Appendix: Analytical Proof of Graphical Results
Leibold (1996) used the quantitative analysis of feedback loops (Puccia and Levins
1985) to analyse a simple model of two consumers A and B that share a single
5
limiting resource R and a single predator P, described by equations 1a-c. He showed
that coexistence is feasible when the following trade-off holds true:
If
f A aA
f a
m c
m c
< B B then A A  B B
fA
fB
mA
mB
Since
(A.1)
fi ai
mc
is the slope of ZNGI in the graphical model shown in Figure 2a and i i
mi
fi
is the slope of the impact vector, these conditions suggest that consumer A, which is
10
more vulnerable to predation (i.e. shallower ZNGI), must have a greater impact on the
predator’s abundance P (i.e. steeper impact vector).
Consumer A is assumed to modify, via niche construction, the slope of species
impact vectors and ZNGIs in various ways as described in sections 2.2 and 3,
respectively, in the main text. Following Leibold’s method for deriving the stability
15
conditions (A.1), we prove here that the conditions for the stability of the new
coexistence equilibrium modified by niche construction correspond to the trade-off
between the slopes of species impact vectors and ZNGIs shown in figure 5.
Consider the case of niche construction where species A enhances the resource
turnover rate k. The following equations then describe the dynamics of the model:
20
dP
 P (mA c A N A  mB cB N B  d P )
dt
dN A
 N A ( f A a A R  mA P  d N A )  nR' N A R
dt
(A.2)
(A.3)
1
dN B
 N B ( f B aB R  mB P  d N B )
dt
(A.4)
dR
 (k nR N A )(S  R)  f A N A R  f B N B R
dt
(A.5)
The parameters nR and nR’ represent the rates at which species A improves the
resource R and receives a benefit from this niche construction activity, respectively.
5
The system of equations (A2-A5) has one internal (coexistence) equilibrium, where
the four compartments have the following abundances:
P 
*
( f AaA  nR' )d NB  f B aB d N A
(A.6)
mA f B aB  mB ( f AaA  nR' )
dP fB *
R  k ( S  R* )
m
c
B B
N A* 
mAc A *
fB
R  ( f A R*  nR ( S  R* ))
mB cB
(A.7)
dP
( f A R*  nR ( S  R* ))
mAc A
mc
f B R*  B B ( f A R*  nR ( S  R* ))
mA c A
k ( S  R* ) 
N B* 
10
R* 
(A.8)
mA d N B  mB d N A
(A.9)
mA f B aB  mB ( f A a A  nR' )
The necessary condition for the equilibrium abundances P* and R* to be positive, and
hence for the ZNGI’s of the two species to cross in the positive quadrant of the (R, P)
plane, is that the nominators and denominators in (A.6) and (A.9) are either all
positive or all negative. Assume first that they are all positive. In this case, the three
15
feasibility
(iii)
conditions
are:
f A a A  nR'
f a
 B B,
(i)
mA
mB
(ii)
d NB
f B aB

dNA
f A a A  nR'
,
mA
m
 B . These conditions determine the relative position of the ZNGIs (see
d N A d NB
fig. 5). If the x-intercept of ZNGIA is smaller than that of ZNGIB (condition (ii)), then
2
the slope of ZNGIA has to be shallower than that of ZNGIB (condition (i)). Notice also
that P* and R* increase with nR’, as shown in figure 5a. When the nominators and
denominators are all negative, however, then the direction of inequalities in conditions
(i)-(iii) is reversed, and so are the relative positions of species ZNGI’s.
Similarly, the equilibrium abundances N*A and N*B are positive when the
5
nominators and denominators in (A.7) and (A.8) are either all positive or all negative.
In the first case, we derive the following conditions: (iv)
and (v)
mA c A
m c
 B B*
*
f A R  nR ( S  R ) f B R
*
dP
d f
f A R*  nR ( S  R* )  R* < S < ( B B  1) R* . Condition (iv) means that
k mAc A
k mB cB
the modified impact vector of species A is steeper than that of species B, as shown in
10
figure 3. In the second case, the direction of inequalities in conditions (iv) and (v) is
reversed, and so are the relative slopes of species impact vectors.
The local stability of the coexistence equilibrium can be assessed using the
classical Routh-Hurwitz criteria, or, equivalently, by the signs of the feedbacks
defined at various levels in Puccia and Levins’s (1975) loop analysis. These feedbacks
15
are derived from the elements aij (i.e., the effect of compartment j on the population
growth rate of compartment i) of the Jacobian matrix of system (A2-A5) at
equilibrium:
 a PP

