Models of lake invasibility by Bythotrephes longimanus, a non-indigenous zooplankton

Abstract

We built a family of hierarchical risk models for the spread of invasions by the spiny waterflea (Bythotrephes longimanus) in lakes in Ontario, Canada. Knowledge of covariates determining lake invasibility and ability to predict risk of future invasions may help to develop management policy and slow the invasions in the future. The models are based on two component submodels. The first component was a stochastic gravity submodel for the propagule pressure between lakes via recreational boaters. The second component was a submodel for establishment risk, given that the invader has already been introduced to a lake. This component was a logistic regression model, incorporating up to 17 measured covariates that describe the physical and chemical condition of the lake. Variants of the risk model, each incorporating different subsets of the covariates, were calibrated using presence/absence data from a 300-lake survey conducted in 2005–2006 by the Canadian Aquatic Invasive Species Network (CAISN). The predictive capacity of the best model was high, giving AUC values close to 0.94. Of the model covariates considered, the most important predictors of existing invasions were propagule pressure and lake pH, and, to lesser extents, phosphorus (P) and lake elevation. Our fitting of the propagule pressure submodel demonstrated a significant Allee effect for Bythotrephes. Our development of the establishment risk predictor showed that it is essential to account for temporal variability in lake physico-chemistry. We demonstrated that invasions of lake networks by the spiny waterflea follow highly predictable patterns which can be understood with a properly calibrated, hierarchical risk model.

Appendices

Appendix A

For the function Q in Eq. 7 we have the following expression:

$$ Q\left( {\mu ,m} \right) = \sum\limits_{j = m}^{\infty } {P\left( {j|\mu } \right)} = \sum\limits_{j = m}^{\infty } {\frac{{\mu^{j} }}{j!}\exp \left( { - \mu } \right)} = \frac{{\mu^{m} }}{m!}\exp \left( { - \mu } \right)\sum\limits_{j = 0}^{\infty } {\frac{{m!\mu^{j} }}{{\left( {j + m} \right)!}}} $$

For small μ $Q\left( {\mu ,m} \right) \approx \mu^{m} /m!$, for μ big Q(μ, m) approaches 1. It is convenient to replace the sum by an empirical approximation with qualitatively similar behavior: $Q\left( {\mu ,m} \right) \approx Q_{1} \left( {\mu ,m} \right) = 1 - \exp \left( { - \frac{{\mu^{m} }}{m!}} \right).$

Or, in terms of the relative boater flow λ,

$$ Q\left( {\lambda ,m} \right) \approx 1 - \exp \left( { - \frac{{\left( {C_{\mu } \lambda } \right)^{m} }}{m!}} \right) = 1 - \exp \left( { - \kappa \lambda^{m} } \right),\quad \kappa = \frac{{C_{\mu }^{m} }}{m!}, $$

where κ should be fitted from data.

We have tried another possible approximation to Q, $Q_{2} \left( {\mu ,m} \right) = \left[ {1 - \exp \left( { - \mu /\left( {m!} \right)^{1/m} } \right)} \right]^{m} .$ It have shown slightly worse model performance, though formally it provides closer approximation to Q(μ, m). Both approximations are compared in Fig. 9.

Fig. 9

Comparison of exact Q(λ, m) and its approximations for m = 2 and 3: black solid line—exact Q(λ, m); dashed—$Q_{1} \left( {\lambda ,m} \right) = 1 - \exp \left( { - \lambda^{m} /m!} \right)$; dotted line—$Q_{1} \left( {\lambda ,m} \right) = 1 - \exp \left( { - \alpha \lambda^{m} /m!} \right)$ where α is obtained by fitting Q ₁ to Q, (actual coefficient will be fitted anyway); gray line $Q_{2} \left( {\lambda ,m} \right) = \left[ {1 - \exp \left( { - \lambda /\left( {m!} \right)^{1/m} } \right)} \right]^{m}$

Appendix B

With the help of Eq. 11, formula (10) can be written as

$$ P\left( {X = 1|Y = m} \right) = \int\limits_{ - \infty }^{\infty } {S\left( {v\left( {\xi_{0} } \right)} \right)\exp \left( { - \frac{{\xi_{0}^{2} }}{{2\sigma_{0}^{2} }}} \right)d\xi_{0} } , $$

(B1)

where

$$ \nu \left( {\xi_{0} } \right) = a_{0} + \sum\limits_{{k \in {\mathbf{K}}}} {a_{k} x_{k} } + \xi_{0} \sigma_{0} ,\quad \sigma_{0} = \sqrt {\sum\limits_{{k \in {\mathbf{K}}}} {a_{k}^{2} \sigma_{k}^{2} } } . $$

(B2)

Introduce a change of variables,

$$ \eta = \int\limits_{ - \infty }^{{\xi_{0} }} {\exp \left( { - \frac{{u^{2} }}{{2\sigma_{0}^{2} }}} \right)du,\quad d\eta = } \exp \left( { - \frac{{\xi_{0}^{2} }}{{2\sigma_{0}^{2} }}} \right)d\xi_{0} , $$

(B3)

then we have one-to-one relation between ξ ₀ and η, there exists the inverse change ξ ₀(η), and hence we can write

$$ P\left( {X = 1|Y = m} \right) = \int\limits_{0}^{1} {S\left( {v\left( {\xi_{0} \left( \eta \right)} \right)} \right)d\eta } . $$

(B4)

We approximate this integral by a finite sum over n ₀ points using midpoint rule. The interval [0, 1] we split into n ₀ segments of the length 1/n ₀, with the middle in the points $\eta_{j} = (j - 0.5)/n_{0}$, $j = 1, \ldots ,n_{0}$. The corresponding values of ξ _0j can be obtained from (B3), and they coincide with the probabilities that ${\text{Prob}}(\xi_{0} < \xi_{0j} ) = \eta_{j} = (j - 0.5)/n_{0}$. Substituting these values of ξ _0j into (B4), we obtain Eq. 12.