A maximum likelihood (ML -based phylogenetic tree was constructed using RAxML -basedm [ 48] with the JTT + GAMMA + Invariable sites model, based on model selection by phylogenetic2], and treereliability was assessed by bootstrapping 100,000 pseudorepliconstructed
Furthermore, from Lemma 1 and (10), we easily see that Γ is positive invariable for model (4).
The GTR (General Time Reversible)+I (Invariable sites) model with gamma-distributed rate variation across sites and a proportion of invariable sites, was specifically used and other parameters were kept as default.
Maximum-likelihood (ML) phylogenetic trees were constructed by using the best-fit general time reversible + gamma 4 + invariable sites model, which was identified by using MODELTEST version 3.7 (28 ).
From each concatenation set, a maximum-likelihood tree was reconstructed with RaxML version 7.4.2 [ 42] using a GTR plus invariable sites model, as described before.
These resampled datasets were analysed using PHYML [ 57] with alpha and invariant sites parameters optimized on the Bayesian tree in TREE-PUZZLE 5.0 [ 58] with a mixed four-category discrete gamma plus invariable sites model of rate heterogeneity.
From the previous discussion we see that the region Gamma=bigl{ (S,E,I,R): Sgeq0, Egeq0, Igeq0, Rgeq0, S+E+I+Rleq S_{0}bigr} is a positive invariable set for model (2) and absorbs all nonnegative solutions of model (2).
Then R + 3 is the invariable set of model (1), that is, any solution ( J ( n ), S ( n ), I ( n ) ) of model (1) with initial condition (2) is positive for any n ∈ Z.
The General Time Reversible (GTR) + Invariable sites (I) model with gamma-distributed rate variation across sites and a proportion of invariable sites was specifically used with other parameters kept at default.
For this leopard cat sequence data the General Time Reversible with Invariable sites (GTR+I) model was selected by the AIC using ModelTest.
