1 ON-LINE APPENDICES
2
3 APPENDIX 1: Analytic Results for Fixed Allocation of Foraging Effort, σ1 and σ2 .
4 (I) Analysis of equations 4a-4c, in absence of toxins
5 Here we analyze the model for herbivory in the absence of toxins, assuming that the herbivore has a
6 strategy of fixed feeding effort (σ1 and σ2), rather than feeding adaptively, as in the simulations described
7 in the main text. As is for the case of no herbivore, it is assumed that c12<1< c21 (under which species 1
8 will always exclude species 2 if K1=K2). Here, the analysis is conducted for a more general case that
9 allows for different Ki (i=1,2).
10
To explore the conditions under which species 2 can invade in an environment where species 1 is
11 already established, we consider the non-trivial equilibrium Eh*  ( N1*h ,0, Ph* ) (the subscript h for Holling)
12 where
13
N1*h 
mp
e11 ( B1  h1m p )
, P* 
r1 (1  N1*h / K1 )
e11 /(1  h1e11N1*h )
.
14 Clearly, N1*h  0 when m p  B1 / h1 . Thus, Eh* is biologically feasible if N1*h  K1 . This requires that 1 be
15 bounded below by some positive number that we denote by 1a . The Jacobian matrix at Eh* has two
16 eigenvalues (either real or complex) with a negative real part. The third eigenvalue is negative if and only
17 if
18
r1 e11 (1  c21N1*h / K 2 ) .

r2
e2 2 (1  N1*h / K1 )
(A1)
19 Therefore, the equilibrium Eh* is locally asymptotically stable if equation (A1) holds. If the inequality
20 equation (A1) is reversed then Eh* becomes unstable, in which case the invasion of species 2 is expected.
21
Notice that N1*h  N1*h (1 ) is a function of 1 and that 2=11. Thus, the right-hand-side of
22 equation A1 defines a function of 1, which we denote by H1 (1 ) . The curve corresponding to H1 (1 ) is
1
23 shown in Figure A1, which intersects the 1 axis at 1* . The threshold condition equation (A1) implies that
24 in the region above the curve H1 (1 ) invasion of species 2 is impossible if it starts at a low density.
25
Using a symmetric argument (switching between N1 and N2) we can identify another function
26 H2(1), which determines the stability for the symmetric equilibrium at which species 1 is absent. It
27 requires that 1 be bounded above by some positive number less than 1 which we denote as 1b . The
28 curve of H2(1) is shown in Figure A1. Thus, species 2 can exclude species 1 for (1, r1/ r2) below the
29 curve. In the region between the curves of H1(1) and H2(1), and above H2(σ1), coexistence of the two
30 plant species is expected.
31
32 (I) Analysis of equations 4a-4c, with toxins included
33 Now we show the results of analysis for the case in which there is herbivory and plant toxicity, where, as
34 above, the herbivore has a strategy of fixed σ1 and σ2. (Again, these results are intended to illustrate the
35 alternative assumption to that in the main text, where the herbivores foraged adaptively.) Here, we present
36 the analysis on invasion criteria for equations 4a-4c. Consider the non-trivial equilibrium Et*  ( N1*t ,0, Pt* )
37 (the subscript t for toxin) where
38
N1*t 

2 G1  G1 (G1  m p / B1 )



e11 1  2h1 G1  G1 (G1  m p / B1 ) 


, Pt* 
r1 (1  N1*t / K1 )

e11 /(1  h1e11N1*t ) 1  f1 ( N1*t , 0) / 4G1

.
(A2)
39 It can be verified that N1*t  0 if m p  B1C1 ( K1, 0) , and that Et* is biologically feasible if N1*t  K1 .
40 The Jacobian matrix at Et* has two eigenvalues (either real or complex) with a negative real part. The third
41 eigenvalue is negative if and only if
42

e11 N1*t
r1 e1 1 (1  c21 N1*t / K 2 ) 

