ON-LINE APPENDICES

APPENDIX 1: Analytic Results for Fixed Allocation of Foraging Effort, σ1 and σ2 .

(I) Analysis of equations 4a-4c, in absence of toxins

Here we analyze the model for herbivory in the absence of toxins, assuming that the herbivore has a strategy of fixed feeding effort (σ1and σ2), rather than feeding adaptively, as in the simulations described in the main text.As is for the case of no herbivore, it is assumed that c121 c21 (under which species 1 will always exclude species 2 if K1=K2). Here, the analysis is conducted for a more general case that allows for different Ki(i=1,2).

To explore the conditions under which species 2 can invade in an environment where species 1 is already established, we consider the non-trivial equilibrium (the subscript h for Holling) where

.

Clearly, when . Thus, is biologically feasible if . This requires that 1 be bounded below by some positive number that we denote by . The Jacobian matrix at has two eigenvalues (either real or complex) with a negative real part. The third eigenvalue is negative if and only if

. (A1)

Therefore, the equilibrium is locally asymptotically stable if equation (A1) holds. If the inequality equation (A1) is reversed then becomes unstable, in which case the invasion of species 2 is expected.

Notice that is a function of 1 and that 2=11. Thus, the right-hand-side of equation A1 defines a function of 1, which we denote by. The curve corresponding to is shown in Figure A1, which intersects the 1 axis at. The threshold condition equation (A1) implies that in the region above the curve invasion of species 2 is impossible if it starts at a low density.

Using a symmetric argument (switching between N1 and N2) we can identify another function H2(1), which determines the stability for the symmetric equilibrium at which species 1 is absent. It requires that 1be bounded above by some positive number less than 1 which we denote as. The curve of H2(1)is shown in Figure A1. Thus, species 2 can exclude species 1 for (1, r1/ r2) below the curve. In the region between the curves of H1(1)and H2(1), and above H2(σ1),coexistence of the two plant species is expected.

(I) Analysis of equations 4a-4c, with toxins included

Now we show the results of analysis for the case in which there is herbivory and plant toxicity, where, as above, the herbivore has a strategy of fixed σ1 and σ2. (Again, these results are intended to illustrate the alternative assumption to that in the main text, where the herbivores foraged adaptively.) Here, we present the analysis on invasion criteria for equations 4a-4c. Consider the non-trivial equilibrium (the subscript t for toxin) where

. (A2)

It can be verified that if , and that is biologically feasible if .

The Jacobian matrix at has two eigenvalues (either real or complex) with a negative real part. The third eigenvalue is negative if and only if

(A3) It follows that the equilibriumis locally asymptotically stable if equation (A3) holds. If the inequality equation (A3) is reversed then becomes unstable, in which case the invasion of species 2 is expected.

Notice that is a function of G1 (and that it does not depend on G2). Thus, the right-hand-side of equation (A3) defines a function of G1, which we denote by. The curve corresponding to is shown in Figure A2. In the region above the curve, species 2 will be excluded if starts at a low density, and in the region below the curve it is possible for species 2 to invade.

APPENDIX 2: TananaRiverFloodplain Hare-Moose Exclusion Experiment

In 1987 paired herbivore-exclosure and control plots (20m×30m plots separated by 20m buffer strip; replicated 7 times) were randomly located in the mid-colonization stage of succession (Chapin et al. 2006). The river terrace on which the plots were established was about 12yrs old (year 12 in Fig. 8). The plots were subdivided into 5 strata (4m wide) oriented parallel to the river, and a permanent 2m2 rectangular quadrat was randomly located within each strata so subsequent vegetation surveys were done at the same spot. The quadrats were used to obtain the ratios of willow/alder used to test the TDFRM and the Holling Type 2 Functional response predictions. Because there was very little litter in the first few years after sedimentation began, twig counts were used to obtain the ratios in “model” years 14, 17 and 20. Because of the time required to census twigs, leaf litter was used in years 20, 26 and 32. In year 20 the estimates of the alder/willow ratios obtained by the twig census method and the leaf litter method were similar, indicating that the change in methods did not affect our conclusions.

The simulation results shown inFigure 8reflect parameter values consistent with field measurements. We first estimated the values for plant growth r1 and r2, the carrying capacity K1 and K2 and the competition coefficients cijby fitting the model to the plant ratio data without herbivore browsing (in which case the two models are identical). We then compared the two models with herbivore browsing (the difference between the two models is that one model does not include plant-toxin effect on herbivory, while the other does) by looking at the plant ratios in comparison with the plant ratio data with browsing. The parameter values used in the simulations are r1= 0.0016, r2=0.0017, e1=0.0001, e2 =0.0001, K1=50000, 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, G2=4.5.

APPENDIX 3: Use of Functional Response for Additive Toxic Effects (Equation 6)

Two scenarios considered in the main text are reconsidered with equation 6 substituted for equation 5. The first scenario is that shown in Figure 2, the results of which are shown again in Figure A3a-b. When the simulation is repeated using equation (6), the results are shown in Figure A3c-d. The second scenario is that shown in Figure 3, the results of which are shown again in Figure A4a-b. When the simulation is performed again using equation (6), the new results are shown in Figure A4c-d.

FIGURE CAPTIONS

Figure A1. A bifurcation diagram for equations 6a-6c as discussed in Appendix 2.

Figure A2. A bifurcation diagram for equations 4a-4c as discussed in Appendix 2.

Figure A3.Simulation results of the TDFRM for an adaptively foraging herbivore when the resident species (1) is more toxic than the invading species(2): G1=35, G2=60. Initial plant densities are N1,0=5×105 and N2,0=5×103. Parameter values used for this figure are the same as in Figure 2. (a-b) Results for the non-interacting toxins (equation 5). (c-d) Results for the additive toxins (equation 6).

Figure A4. Simulation results for the TDFRM for an adaptively foraging herbivore when the competition coefficients are equal c12 = c21 and the resident species 1 is less toxic (G1 = 50) than in the scenario in Figure 2, and other parameter values being the same except where stated. G1 = 50, G2 =35, c12=0.9, c21=0.9. (a-b) Results for the non-interacting toxins (equation 5). (c-d) Results for the additive toxins (equation 6).


Figure A1.


Figure A2.

(a) (b)

(c) (d)

Figure A3

(a) (b)

(c) (d)

Figure A4

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