and Young, 1972). Cheatgrass is
not valueless in livestock production, but several
drawbacks render it less productive than perennial
grasses. Moreover, cheatgrass promotes several environmental problems. It is more superficially rooted
0304-3800/97/$17.00 © 1997ElsevierScienceB.V. All fights reserved.
PH S0304-3 800(96)00072-5
60
K. Cooper, R. Huffaker / Ecological Modelling 97 (1997) 59-73
than the relatively large fibrous root systems of
perennials, and thus, is not well-suited for binding
soil. This promotes erosion that, among other problems, harms riparian habitat for fish and wildlife
(Steward and Hull, 1949).
The rangeland economics literature has puzzled
over the reasons for historic overgrazing since
"ranchers have no economic incentive as profit maximizers to overgraze continually" (Torell et al., 1991,
p. 805). However, private economic incentives may
not curb chronic overgrazing from a social point of
view. Producers may have little private economic
incentive to improve socially-desirable environmental performance for which they are not legally liable.
For example, the cost of on-site environmental
degradation (in terms of lost productivity) may occur
slowly over a long period, and thus be so heavily
discounted that it does not have a significant impact
on curbing resource use (Taylor and Young, 1985).
This may explain overgrazing leading to adverse
plant succession on private rangeland, since empirical evidence shows that succession occurs relatively
slowly on a decade-long time scale (McLean and
Tisdale, 1972).
National agricultural policy traditionally has relied on financial incentives to encourage more environmentally sound production practices in other resource-use situations (e.g., the U.S. Department of
Agriculture's Conservation Reserve Program). Similarly, ranchers can be induced to curb overgrazing to
socially-desired levels through either production subsidies or taxes on output (e.g., an ad volorem tax on
beef price). The ultimate environmental success of
economic grazing policies depends on the extent to
which induced stocking reductions can reverse undesired changes in vegetation types (i.e., the 'resilience' of the rangeland ecosystem).
The traditional 'range-succession' (RS) theory optimistically predicts that any grazing-induced retrogression in vegetation type away from the single
stable 'climax' state can be reversed continuously
along the same pathway (i.e., 'secondary succession')
by reducing grazing. However, the RS theory is
losing ground to the 'state-and-transition' (ST) theory due to empirical evidence that rangeland ecosystems are not so resilient. The ST theory predicts less
optimistically that, due to discontinuous and irreversible hysteresis effects, vegetation types may be
locked into 'basins-of-attraction' compelling them
toward stable lower-successional-states over time
(Westoby et al., 1989; Laycock, 1991). Rangeland
conditions are described " b y means of catalogues of
alternative states and catalogues of possible transitions between states" (Westoby et al., 1989, p. 266).
The ST theory implies that the success of stocking
reductions in promoting secondary succession depends on whether rangeland conditions can be pushed
across 'thresholds' of environmental change to more
socially-desirable stable plant states (i.e., those with
a large degree of perennial grasses). Thus, successional thresholds become the key analytical tool in
characterizing the resilience of the rangeland ecosystem.
1.1. Purpose and approach
Thus far, successional thresholds between plant
states have been studied in a purely-ecological context where wildlife numbers are regulated solely by
natural forces (Boyd, 1991). The purpose of this
paper is to build in the additional economic forces
necessary to approximate successional thresholds in
the bioeconomic context relevant to for-profit livestock enterprises operating on rangeland. Grazing
management in a bioeconomic context is controlled
by the interaction of economic forces inducing producers to stock animals on rangeland, and ecological
forces determining the response of the rangeland
resource to grazing, and thus future stocking possibilities. Consequently, the bioeconomic grazing system is in equilibrium only when the producer has no
economic incentive to stock an additional animal
(economic balance), and the stocking rate interacts
with successional forces to sustain a given vegetation
type over time (ecological balance). Successional
thresholds become indirectly dependent on economic
parameters, and thus useful in determining the environmental success of economic grazing policies applied to for-profit livestock enterprises.
The paper applies slow-manifold theory (Wasow,
1976) to link the fast (annual) dynamics of a grazing-decision submodel (formulating the predatorprey relationship between livestock and vegetation in
an economic context), with the slow (decade-long)
dynamics of a plant-succession submodel (formulating the competitive relationship between perennial
K. Cooper, R. Huffaker / Ecological Modelling 97 (1997)59-73
grasses and less ecologically desired annual grasses).
A stable-manifold (partitioning phase space into
basins-of-attraction to equilibria representing desirable and less-desirable plant states) is analytically
approximated, and the approximation is analyzed for
its mathematical accuracy under various bioeconomic conditions. The approximated stable manifold
represents the 'successional threshold' for the bioeconomic grazing system, and measures the resilience
of the rangeland ecosystem in recovering from historic overgrazing. The successional threshold provides a means of evaluating the environmental efficacy of agricultural programs which would promote
recovery of private rangeland by offering financial
incentives to induce for-profit livestock enterprises
to reduce grazing.
The paper proceeds by setting out the overall
agroecological grazing model. The solution is then
discussed, emphasizing procedures leading directly
to the analytical approximation of successional
thresholds.
2. The agroecological grazing model
The model developed in this paper extends the
purely ecological formulation of successional thresholds of Boyd (1991) into a bioeconomic context. He
conceptualizes the overall rangeland ecosystem as an
interdependent pair of interrelated-species models
defined over different time scales. A two-equation
species-competition model specifies the long-term
relationships between the portion of habitat in perennial grasses (suitable for grazing) and in 'weeds'
(alien annual grasses not suitable for grazing) on a
decade-long 'slow' time scale. A two-equation
predator-prey model formulates how the annual production of grasses (used as a proxy for the grasses'
vigor) depends on the density of naturally-grazing
herbivores, and vice-versa, on an annual 'fast' time
scale. The grasses' annual productivity, or vigor,
(determined in the predator-prey model) affects their
long-term success in competing with weeds. Reciprocally, the long-term portion of available habitat in
grasses (i.e., 'range condition' determined in the
competition model) affects annual forage productivity, and thus the carrying capacity of the habitat for
herbivores.