1


*
* 

r2
e2 2 (1  N1t / K1 )  4G1 (1  h1e11 N1t ) 
(A3)
43 It follows that the equilibrium Et* is locally asymptotically stable if equation (A3) holds. If the inequality
44 equation (A3) is reversed then Et* becomes unstable, in which case the invasion of species 2 is expected.
2
45
Notice that N1*t  N1*t (G1 ) is a function of G1 (and that it does not depend on G2). Thus, the right-
46 hand-side of equation (A3) defines a function of G1, which we denote by L(G1 ) . The curve corresponding
47 to L(G1 ) is shown in Figure A2. In the region above the curve, species 2 will be excluded if starts at a low
48 density, and in the region below the curve it is possible for species 2 to invade.
49
50 APPENDIX 2: Tanana River Floodplain Hare-Moose Exclusion Experiment
51 In 1987 paired herbivore-exclosure and control plots (20m×30m plots separated by 20m buffer strip;
52 replicated 7 times) were randomly located in the mid-colonization stage of succession (Chapin et al. 2006).
53 The river terrace on which the plots were established was about 12yrs old (year 12 in Fig. 8). The plots
54 were subdivided into 5 strata (4m wide) oriented parallel to the river, and a permanent 2m2 rectangular
55 quadrat was randomly located within each strata so subsequent vegetation surveys were done at the same
56 spot. The quadrats were used to obtain the ratios of willow/alder used to test the TDFRM and the Holling
57 Type 2 Functional response predictions. Because there was very little litter in the first few years after
58 sedimentation began, twig counts were used to obtain the ratios in “model” years 14, 17 and 20. Because
59 of the time required to census twigs, leaf litter was used in years 20, 26 and 32. In year 20 the estimates of
60 the alder/willow ratios obtained by the twig census method and the leaf litter method were similar,
61 indicating that the change in methods did not affect our conclusions.
62
The simulation results shown in Figure 8 reflect parameter values consistent with field measurements.
63 We first estimated the values for plant growth r1 and r2, the carrying capacity K1 and K2 and the
64 competition coefficients cij by fitting the model to the plant ratio data without herbivore browsing (in
65 which case the two models are identical). We then compared the two models with herbivore browsing (the
66 difference between the two models is that one model does not include plant-toxin effect on herbivory,
67 while the other does) by looking at the plant ratios in comparison with the plant ratio data with browsing.
68 The parameter values used in the simulations are r1= 0.0016, r2=0.0017, e1=0.0001, e2 =0.0001, K1=50000,
3
69 K2=170000, h1=0.01, h2=0.06, B1=0.00034, B2=0.00031, c12= 0.17; c21= 0.1, mp= 1/(3.5*365) G1 = 100,
70 G2=4.5.
71
72 APPENDIX 3: Use of Functional Response for Additive Toxic Effects (Equation 6)
73 Two scenarios considered in the main text are reconsidered with equation 6 substituted for equation 5.
74 The first scenario is that shown in Figure 2, the results of which are shown again in Figure A3a-b. When
75 the simulation is repeated using equation (6), the results are shown in Figure A3c-d. The second scenario
76 is that shown in Figure 3, the results of which are shown again in Figure A4a-b. When the simulation is
77 performed again using equation (6), the new results are shown in Figure A4c-d.
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4
94 FIGURE CAPTIONS
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96 Figure A1. A bifurcation diagram for equations 6a-6c as discussed in Appendix 2.
97
98 Figure A2. A bifurcation diagram for equations 4a-4c as discussed in Appendix 2.
99
100 Figure A3. Simulation results of the TDFRM for an adaptively foraging herbivore when the resident
101 species (1) is more toxic than the invading species (2): G1=35, G2=60. Initial plant densities are
102 N1,0=5×105 and N2,0=5×103. Parameter values used for this figure are the same as in Figure 2. (a-b)
103 Results for the non-interacting toxins (equation 5). (c-d) Results for the additive toxins (equation 6).
104
105 Figure A4. Simulation results for the TDFRM for an adaptively foraging herbivore when the
106 competition coefficients are equal c12 = c21 and the resident species 1 is less toxic (G1 = 50) than in the
107 scenario in Figure 2, and other parameter values being the same except where stated. G1 = 50, G2 =35,
108 c12=0.9, c21=0.9. (a-b) Results for the non-interacting toxins (equation 5). (c-d) Results for the additive
109 toxins (equation 6).
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Figure A3
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Figure A4
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