Ecological Archives E094-205-A2
Gretchen J. A. Hansen, Anthony R. Ives, M. Jake Vander Zanden, Stephen R. Carpenter. 2013. Are rapid transitions between invasive and native species caused by alternative stable states, and does it matter?. Ecology 94:2207–2219. http://dx.doi.org/10.1890/13-0093.1
Appendix B. Model results when Eq. 6 was used to describe rusty crayfish abundance, separating effects of water level on crayfish recruitment from other lake-specific effects.
Table B1. Maximum likelihood parameter estimates with mean and 95% confidence intervals from parametric bootstrapping when Eq. 6 is used to describe age-0 rusty crayfish dynamics, which includes the parameter b to estimate differences in crayfish recruitment among the two lakes that is not due to water level fluctuations.
Parameter |
Description |
MLL estimate |
Bootstrap mean |
Lower 95% CI |
Upper 95% CI |
a_{x} |
Lepomis effect on juvenile crayfish |
1.03 |
0.65 |
0 |
1.8 |
a_{y} |
Crayfish effect on juvenile Lepomis |
0.02 |
0.02 |
0.01 |
0.04 |
b |
Fixed difference between Sparkling and Trout Lakes |
1.88 |
0.97 |
-2.29 |
3.71 |
c |
Water level effect on crayfish recruitment (Sparkling) |
3.72 |
9.64 |
2.41 |
15.01 |
f_{x} |
Maximum crayfish recruitment rate |
19.44 |
11.41 |
2.31 |
143.35 |
k_{x} |
Rate of adult-density-dependent reduction in recruitment (crayfish) |
0.06 |
0.05 |
0.02 |
0.17 |
k_{y} |
Rate of adult-density-dependent reduction in recruitment (Lepomis) |
0.24 |
0.25 |
0.01 |
0.6 |
m |
Effect of trapping on adult crayfish |
1 |
0.89 |
0.09 |
1 |
q |
Nonlinearity in the effect of water level on crayfish |
0.37 |
0.72 |
0.13 |
2.86 |
v |
Effect of predatory fish on juvenile crayfish |
0 |
0.06 |
0 |
0.32 |
w_{x} |
Measurement error rescaling for crayfish CPUE |
0.87 |
0.82 |
0 |
1.87 |
w_{y} |
Measurement error rescaling for Lepomis CPUE |
3.82 |
3.76 |
2.09 |
5.95 |
σ_{x} |
Process error standard deviation in adult crayfish survival |
0.35 |
0.37 |
0.21 |
0.76 |
σ_{y} |
Process error standard deviation in adult Lepomis survival |
0 |
0.29 |
0 |
1.02 |
Fig. B1. Proportion of total abundance composed of rusty crayfish (rusty crayfish CPUE/ rusty crayfish CPUE + Lepomis CPUE) predicted from invasibility analysis as a function of the bifurcation parameter U (reduction in crayfish recruitment) when Eq. 6 was used to describe rusty crayfish abundance (parameter estimates from Table S1). Predicted abundances were generated from invasibility analysis (see main text) using the maximum likelihood parameter estimates, which produce alternative stable states for U values from -0.74 to 0.52 (gray dotted line). Rusty crayfish always exclude Lepomis at low U values, and Lepomis always exclude rusty crayfish at high U values (solid black lines).
Fig. B2. For the 1000 bootstrapped parameter sets generated using Eq. 6 and the parameters in Table B1, the range of values in the reduction of crayfish recruitment (U) from the point at which rusty crayfish exclude Lepomis to the point at which Lepomis exclude crayfish. Negative values (white) correspond to alternative stable states in which Lepomis can exclude crayfish and crayfish exclude Lepomis depending on initial conditions (61% of parameter sets; e.g., Fig. A1.1); positive values (black) correspond to coexistence parameter sets. Values with absolute values >10 were binned in the maximum (or minimum) bin for clarity of presentation.