Plant–microbe interactions play crucial roles in species invasions but are rarely investigated at the intraspecific level. Here, we study these interactions in three lineages of a globally distributed plant, Phragmites australis. We use field surveys and a common garden experiment to analyze bacterial communities in the rhizosphere of P. australis stands from native, introduced, and Gulf lineages to determine lineage-specific controls on rhizosphere bacteria. We show that within-lineage bacterial communities are similar, but are distinct among lineages, which is consistent with our results in a complementary common garden experiment. Introduced P. australis rhizosphere bacterial communities have lower abundances of pathways involved in antimicrobial biosynthesis and degradation, suggesting a lower exposure to enemy attack than native and Gulf lineages. However, lineage and not rhizosphere bacterial communities dictate individual plant growth in the common garden experiment. We conclude that lineage is crucial for determination of both rhizosphere bacterial communities and plant fitness.
Let Hm(z) be a sequence of polynomials whose generating function ∞ m=0 Hm(z)t m is the reciprocal of a bivariate polynomial D(t, z). We show that in the three cases D(t, z) = 1 + B(z)t + A(z)t 2 , D(t, z) = 1 + B(z)t + A(z)t 3 and D(t, z) = 1 + B(z)t + A(z)t 4 , where A(z) and B(z) are any polynomials in z with complex coefficients, the roots of Hm(z) lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the q-analogue of the discriminant, a concept introduced by Mourad Ismail.
For any fixed positive integer $n$, we study the root distribution of a
sequence of polynomials $H_{m}(z)$ satisfying the rational generating function
\[ \sum_{m=0}^{\infty}H_{m}(z)t^{m}=\frac{1}{1+B(z)t+A(z)t^{n}} \] where $A(z)$
and $B(z)$ are any polynomials in $z$ with complex coefficients. We show that
the roots of $H_{m}(z)$ which satisfy $A(z)\ne0$ lie on a specific fixed real
algebraic curve for all large $m$
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