a N A P
a
 NB P
 a RP

20
a PN A
a PN B
a NANA
a N ANB
a NB N A
a NB NB
a RN A
a RN B
a PR   0
*
 
a N A R    mA N A
  m N *
a NB R   B B


a RR   0

mA c A P*
mB cB P*
0
0
0
0
nR ( S  R* )  f A R*
 f B R*


( f A a A  nR' ) N A* 

aB f B N B*

k  nR N A* 


R*
0
The feedbacks at the various levels are then computed as follows:
F1= a RR = (
k  nR N A*
)S
R*
(A.10)
3
F2= a RN A a N A R  a RN B a N B R  a N B P a PN B  a N A P a PN A =
( f A R*  nR ( S  R* )) ( f A a A  nR' ) N A*  aB f B2 N B* R*  mB2 cB N B* P*  mA2 c A N A* P*
(A.11)
F3= -a RR a N A P a PN A - a RR a N B P a PN B =
(
5
k  nR N A*
) (mA2 cA N A*  mB2 cB N B* ) S P*
*
R
(A.12)
F4= a N A R a PN A a N B P a RN B + a N B R a PN B a N A P a RN A  a RN A a N A R a N B P a PN B  a RN B a N B R a PN A a N A P =
[mA f B aB  mB ( f A a A  nR' )][mB cB ( f A R*  nR (S  R* ))  mAc A f B R* ]P* N A* N B*
(A.13)
The stability conditions are that all the above feedbacks are negative (this is
equivalent to the first Routh-Hurwitz criterion) and that:
F1 F2+ F3 = [( f A R*  nR ( S  R* )) ( f Aa A  nR' ) N *A  aB f B2 N B* R* ](
10
k  nR N A*
) S >0 (A.14)
R*
(this is equivalent to the second Routh-Hurwitz criterion). Given positive values for
all the model parameters and nR 
f A R*
, then the stability of the coexistence
S  R*
equilibrium hinges entirely on F4. If it holds that
condition (i)), then it is necessary that
f A a A  nR'
f a
 B B (feasibility
mA
mB
mA c A
m c
 B B*
*
f A R  nR ( S  R ) f B R
*
(feasibility
condition (iv)) for F4 to be negative and for local stability to be guaranteed. Thus, we
15
have proved that the conditions for local stability of the new coexistence equilibrium
modified by niche construction coincide with the trade-off between the slopes of
species impact vectors and ZNGIs (see fig. 5). Note, however, that when nR 
f A R*
,
S  R*
it is possible that the values of F2 and F1 F2+ F3 become positive and negative,
respectively, which leads to an unstable coexistence equilibrium. In other words, a
20
strong enough niche improving impact (nR) may destabilise the system.
4
In the same fashion, it is possible to prove the local stability of the equilibrium
in all the other instances of niche construction (2b-2d) studied in the main text. The
only difference is that these instances generate two internal equilibria; however, we
find that one of them is always unfeasible (i.e. non-positive equilibrium abundance of
5
at least one of the model compartments).
REFERENCES
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Leibold, M.A. (1996). A graphical model of keystone predators in food webs:
trophic regulation of abundance, incidence, and diversity patterns in
communities. Am. Nat., 147, 784–812.
Puccia C.J. & Levins R. (1985). Qualitative modeling of complex systems: an
introduction to lool analysis and time averaging. Harvard University
Press, Cambridge, Mass.
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