61
Boyd's species-competition model is employed
intact. However his 'naturally-regulated' vegetationherbivore submodel is modified in two substantial
ways. First, the submodel is reformulated in an
optimization-based decision-making context. This
necessitates introducing a control variable (herbivore
stocking density) that responds to economic parameters (e.g., interest rates, meat prices, purchasing and
handling costs, etc.) as well as biological parameters.
Second, an additional state variable (annual productivity of 'weeds') is introduced to reflect the reality
that annual grasses have some grazing value.
2.1. The grazing-decision component
The grazing-decision submodel focuses attention
on the livestock densities that a hectare of rangeland
can most profitably service each year. The objective
is to maximize the value - - added on rangeland as
livestock increase toward a desired weight that triggers removal to dry-lot finishing. Since animals are
assumed to be stocked on the rangeland to gain
weight - - and not to reproduce - - the model makes
no provision for their fecundity. In short, the model
is applicable to the 'stocker' operations prevalent in
the Intermountain West.
Let the variables 0 _< G < 1 and 0 < W < 1 represent the portion of the area available for colonization
by 'grasses' (perennial grasses) and 'weeds' (annual
grasses), respectively, that is in fact inhabited by
them. Areas may be barren or have overlapping
vegetation, thus the sum of G and W need not equal
one. Let X ( k g / G / h a ) represent the annual herbage
productivity of grasses per unit G per hectare as a
proxy for the grasses' vigor; and Y ( k g / W / h a )
represent the annual production of weeds per unit W
per hectare. Thus, variables O x = XG and O r = YW
denote the total amount (kg/ha) of grasses and
weeds, respectively, available to livestock. The variables O x and O r are important because they decompose total vegetation into a component moving
on a fast (annual) time scale (X and Y) and a
component moving on a slow (decade-long) time
scale (G and W). This allows for feedback between
the fast-dynamics of the grazing-decision submodel
and the slow-dynamics of the plant-succession submodel.
62
K. Cooper, R. Huffaker/ Ecological Modelling 97 (1997) 59-73
The growth of rangeland vegetation each period is
typically modeled by ecologists as a logistic function
(see e.g., Metzgar and Boyd, 1988). Therefore, the
average annual growth functions for grasses
(kg/G/ha/year (yr)) and weeds (kg/W/ha/yr),
respectively, are:
rate of grasses (weeds), then the annual flow of
stocking profits ( $ / h a / y r ) is:
Fo( X) = RxX(1 - X / r x )
(1)
F2(Y) = RrY(1 - Y / K r )
(2)
where S (control variable, hd/ha) is the animal
stocking density; P ($/kg) is the meat price; 7x and
Yr are forage conversion ratios when livestock forage on grasses and weeds, respectively; and C
( $ / h d / y r ) represents the annual costs of the fattening program on rangeland per animal.
The producer is assumed to select stocking rates
each year to maximize the discounted flow of annual
profits subject to the biological capabilities of the
rangeland ecosystem:
R x (1/yr) and R r (1/yr) are intrinsic growth rates,
and K x (kg/G/ha) and K r (kg/W/ha) are maximum productions of X and Y. Fo(X) and FE(Y) are
symmetric concave-down functions with zeros at the
origin and K x and Kr, and maxima at K x / 2 and
Kr/2, respectively.
The per capita annual consumption of rangeland
vegetation by herbivores is typically modelled as a
Michaelis-Menten 'satiation' function (see, e.g.,
Metzgar and Boyd, 1988). Hence, average annual
per capita consumption is given by:
QOx
F,(Ox) = 0.05Dx + Ox
(3)
where Q (kg/head (hd)/yr) measures the maximum
annual forage consumption rate per animal; and D x
(kg/ha) represents the level of O x at which an
animal is 95% satiated, i.e., FI(Dx) = (0.95)Q. Thus,
D x is inversely related to an animal's grazing efficiency. F~(O x) begins at the origin and increases at
a decreasing rate toward the horizontal asymptote
given by Q.
Grazing preference for grasses is accounted for by
assuming that an animal forages weeds only when
necessary to continue feeding toward Q:
[ Q - Fl( Ox)] Or
F3(Ox, O r ) =
0.05Or + O r
(4)
where F 3 ( k g / h d / y r ) denotes the average annual
consumption of weeds per animal; [ Q - Fl] is the
maximum annual consumption rate per animal of
weeds as the residual feeding need after foraging on
preferred grasses; and D r is the counterpart of Dx
in F~ with an analogous interpretation.
Let the average annual weight gain per animal on
grasses (weeds) be proportional to the consumption
~r(Ox, Or, S)
= [P[YxF,(Ox) + YrF3(a x, Or) ] - C ] S
(5)
oo
max fo e - r t ~ ( ~ x , 0 r , S) d t
(6)
~( = Fo( X) -- ( S/G)FI( Ox)
(7)
I) = F2(Y) - (S/W)F3( a x , Or)
(8)
0~S~S
max ,
X(t)lt.o=Xo,
Y(t)lt-o= Yo
(9)
where r (1/yr) represents a real annual discount rate
and t represents time, in years. Eq. (7) models the
average annual net rate of change of X as annual
growth less annual total consumption by livestock.
The per capita foraging rate, F1, is multiplied by
(S/G) since X is measured per unit G. Eq. (8)
depicts the annual net rate of change of Y in an
analogous fashion. The producer's selection of stocking rates to maximize Eq. (6) is subject to the control
constraint and initial conditions on the state variables
X and Y, given in Eq. (9). S max is an exogenously
determined limit on the maximum number of animals the producer is willing or able to stock on the
rangeland.
