1 Comparison of richness and diversity

As we can see in figure 1.1, the higher the area the higher the number of species. In this regard the sampling done with a different method (0.25) lies reasonably withing the expected parameters

Species richness for each treatment grouped by the sampled area

Figure 1.1: Species richness for each treatment grouped by the sampled area

A closer look at the curve that we see for each block separated by initial habitat shows us again that the number of species at 0.25 lies closely to what we would expect for a natural progression (figure 1.2).

Figure 1.2: Curve of acumulation of species with area by block number and treatment

1.1 Model

For all plot sizes we fitted the following general equation as a generalized mixed effect linear model:

glmer(richness ~  initial_habitat + treatment + (1 | block_no), family = poisson)

For which we get the results shown in tables 1.1 and 1.2, there we can see that generally the \(R^2\) values are higher the larger the plot size is. Also in 1.2 we can see that the initial habitat is significant at all distances, after that, when we compare to permanent exclosure, summer exclosure is significant in 4 out of the 5 distances, winter and all year grazing are significant in 3 out of the 5 distances, and mowing is never shown to be significantly different from permanent exclosure. Finally the equations of all the final models are shown in the Equations section bellow

Table 1.1: General parameters of the models fitted for each plot size
loglik aic df.residual r2.conditional r2.marginal rmse plot_size
-52.369 114.738 21 NA 0.315 1.761 0.01
-106.398 226.796 37 0.446 0.378 2.473 0.10
-107.996 229.991 37 0.479 0.396 2.384 0.25
-115.746 245.492 37 0.501 0.389 2.843 1.00
-127.691 269.382 37 0.613 0.468 3.631 10.00
Table 1.2: Estimates of the slope an intercept for the different models, we only show the significant parameters
term estimate std.error statistic p.value plot_size
(Intercept) 1.159 0.237 4.882 0.000 0.01
initial_habitatRangeland 0.594 0.212 2.808 0.005 0.01
(Intercept) 1.649 0.166 9.907 0.000 0.10
initial_habitatRangeland 0.471 0.144 3.281 0.001 0.10
treatmentSummerExclosure 0.405 0.159 2.553 0.011 0.10
treatmentWinterExclosure 0.343 0.161 2.132 0.033 0.10
(Intercept) 1.676 0.168 9.999 0.000 0.25
initial_habitatRangeland 0.510 0.149 3.428 0.001 0.25
treatmentControl 0.326 0.160 2.041 0.041 0.25
treatmentSummerExclosure 0.357 0.156 2.291 0.022 0.25
(Intercept) 2.154 0.140 15.334 0.000 1.00
initial_habitatRangeland 0.402 0.129 3.123 0.002 1.00
treatmentControl 0.302 0.132 2.289 0.022 1.00
treatmentSummerExclosure 0.304 0.129 2.357 0.018 1.00
treatmentWinterExclosure 0.318 0.129 2.473 0.013 1.00
(Intercept) 2.634 0.120 21.877 0.000 10.00
initial_habitatRangeland 0.385 0.117 3.304 0.001 10.00
treatmentControl 0.333 0.104 3.209 0.001 10.00
treatmentSummerExclosure 0.273 0.103 2.648 0.008 10.00
treatmentWinterExclosure 0.348 0.101 3.432 0.001 10.00

2 Equations

2.1 Equation for plot size 0.01

\[ \begin{aligned} \operatorname{\widehat{richness}}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=1.16_{\alpha_{j[i]}} + 0.16_{\beta_{1}}(\operatorname{treatment}_{\operatorname{SummerExclosure}}) + 0.08_{\beta_{2}}(\operatorname{treatment}_{\operatorname{WinterExclosure}}) \\ \alpha_{j} &\sim N \left(0.59_{\gamma_{1}^{\alpha}}(\operatorname{initial\_habitat}_{\operatorname{Rangeland}}), 0 \right) \text{, for block\_no j = 1,} \dots \text{,J} \end{aligned} \]

2.2 Equation for plot size 0.1

\[ \begin{aligned} \operatorname{\widehat{richness}}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=1.65_{\alpha_{j[i]}} + 0.22_{\beta_{1}}(\operatorname{treatment}_{\operatorname{Control}}) + 0.11_{\beta_{2}}(\operatorname{treatment}_{\operatorname{Mowing}}) + 0.41_{\beta_{3}}(\operatorname{treatment}_{\operatorname{SummerExclosure}}) + 0.34_{\beta_{4}}(\operatorname{treatment}_{\operatorname{WinterExclosure}}) \\ \alpha_{j} &\sim N \left(0.47_{\gamma_{1}^{\alpha}}(\operatorname{initial\_habitat}_{\operatorname{Rangeland}}), 0.11 \right) \text{, for block\_no j = 1,} \dots \text{,J} \end{aligned} \]

2.3 Equation for plot size 0.25

\[ \begin{aligned} \operatorname{\widehat{richness}}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=1.68_{\alpha_{j[i]}} + 0.33_{\beta_{1}}(\operatorname{treatment}_{\operatorname{Control}}) + 0.12_{\beta_{2}}(\operatorname{treatment}_{\operatorname{Mowing}}) + 0.36_{\beta_{3}}(\operatorname{treatment}_{\operatorname{SummerExclosure}}) + 0.22_{\beta_{4}}(\operatorname{treatment}_{\operatorname{WinterExclosure}}) \\ \alpha_{j} &\sim N \left(0.51_{\gamma_{1}^{\alpha}}(\operatorname{initial\_habitat}_{\operatorname{Rangeland}}), 0.13 \right) \text{, for block\_no j = 1,} \dots \text{,J} \end{aligned} \]

2.4 Equation for plot size 1

\[ \begin{aligned} \operatorname{\widehat{richness}}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=2.15_{\alpha_{j[i]}} + 0.3_{\beta_{1}}(\operatorname{treatment}_{\operatorname{Control}}) + 0.13_{\beta_{2}}(\operatorname{treatment}_{\operatorname{Mowing}}) + 0.3_{\beta_{3}}(\operatorname{treatment}_{\operatorname{SummerExclosure}}) + 0.32_{\beta_{4}}(\operatorname{treatment}_{\operatorname{WinterExclosure}}) \\ \alpha_{j} &\sim N \left(0.4_{\gamma_{1}^{\alpha}}(\operatorname{initial\_habitat}_{\operatorname{Rangeland}}), 0.12 \right) \text{, for block\_no j = 1,} \dots \text{,J} \end{aligned} \]

2.5 Equation for plot size 10

\[ \begin{aligned} \operatorname{\widehat{richness}}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=2.63_{\alpha_{j[i]}} + 0.33_{\beta_{1}}(\operatorname{treatment}_{\operatorname{Control}}) + 0.13_{\beta_{2}}(\operatorname{treatment}_{\operatorname{Mowing}}) + 0.27_{\beta_{3}}(\operatorname{treatment}_{\operatorname{SummerExclosure}}) + 0.35_{\beta_{4}}(\operatorname{treatment}_{\operatorname{WinterExclosure}}) \\ \alpha_{j} &\sim N \left(0.39_{\gamma_{1}^{\alpha}}(\operatorname{initial\_habitat}_{\operatorname{Rangeland}}), 0.13 \right) \text{, for block\_no j = 1,} \dots \text{,J} \end{aligned} \]