Time taken for breakaway rider to complete a course compared to the peloton, for a course profile given by \(h=h_0\text exp(-\beta (x-x_\text max)^2)\) (shown schematically in inset), with \(h_0=0.02,0.025,0.03,0.035,0.04\), \(\beta =0.5\) and \(x_\text max=6\). Here, \(m_\text b=1\), \(F_0=0.028\) and \(F_\textrmb0=0.011\). We choose \(v_i=\sqrtF_0/\gamma =0.2\) to give \(\ddotx(0)=0\), so that the peloton is travelling at a constant speed
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Effect of valley position on breakaway success. Motion along the last 20 length units (\(\approx 10\)km) of a valley course profile given by \(h=h_0\exp \bigl (-\beta (x-x_\text min)^2\bigr )\) with \(\text h_0=-\,0.04\), \(\beta =0.5\), \(m_\text b=1\), \(\text F_\text s=0.032\), \(F_\textb0=0.8F_\text s\), \(p_1=0.1\), \(a=0.5\) and \(b=0.5\). The solution is given by (16), (17), and (32)
Effect of peak position on breakaway success. Motion along the last 20 length units (\(\approx 10\) km) of a hill course profile given by \(h=h_0\exp \bigl (-\beta (x-x_\text max)^2\bigr )\) with \(h_0=0.04\), \(\beta =0.5\), \(x_\text max=6,10,14,18\), and \(m_\text b=1\), \(F_\text s=0.032\), \(F_\textrmb0=0.8F_\text s\), \(p_1=0.1\), \(a=0.5\) and \(b=0.5\). We choose \(v_i=0.2\) so that initially the peloton is moving at constant velocity on a flat course (given by (16) with \(\ddotx=0\)). The solution is given by (16), (17), and (32)
Effect of hill gradient on breakaway success. Motion along the last 20 length units (\(\approx 10\) km) of a hill course profile given by \(h=h_0\exp \bigl (-\beta (x-x_\text max)^2\bigr )\) with \(h_0=0.04\), \(\beta =0.1,0.3,0.5,0.7\), \(x_\text max=12\), and \(m_\text b=1\), \(F_\text s=0.032\), \(\text F_\textrmb0=0.8\text F_\text s\), \(p_1=0.1\), \(a=0.5\) and \(b=0.5\). We choose \(v_i=0.2\) so that initially the peloton is moving at constant velocity on a flat course (given by (16) with \(\ddotx=0\)). The solution is given by (16), (17), and (32)
Effect of hill height on breakaway success. Motion along the last 20 length units (\(\approx 10\) km) of a hill course profile given by \(h=h_0\exp \big (-\beta (x-x_\text max)^2\bigr )\) with \(h_0=0.02,0.03,0.04,0.05\), \(\beta =0.5\), \(x_\text max=12\), and \(m_\text b=1\), \(F_\text s=0.032\), \(F_\textrmb0=0.8F_\text s\), \(p_1=0.1\), \(a=0.5\) and \(b=0.5\). We choose \(v_i=0.2\) so that initially the peloton is moving at constant velocity on a flat course (given by (16) with \(\ddotx=0\)). The solution is given by (16), (17), and (32)
To estimate alpha diversity and beta diversity we used the breakaway v4.6 and divnet v0.34 packages, respectively [47, 48]. These approaches avoid common pitfalls from applying classical ecology indexes (i.e., Chao1, Shannon) to microbiome data, such as the influence of sequencing depth and data compositionality. 2ff7e9595c
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