2.2. The successional-change component
Boyd models plant succession as a special case of
Gause's interspecies-competition equations:
=R~G[1 - G - qw(G, X)W]
(10)
I/V= R w W [ 1 - W - qc( G, X)G]
(11)
K. Cooper, R. Huffaker / Ecological Modelling 97 (1997) 59-73
where G, W, and X are defined above; R e ( l / t )
and R w ( l / t ) are the intrinsic growth rates of G and
W; and qe > 0 and qw > 0 are unitless competition
coefficients. Each equation models competitive losses
as a drag on the net per-capita growth rate term. The
values of R e and R w are assumed to be small
relative to R x and R r (i.e., the fractional composition of the rangeland in grasses and weeds changes
slowly compared with changes in the annual productivities X and Y). Although time continues to represent years in the successional-change submodel, R e
and R w are sufficiently small that G and W change
significantly only over decades. Consequently, they
are effectively 'slow' variables moving on a decadelong time scale.
The competition coefficients are functions of G
and X because the density and vigor of grasses
determine their ability to compete with weeds for
habitat. Grasses compete more favorably when G
increases, forcing qe toward an upper bound q~,
and qw toward a lower bound q~,. Boyd models the
direct (inverse) bounded relationship between G and
qe (qw) with Michaelis-Menten functions:
u[ B + G / E ( X )
qe = qe -~ + G / E ( X)
(12)
, [ I +G/E(X) ]
qw = qw ~ + G / E ( X )
(13)
E(X)
(14)
=
3,+ 8(1
-X/Kx)
As G increases, Eq. (12) increases qe from its lower
bound (q~ - B qe,
u when G = 0) asymptotically toward its upper bound (q~ as G ~ oo). Conversely, as
G increases, Eq. (13) decreases qw from its upper
bound (q~v = qlw/B, when G = 0) asymptotically toward its lower bound (q~v as G ~ oo). The parameter
B < 1 is simply the ratio between the upper and
lower bounds on qe and qw. The competition coefficients respond more rapidly to increases in G (i.e.,
qe and qw approach their maximum and minimum
values, respectively, at lower levels of G) for smaller
levels of E(X). Because the competitiveness of
grasses directly depends on the grasses' vigor, X,
E(X) is an inverse function of X in Eq. (14). E(X)
approaches its minimum value, 3¢, as X approaches
its maximum production K x, and 8 is a positive
constant fixing the rate of decrease.
63
3. Solution of the agroecological grazing m o d e l
The combination of analytical and numerical work
needed to analyze the agroecological grazing model
is facilitated by converting the formulation to one
using dimensionless variables and parameters. Fast
variables X and Y are scaled between zero and one;
functional forms are greatly simplified, especially in
the grazing-decision submodel; and the number of
parameters decreases from eleven to seven in the
grazing-decision submodel, and from eight to seven
in the plant-succession submodel.
The dimensionless grazing-decision submodel is:
maxf e- ,'[
p5 ~,( x)
+
p6 yf3( x ,
y) -
1] s dr
(15)
x' = x [ y 0 ( x )
- ss,(x)]
Y' = Y[ P3f2(Y) - p4sf3( x, y)]
0 ~ S <: S max ,
x(t)l,= 0 = x0,
(16)
(17)
y(t)l,=0 =Y0
(18)
where scaled variables are x = X / K x (fraction of
maximum annual production of grasses); y = Y / K r
(fraction of maximum annual production of weeds);
r = R x t (scaled time variable); and s =
( Q / R x K x G ) S (scaled annual stocking density).
Scaled parameters are p~ =O.05Dr/KrW; p2 =
O.05Dx/KxG; P3 = Rr/Rx; P4 = KxGP2/KyW;
P5 = PQTx/c; P6 = PQTrP2/C; and P7 = r/Rx.
The slow variables G and W enter the grazing-decision submodel indirectly through s, Pl, P2 and P4.
Scaled functions are fo(x)= 1 - x ; f l ( x ) = 1/(p2
+ x ) ; fE(Y) = 1 - y ; and fa(x, y ) = 1/(p2 +x)(pt
+ y). The prime symbol represents a dimensionless
time derivative, e.g., d x / d r = x'.
The dimensionless successional-change submodel
is"
G'=roG[1-G-qw(G,E)W l
(19)
W '= rwW[1 - W - qe(G, E)G]
(20)
E( x) = y + t ~ [ 1 - x ]
(21)
where rc = R c / R x , r w = R w / R x ; and qG(G, E)
and qw(G, E) are given in Eqs. (12) and (13).
64
K. Cooper, R. Huffaker / Ecological Modelling 97 (1997) 59-73
The transformed Hamiltonian and adjoined system
3.1. Solution of the grazing-decision submodel
are:
Solution of the fast grazing-decision submodel is
initiated by setting the slow variables in scaled parameters Pl, Pz and P4 at fixed levels, (G, W ) =
(Gf, Wf). Techniques developed by Mesterton-Gibbons (1988) are modified to derive the optimal stocking rule in a competitive-species and linear-control
setting: s* = s * ( x , ylG e, We).
The Hamiltonian and adjoined system associated
with Eqs. (15)-(18) are:
H( x, y, s, W l, W2, ~')
H ( x , y, s, A1, A2, r )
where f0~, fl~, f3~, ~ , f2y, f3y, and ~y are the
first derivatives of fo, ft, f2, f3, and ~ with respect
to their arguments. The transformed adjoined system
Eqs. (27) and (28) is derived by taking the dimensionless time derivatives of the transformed adjoined
variables Eq. (24), and then substituting in the previous adjoined system Eq. (23) and the equations of
motion Eqs. (16) and (17).
