Fig. 2

Microbial occurrence in hosts under microbial inheritance. A Starting from a condition where all hosts are initially empty, the microbial occurrence increases through time. At first sight, this increase is largely independent of \(\alpha _0\) and the inheritance of microbes. A closer look at equilibrium abundance reveals that inheritance increases the occurrence, in this case, regardless of how rapidly hosts are occupied (\(\alpha _0\)). B The increase results from a distribution of microbial load across the host population where the microbe-free state is less common. A microbial load of \(10^{-5}\) corresponds to 1 microbe per host. In (C–E), single parameters are modified from the case shown in (A-B) (with parameters of immigration \(m=10^{-2}\), host death \(\tau =10^{-4}\), and the carrying capacity \(N=10^{5}\), indicated by the triangles in (C–E)). C A large migration from the pool of colonizers \(m \rightarrow 1\), hinders any effect of inheritance on occurrence as hosts are readily colonized. The change peaks and decreases for smaller immigration, as for \(m \rightarrow 0\) hosts are less likely to be colonized. The change can even be negative for slowly occupied hosts where the few colonizing microbes are lost to stochasticity. D The gain from inheritance is maximal for intermediate values of host death probability, \(\tau\). Long living hosts, \(\tau \rightarrow 0\), are colonized even without inheritance. Short living hosts, \(\tau \rightarrow 1\), are less likely to be colonized and thus transmit microbes through inheritance. E The carrying capacity for microbes of a host, N, and \(\alpha _0\) do not alter the gain from inheritance. Points and bars in (C–E) indicate the average and standard deviation of 6 simulation pairs, with vs. without inheritance, with \(10^4\) hosts each. Offspring receive \(9\%\) of their parent’s microbiome on average, \(a_i = 0\) and \(b_i = 9\) in Eq. (4). The whole distributions are shown in Additional file 1: Fig. S2