Fig. 4

Effect of asymmetric inheritance on the average frequency of a taxon in hosts. Cases without inheritance and inheritance are compared. Inheritance is symmetric if offspring receive \(9 \%\) of their parent’s microbiome on average (\(a_i = 0\) and \(b_i = 9\)). Inheritance is asymmetric if offspring receive \(9 \%\) of taxon 1 and \(1 \%\) of other taxa (\(a_i = 0\) and \(b_1 = 9\), \(b_{i\ne 1} = 99\) in Eq. (4)). Available space within hosts is occupied more easily for \(\alpha _0 \rightarrow 0\). Single parameters are modified from the condition \(p_1 = 10^{-2}\) for the frequency of taxa 1 in the pool of colonizers, \(m = 10^{-2}\) for immigration, \(\tau = 10^{-4}\) for host death, and \(N = 10^5\) for carrying capacity. A, B The average frequency increases for larger abundances in the pool of colonizers (\(p_1\)), immigration (m), and \(\alpha _0 \rightarrow 0\). An asymmetric inheritance has no effect, as hosts are not fully occupied within their lifetime (Additional file 1: Figs. S2 and  S3). C Longer host lifespans, \(\tau \rightarrow 0\), increase the average frequency and effect of asymmetric inheritance. The gain is maximal at intermediate \(\tau\). Inheritance has more influence before hosts are fully occupied. After this, hosts resemble the colonizers pool. D The average frequency increases with competition for space (smaller N). While the symmetry of inheritance decreases the average frequency as a result of the reduced initial microbiome variability, asymmetry increases it. Each simulation included \(10^4\) hosts