The optimal stocking rule s * (x, y) can be solved
for explicitly by taking the first and second dimensionless time derivatives of r/(x, y, Wt, W2, ~-) in
Eq. (25) and substituting in the transformed adjoined
system W( and W~ in Eqs. (27) and (28):
= e-PTrtp(x, y ) S + A l x [ f o ( X
+ A2y[p3f2(y ) -pasf3(x,
A'~ = - OH/Ox,
) -sf,(x)]
y)]
A'~ = - d H / O y
(22)
(23)
where ~p(x, y ) = psxf~(x) + p6Yfa(x, y ) - 1 .
Costate variables A~ and A2 ($/lb consumed) measure the marginal present values of the grass and
weed forage stocks, respectively, and thus the opportunity costs of consuming the stocks presently by
marginally increasing the stocking rate.
Following Mesterton-Gibbons, the following new
variables are defined:
W~ = eP~9,~x,
W2 = eP79t2 y
+ e-P: [f0(x)Wl +p3f2(Y)W2]
(26)
W; = [ P7 - X( fox - sf,~)]Wl + P, f3xxsW2 - q~xxs
(27)
W~=[p7-y(p3f2y-p4sf3y)]W2-~yys
y)W2-M(x,
(28)
y)]
(29)
rl'= e - P ' r [ K ( x ,
y)W l +L(x,
7/" = e - ' : [ A ( x ,
y, W , , W 2 ) s - B ( x, y, Wi, W2) ]
(24)
*/( x, y, W,, W2 , r )
= e - P " [ tp(x, y) - f , ( x ) W , - p j 3 ( x ,
= *l( x , y , Wl, W 2, r ) s
(3o)
y)W~]
(25)
where functions K, L, M, A and B are defined in
Table 1.
Table 1
Functions needed to calculate s *(x, y)
/¢(x) = (fo~fl - fof~x)x
L(x, y)= P3P4(Ayf3 - f2f3y)Y - P4fofax x
g ( x , y) = pTtp - foq~xx - Paf2 ~yy
A(x, y, W~, W2) = A x[Mx - ff~ ~x K -(Kx - ff~ ~f~xK)W~-(L~ - P4ff~ ~AxK)W~]+pjay[My - ( P J a ) - ~%L-(Ly - ~ JAyL)W2]
B(x, y, W~, W~) = (fox K - fo gx )xWj + [ Pa(A y L - f2 Ly) y - fo Lx x]W2 + P3A My y + fo Mx x -- P7 M
A = A L-- p4faK
Wl = wl(x, y) = (q~L - p g f a M ) / A
W2 = wz(x, y ) = ( f i M - ~oK)/A
QI( x, y) = Mx - J~z t~xX - ( K x - fill 1A x K ) w t ( x , Y ) - ( L x -- P4f-I lAXK)W2(X'Y)
Q2(x, y) = My -(P4f3)- l~°yL - (Ly - ~ lfayL)wz(X, y)
O(x, Y)= (fo~-fof~x/A)xKw~( x, Y ) + [ P 3 ( f 2 y - f z f 3 y / f 3 ) L Y - P4fof3xKx/fl]w2( x,
y)+ fo~.Kx/A + p3f2%Ly/p4A- pTM
K. Cooper,R. Huffaker/ EcologicalModelling 97 (1997)59-73
Since r/(x, y, W1, w 2, ~') = 0 along the singular
path, the first and second dimensionless time derivafives in Eqs. (29) and (30) also vanish. Forming the
two equation system 7/= 0 = ~/' allows solution of
the transformed adjoined variables along the singular
path as functions of the variables and parameters of
the system: W1 = wt( x, y) and W 2 = w2( x, y), where
wl(x, y) and w2(x, y) are given in Table 1. Finally,
setting 7/" = 0 in Eq. (30) yields the optimal stocking
rule:
s*(x, y)
b( x, y)
a( x, y)
f o x a l ( x, y) + p3f2YQ2( x, y) + t~( x, y)
f l x Q , ( x, Y) + p+f3YQ2( x, y)
(31)
X
where a(x, y) and b(x, y) result from substituting
W l = Wl(X, y) and W e = w 2 ( x , y ) into A(x, y, W 1,
WE) and B(x, y, W l, W2); and Ql(x, y), Q2(x, y)
and qJ(x, y) are defined in Table 1.
The optimal sequence of stocking rates from given
initial levels of x and y is synthesized from three
phase planes in (x, y)-space. The 'singular' phase
plane is derived by substituting s* (x, y) into x' and
y' in Eqs. (16) and (17)), respectively. Numerical
work shows it to be characterized by a net-presentvalue-maximizing saddle point equilibrium that depends on fixed values of the slow variables Gf and
We, i.e., [x*(Gf, Wf), y*(Gf, We)]. The associated
convergent separatrices represent the singular path
along which the stocking rate follows the optimal
feedback control rule s*(x, ylGf, Wf), and x and y
are adjusted toward their optimal sustained levels,
x * and y *. The saddle point equilibrium (x *, y *)
y
•
65
+III
.I)41
.I),I
.II
%
+I
$
.x
%
Fig. I. Equilibriumlevelsfor (a) productivityof grasses, x; (b) productivityof weeds, y; and (c) the stockingdensity,s; on the fast time
scale plottedagainst G and W.
66
K. Cooper, R. Huffaker / Ecological Modelling 97 (1997) 5 9 - 7 3
must satisfy x' = y' = 0, which calculations show to
occur when z(x, y)= P4fo(x)f3(x, y ) p3fl(x)f2(y) = 0. The equation z(x, y ) = 0 is the
implicit representation of a curve in phase space that
connects the saddle points associated with different
fixed stocking rates, s. When the economics are
blended in by substituting the optimal stocking rule
s*(x, y) from Eq. (31) into x' and y', calculation
shows that equilibrium must also satisfy ~(x, y) = 0,
where tp appears in Eq. (31). Thus, the saddle point
equilibrium can be found numerically by scanning
values of x and y along the curve z(x, y ) = 0 for
those satisfying O(x, y ) = 0.
