Ecological Archives E085-043-A4

Andrew R. Jacobson, Antonello Provenzale, Achaz von Hardenberg, Bruno Bassano, and Marco Festa-Bianchet. 2004. Climate forcing and density dependence in a mountain ungulate population. Ecology 85:1598–1610.

Appendix D. Model equations.

We provide here expressions for all the models listed in Table 3.

$\displaystyle y_i = a + b n_i + c v_i + e v_i n_i + \sigma\epsilon_i$ (D.1) (same as Eq. 10)

$\displaystyle y_i = a + b x_i + c v_i + e v_i x_i + \sigma \epsilon_i$

(D.2)
(same as Eq. 11)

$\displaystyle y_i = a + c v_i + e v_i n_i + \sigma\epsilon_i$ (D.3)

$\displaystyle y_i = a + c v_i + e v_i x_i + \sigma \epsilon_i$ (D.4)

$\displaystyle y_i = a + b n_i + e v_i n_i + \sigma\epsilon_i$ (D.5)

$\displaystyle y_i = a + b x_i + e v_i x_i + \sigma \epsilon_i$ (D.6)

$\displaystyle y_i = a + b n_i + c v_i + \sigma\epsilon_i$ (D.7)

$\displaystyle y_i = a + b x_i + c v_i + \sigma \epsilon_i$ (D.8)

$\displaystyle y_i = a + e v_i n_i + \sigma\epsilon_i$ (D.9)

$\displaystyle y_i = a + e v_i x_i + \sigma \epsilon_i$ (D.10)

In the following threshold models, the $ l$ and $ h$ indices represent parameters for low and high snow depth years respectively, and $ v_{\text{thresh}}$ is a threshold value of the environmental parameter $ v$.

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_ln_i+c_lv_i+e_ln_iv_i... ... \\ b_hn_i+c_hv_i+e_hn_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.11)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_lx_i+c_lv_i+e_lx_iv_i... ... \\ b_hx_i+c_hv_i+e_hx_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.12)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}c_lv_i+e_ln_iv_i & \tex... ...sh}}$,} \\ c_hv_i+e_hn_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.13)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}c_lv_i+e_lx_iv_i & \tex... ...sh}}$,} \\ c_hv_i+e_hx_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.14)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_ln_i+e_ln_iv_i & \tex... ...sh}}$,} \\ b_hn_i+e_hn_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.15)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_lx_i+e_lx_iv_i & \tex... ...sh}}$,} \\ b_hx_i+e_hx_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.16)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_ln_i+c_lv_i & \text{i... ...hresh}}$,} \\ b_hn_i+c_hv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.17)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}b_lx_i+c_lv_i & \text{i... ...hresh}}$,} \\ b_hx_i+c_hv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.18)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}e_ln_iv_i & \text{if $v... ...xt{thresh}}$,} \\ e_hn_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.19)

$\displaystyle y_i = a + \sigma\epsilon_i + \begin{cases}e_lx_iv_i & \text{if $v... ...xt{thresh}}$,} \\ e_hx_iv_i & \text{if $v_i \ge v_{\text{thresh}}$} \end{cases}$ (D.20)


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