When rangeland conditions are off the singular
path, the producer must engage in 'most-rapid-approach-path' (MRAP) policies. The MRAP recovery
path (used when initial levels of x and y are below
the singular path) is contained in a zero-stocking
phase plane in (x, y)-space, which is derived by
setting s at its lower bound of zero in Eqs. (16) and
(17). The MRAP depletion path (used when initial
levels of x and y are above the singular path) is
contained in a maximum-stocking phase plane in (x,
y)-space, which is derived by setting s at its upper
bound of S max.
Because the fast model is initialized with fixed
Table 2
Baseline parameter valuesa
Parameter
Description
Units
Value
rb
pc
Yx(Yr)d
Ce
Kx(Kr) t
Rx(Rr) g
Qh
Dx(Dr) i
real annual discount rate
meat price
forage conversion ratio on grasses (weeds)
total variable costs of rangeland fattening program
maximum production of X ( Y )
intrinsic growth rate of X ( Y )
maximum consumption rate of forage
level of grasses (weeds) promoting 95% satiation
year- ~
$/kg
-S/bead/year
kg/G (W)/ha
year- t
kg/head/year
kg/ha
0.0515
1.434
0.093 (0.07)
121.44
24.3 (24.3)
3.5 (1.45)
938.95
4.85 (4.85)
aGeneral notes: The baseline values represent a typical 200 head cattle stocker operation in southern Idaho, as reported in Smathers and
Gibson (1990). Yearling steers are purchased on March 15, fed on alfalfa and spring range for about 60 days, and then placed on rangeland
from May 15 to September 30 (about 138 of the total 198 days of the season).
br is calculated as the annual interest rate on AAA corporate bonds for March 1992 (0.0835) less the percentage change in prices over the
previous year (0.032) (Federal Reserve, 1992).
Cp is calculated as the average of the reported price at the end of the grazing season (September 30) over the years 1989 and 1990 (USDA,
1990).
ayx is calculated as the ratio of the average daily gain over the 138 day grazing season (0.635 kg/head/day, Smathers and Gibson, 1990)
to average daily consumption (6.8 kg/head/day, Holochek). Yw is assumed to equal 0.75y x, reflecting the reality that weeds are less
productive in livestock production.
eThe paper prorates total variable costs reported in Smathers and Gibson 0990) over the fractional time that livestock are on rang¢land. The
costs of initially purchasing steers weighing 272.16 kg/head at $ 1.74/kg are assumed to be sunk for the purposes of calculating net value
added on rangeland. However, the model does prorate the loss that the producer takes on the first 272.16 kg of each animal, since each
kilogram is sold at the end of the season at only $1.43/kg.
fNoy-Meir (Noy-Meir, 1976, p. 95) reports that a forage carrying capacity of 500 g / m squared (24.3 kg/ha) is reasonable for rangeland of
high productivity. For lack of better information, the paper adopts this value for both grasses and weeds.
gSteward and Hull (Steward and Hull, 1949, p. 67) report a spring clipping for crested wheatgrass (perennial) of 21.33 kg/ha. For lack of
better information, the paper assumes that the clipping occurred at the maximum sustained yield stock level ( = 0.5Kx). Substituting this
information into (l) gives: 21.33 = R x ( K x / 2 ) [ l - K x / 2 / K x ]. Solving for R x yields the baseline value. R r was solved for in the same
way, except that the spring clipping for cbeatgrass (weeds) was reported to be 11.81 kg/ha.
hQ is calculated as the product of the average daily consumption rate of vegetation by an animal (6.8 kg/head/day, Holechek, 1988) and
the number of days in the rangeland grazing season (138).
iThe grazing efficiency parameters D x and D r are assumed to be 20% of maximum annual production of X and Y, respectively
(Noy-Meir, 1976).
67
K. Cooper, R. Huffaker/ Ecological Modelling 97 (1997)59-73
values of the slow variables, it is important to determine how the net-present-value-maximizing equilibrium [x * (Gf, Wf), y * (Gf, Wf)], and optimal annual
stocking densities s* change over the grid of possible values for the slow variables, 0 < G, W < 1. Fig.
1 displays these results generated with baseline values for the grazing-decision parameters described in
Table 2. When grasses and weeds are colonized at
very low levels (e.g., G - - - W = 0), there is little
economic incentive to stock (s = 0), and x and y
w
(a)
o
" - i "r~
0
G,w~
(C)
wjGl
are unconsumed and sustained at maximum capacity
( x = y = 1). As G increases to one, the producer
increases livestock to approximately S * = 3.73
head/ha (s* = 0.2), and x* is preferentially consumed down to a sustained level of about 6% capacity. Given the parameters underlying the simulation,
there is sufficient annual production of grasses for an
animal to draw close to the satiation consumption
level Q, and thus weeds remain largely unconsumed.
The trends are dampened when W is at a relatively
G, !
/
o
G= G~wl
(d)~/
w,Gv,,
il
Gj~b
0~ ~ ~ - ~ ~ W L
(e)
~P
w,
GI
l"
o
co)
/
6, I
6,wl
Fig. 2. Arrayof slow-systemphasediagrams.
K. Cooper,R. Huffaker/ EcologicalModelling 97 (1997)59-73
68
low level (W--0.2), so that s* increases less, and
x* decreases less, in response to increased colonization of grasses.
3.2. Solution of the successional-change submodel
Solution of the successional-change submodel Eqs.
(19)-(21) on a slow time scale is initiated by fixing
the fast variable x in E ( x ) at a constant level. The
competitive dynamics of G and W are then investigated in (G, W)-space using conventional phase-diagram techniques. The dynamics are discussed in
detail since they lead directly to analytically approximating successional thresholds.
Nullclines setting G' = 0 and W' = 0 are denoted
as GI and WI, respectively. Two nullclines are the
W-axis (GI) and the G-axis (WI). The other two are
the graphs of"
GI(G) =
W]c'=0
[G2+ [E( x)B- I]G- E(x)B]
qlw[E( x ) + G ]
(32)
w I ( o ) = w Iw,=o
[q~G 2 + [ E ( x ) B q ~ -
1]G- E(x)]
e(x) + C
(33)
Fig. 2 shows that GI Eq. (32) has a G-axis intercept
a t 1, where grasses are 100% colonized and weeds
are extinct, and a W-axis intercept at critical level
Wc = 1/q~v. WI Eq. (33) has a W-axis intercept at 1,
where weeds are 100% colonized and grasses are
extinct, and a G-axis intercept at:
Fig. 2 shows the range of phase diagrams that can
be generated depending on the relative magnitudes
of Wc and G c to one. Each diagram has an unstable
node equilibrium at the origin (W = G = 0); an 'allweeds' equilibrium along the W-axis at (G = 0, W =
1); and an 'all-grasses' equilibrium along the G-axis
at (G = 1, W = 0). Stability of the 'all-weeds' equilibrium is determined completely by the upper bound
on the competitive ability of weeds, q~,. When
q~ > 1 (i.e., Wc < 1), weeds are a relatively strong
competitor, and the 'all-weeds' equilibrium is a stable node attracting all trajectories in the positive
quadrant (Fig. 2a-c). Alternatively, when q~, < 1
(i.e., Wc > 1), weeds are a relatively weak competitor, and the 'all-weeds' equilibrium is a saddle point
repelling all trajectories in the positive quadrant (Fig.
2d and e). When x > x I (i.e., G c < 1), grasses are a
relatively strong competitor, and the 'all-grasses'
equilibrium is a stable node attracting all trajectories
in the positive quadrant (Fig. 2c and e). Alternatively, when x < x I (i.e., G c > 1), grasses are a
relatively weak competitor, and the 'all-grasses'
equilibrium is a saddle point repelling trajectories in
the positive quadrant (Fig. 2a, b, d). Up to two
interior equilibria are added in the positive quadrant
if the nullclines intersect (Fig. 2b-d).
Because solution of the slow model is initiated
with a fixed value of x, it is important to determine
the systematic transitions among phase diagrams in
Fig. 2 that occur in response to annual changes in x
generated by the fast model. Fig. 3 shows a numeri-
I
saddle (w = O)
Gc
1 - E( x)Bq~ + ~[ E( x)Bq~ - 1] 2 + hE( x)q~U
\
2q~
(34)
node (w ffi 1)
Setting G c = 1 and solving for x yields critical
value, xl, as a function of the remaining parameters:
xl = ( K x / , 3 ) [ T + 8 + ( 1 - q ~ ) / ( 1 - q ~ ) ] .
When
x is greater (less) than x l, G c is less (greater) than
one.
0
(2s)
.
.47
saddle
\ .%.
~"~'"----
X
---- (2b)----~ t 4- (2c)
x2
xl
Fig. 3. Bifurcation diagram o f G against x. Solid lines (including
the x-axis)denotestable-nodeequilibria, and dashed lines denote
saddle-pointequilibria.
K. Cooper,R. Huffaker/ EcologicalModelling97 (1997)59-73
cally-generated bifurcation diagram, plotting equilibrium levels of G against increasing values of fast
variable x. The baseline successional parameter values (R 6 = 0.27, R w = 0.35, 3' = 0.1, ~ = 0.4, q~v =
0.6, q~ = 1.07, and B = 0.3) are selected to make
weeds strongly competitive (i.e., q~ = qtw/B > 1),
because the resulting phase diagrams display successional thresholds (Fig. 2b, c). Successional thresholds appear as a pair of convergent separatrices, or
stable manifold, associated with an interior saddlepoint equilibrium (G sp, WsP). The thresholds partition phase space into disjoint basins-of-attraction
(BOA). All rangeland conditions (i.e., G and W
combinations) to the left of the stable manifold
gravitate over time toward the 'all-weeds' equilibrium, and those to the right gravitate either toward an
interior equilibrium (Fig. 2b), or to the 'all-grasses'
equilibrium (G e, W e) = (1, 0) (Fig. 2c). Thus, a
necessary condition for the existence of successional
thresholds is that the 'all-weeds' equilibrium be stable, which in turn requires q~v > 1. Alternatively, no
disjoint BOA's exist in Fig. 2(a, d, e), where all
initial rangeland conditions in the positive quadrant
gravitate to the 'all-weeds' equilibrium (G e, W e) =
(0, l) (Fig. 2a), an interior grasses-weeds equilibrium (Fig. 2d), and the 'all-grasses' equilibrium (Fig.
2e), respectively.
The dashed line in Fig. 3 at G = 1 represents the
unstable 'all-grasses' equilibrium, and the solid line
at G = 0 represents the stable 'all-weeds' equilibrium. When x is less than a critical value x 2 = 0.47,
Fig. 2(a) governs, and there is eventual competitive
exclusion in favor of the 'all-weeds' equilibrium.
When x 2 < x < x 1 = 1 (where x~ is a critical value
defined above), the nullclines intersect resulting in a
stable higher-grasses equilibrium (solid curved line)
and a lower-grasses saddle point (dashed curved
line) (i.e., Fig. 2(b) governs). At the lower critical
level x 2, the nullclines are tangent at a single interior equilibrium where the interior stable and saddle
point equilibria coalesce (providing a transition between Fig. 2a and b). At the upper critical level x t,
the stable higher-grasses equilibrium coalesces with
the 'all-grasses' equilibrium (providing a transition
between Fig. 2h and c). The 'all-grasses' equilibrium
changes stability from a saddle point to a stable node
and begins to attract initial conditions not in the
'all-weeds' BOA. Thus, competitive exclusion even-
69
tually reigns in favor of grasses as they become more
vital (unless the populations are locked into the
lower-grass saddle point).
Critical value x 2 is especially important because
only values of x above it generate phase diagrams
characterized by successional thresholds. The formula for x 2 can be found by: (a) Equating the
nullclines Eqs. (32) and (33) to derive a quadratic
function whose roots are equilibrium levels of G; (b)
using the quadratic formula to solve for interior
equilibria in terms of fast variable x and the ecological parameters of the slow system; and (c) setting the
discriminant of the quadratic formula equal to zero
to solve for critical value x 2 that generates a single
interior equilibrium.
3.3. The combined dynamics of the fast and slow
subrnodels
Fig. 3 demonstrates that successional thresholds
are not static. They adjust to the parameter x, which
changes through time according to the grazing-decision submodel. Thus, understanding how the slow
dynamics of the successional-change submodel are
coordinated with the fast dynamics of the grazingdecision submodel becomes essential. The solutions
of the two submodels are tied together in (G, W, x,
y)-space via Tihonov's theorem (Wasow, 1976, pp.
249-260).
Recall that x *( G, W) and y* ( G, W) denote the
dependence of the optimal saddle-point equilibrium
in the singular (x, y)-phase plane on G and W. In
(G, W, x, y)-space, x*(G, W) and y*(G, W)
define a two-dimensional 'slow' manifold. The manifold is called 'slow' because the grazing-management system is at its optimal steady state, and thus
only the slow-time-scale dynamics described by the
successional-change system remain operative. Fig. 1
above displays the values of x* (G, W) and y* (G,
W) along the slow manifold.
Tihonov's theorem holds that a trajectory in (G,
W, x, y)-space can be approximated during some
time interval by a curve composed of two contiguous
arcs C l and C 2. The arc C l originates from initial
values (Go, W0, x 0, Y0) and, via the above-described combination of extreme and singular controis, 'rapidly' adjusts to the slow manifold at
[ x * ( G o, W0), y*(G o, W0)]. The arc C 2 is derived
70
K. Cooper, R. Huffaker/ Ecological Modelling 97 (1997) 59-73
by solving the (G, W)-system with E=E[x*(G o,
W0)] in Eq. (21). Arc C2 depicts the movement of
the (G, W, X, Y)-system along the slow manifold
from [x*(G0, W0), y*(Go, W0)] toward [ x * ( G e,
We), y*(G e, We)], where (G 0, W0) is assumed to lie
in the BOA of (G e, We). As x and y adjust on the
fast time scale to their new values along the slow
manifold, the parameter E(x) must be recalculated
for the adjusted values of x, and this may change the
configuration of the (G, W) phase diagram in the
systematic ways discussed above and displayed in
Fig. 3. In short, x and y continue adjusting on the
fast time scale, and G and W continue adjusting on
the slow time scale, until a long-term equilibrium is
reached.
The paper now investigates the analytical approximation of successional thresholds.
4. Approximation of successional thresholds
The successional threshold for the slow system is
formed by three solutions to the G and W equations:
One is a solution that approaches the saddle point
equilibrium as t--* Qo, and approaches the origin as
t~-oo
(i.e., the upward convergent separatrix).
The second is the saddle point itself. The third is the
unique solution that approaches the saddle point as
t ~ 0% and is unbounded as t ~ - ~ (i.e., the downward convergent separatrix). These form the so-called
'stable manifold' of the saddle equilibrium (G 'p,
WsP). A number of techniques have been developed
for approximating stable manifolds locally near saddle points (Hale and Kocak, 1991). These methods
are not satisfactory for our purpose, both because
they are somewhat complicated to use, and because
they provide their most accurate approximations only
near the saddle point itself. In order to obtain an
accurate approximation at a point remote from the
saddle point, one must compute a large number of
terms of a series.
This paper takes a different approach to the approximation from the standard one. Information is
used concerning the stable manifold not only at the
saddle point, but also at the equilibrium at the origin.
Suppose that the stable manifold is the graph of the
function w~m(G). The stable manifold is a concate-
nation of several solutions to the differential equation. Thus at any point that is not an equilibrium
W Sm(G) must satisfy the equation
dW sm
dWSm/dt
dG
dG/dt
rwWSm[1-wsm-q~(G,E)G]
~- r a G [ l _ G _ q w ( G , E ) W S m ]
(35)
This equation has solutions of the form GPX(G),
where limG_,oX(G)=K4~O, and p is a positive
real number. The conditions we use in determining
an approximation of this same form are (i) WS~(G 'p)
= WsP; (ii) dWSm(Gsp)/dG ----/32/01, where (v~, /)2 )
is the eigenvector of the Jacobian matrix corresponding to the negative eigenvalue for the system at
(W 'p, Gsp). The form of the solution forces the
stable manifold to approach the origin as G goes to
zero. The power p determines the behavior of the
curve as G goes to zero. To determine p, note that
pG'-'X(G)
+ G'X'(G)
r w G'X(G)[1 - G'X(G) - qo(G, E)G]
=
rG
G[1 - G - qw( G, E)GPX( G)]
(36)
Multiplying both sides by G ~- P, and taking a limit
as G goes to zero shows that p - - r w / r c. This
means that if the intrinsic growth rate of the weeds is
greater than that of the grass, the stable manifold
approaches the origin tangent to the G-axis (i.e., the
BOA to the 'all-weeds' equilibrium is larger near the
origin), while if the intrinsic growth rate of the grass
is greater, then the stable manifold approaches tangent to the W-axis (i.e., the BOA to the 'somegrasses' equilibrium is larger near the origin). Note
that this behavior of the curve near zero corresponds
to a state in which only very small portions of the
region in question are colonized by either grasses or
weeds, thus it is natural to see that the attraction of
the populations to an equilibrium involving dominance of grasses depends chiefly on the intrinsic
growth rate of the grasses relative to that of the
weeds.
Standard results on stable manifolds establish that
W 'm, and hence X, is smooth at the saddle point
K. Cooper, R. Huffaker/ Ecological Modelling 97 (1997) 59-73
(G sp, W sp). Thus, X may be expanded in a Taylor
series about (G 'p, W sp), in the form
x ( a ) = Xo + x , ( C -
c'p)
+ x2(o-
c'p) +...
(37)
In this form, the conditions (i) and (ii) immediately
give values for the first two coefficients:
x0=(a,p),,
x,= W
lV,
71
other in which rw/r c = 1.3. The parameters chosen
for the figure were B --- 0.3, E(x) = 0.312, qlw = 0.6,
and q~ = 1.07. It is possible to compute an approximarion to the stable manifold numerically, simply by
choosing an initial condition very near to the saddle
point, and then tracing the corresponding trajectory
backwards in rime using a numerical method. In the
figure, this numerical approximation is compared
with the approximation obtained using GPX(G). It is
(38)
Values may be found for coefficients of higher
order terms, but doing so entails evaluating higher
order derivatives of W sm at the equilibrium (G ~p,
W~P). This is done by differentiating Eq. (35) on
both sides with respect to G, substituting the form
for W sp, and solving for x 2. Higher order coefficients may be found by repeating the process with
higher derivatives. Such a procedure is much more
difficult than what has been done up to this point,
and is unnecessary, as the approximation using only
the first two terms of the series is adequate for most
purposes. Indeed, it is easily established that X has
two continuous derivatives for all G > 0, hence the
stable manifold is the graph of WSm(G) = GP[xo +
xl(G GsP) + x2(G)(G- GSp)2].
The error in the approximation obtained by truncating the series after the x~ term is given by
x2GP(G -Gsp) 2, where x 2 is in fact given as
(1/2)X"(3,) at some point ~/ between G and G ~p.
While the second derivative of X is difficult to
compute, as noted earlier, the implications of this
error term for the approximation are nonetheless
easily understood. First, it indicates that the approximation will be at its best near G ~p and near 0. One
expects the approximation to be at its worst as G
moves far to the right of G ~p. The approximation
will be better as the power p increases, so that when
the ratio rw/r o of intrinsic growth rates is small, the
error in the approximation will be larger. It turns out
that the second derivative of X is small, since most
of the curvature of W sm is due to the term G p. Thus
the error in the approximation is very small when
p > I, and is well within acceptable tolerances when
p<l.
The situation is illustrated in Fig. 4, which depicts
two cases: One in which rw/r ~ = 0.675, and an-
0.70
(a)
0.00
0.00
0.70
it
0.50
(b)
.49,0.62)
0.00
0.00
8
0.50
Fig. 4. (a) Numerical and approximate analytical stable manifolds
for p = 0.675. (b) Numerical and approximate analytical stable
manifolds for p = 1.3.
72
K. Cooper, R. Huffaker/ Ecological Modelling 97 (1997) 59-73
evident that the error is larger when p is smaller,
and that the approximation is quite good when p > 1.
This behavior is borne out in numerous other experiments.
5. Discussion
The possession of a simple approximation of successional thresholds facilitates investigating how the
resiliency of the rangeland ecosystem is affected by
changes in bioeconomic circumstances. Changes in
ecological parameters that make weeds more competitive in colonizing the grassland tend to shift the
successional thresholds down and to the right in
phase space. This increases the size of the 'all weeds'
basin of attraction, and decreases the proportion of
phase space occupied by the more socially desirable
basin of attraction leading to an equilibrium with
some positive proportion of grasses. For example,
when the parameter E(x) increases, then weeds become more competitive. E(x) may increase due
either to changes in 8 or 7, or to a decrease in the
annual vitality of grasses, x. Consequently, changes
in economic parameters Mat decrease the optimal
sustained value of x in the grazing-decision model
have an indirect effect on the relative size of socially
desirable basins of attraction. Increases in the livestock price P, or decreases in the annual costs of the
livestock fattening program C are examples of
changes in economic parameters that increase the
optimal sustained stocking density, and hence drive
down the sustained productivity of grasses, x.
As an example of the kind of analysis that the
approximation makes possible, Fig. 5 illustrates relative changes in the size of the desirable 'somegrasses' BOA due to changes in the livestock price.
The figure uses the baseline parameters discussed
above, changing only the value for x. The relative
changes in the BOA are calculated from the ratio
formed by: (1) The difference in areas under the
successional thresholds before and after the price
change within 0 < G, W < 1; and (2) the area under
the old threshold. These areas are calculated as
definite integrals of the approximated successional
threshold. The curve predicts, for example, that a 5%
increase in livestock prices received by producers
will shrink the 'some-grasses' BOA by approxi-
ABOA
\
-!2
-10
, AP*IO0
Fig. 5. Percentage changes in baseline basin-of-attraction to the
interior stable-node equilibrium due to percentage increases/decreases in the livestock price, P.
mately 6.8%. Moving backwards along the curve
from the 5% increase in price, shows the percentage
losses in the 'some-grasses' BOA that can be recouped by taxing away various percentages of the
price increase. Of course, when 100% of the price is
taxed away, 100% of the 'some-grasses' BOA is
recovered (i.e., at the origin).
The ability to make such analyses depends on
having a simple analytical approximation to the
boundary of the BOA. If the only approximation
were numerical, then evaluating the size of the BOA
would require a good estimate of the stable subspace
for the linearized equations, to provide initial conditions for a numerical solver, which would provide a
set of values that could be integrated numerically.
Thus, numerical estimation of the successional
threshold would require much more computation than
that using the approximation.
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