Strangeness production in hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions in the dual parton model

$\Lambda$, $\overline{\Lambda }$, and $K_S^0$ production is studied in a Monte Carlo dual parton model for hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions with an SU(3) symmetric sea for chain formation (chain ends) but strangeness suppression in the chain fragmentation process. Additionally, $\mathrm{}\left(\mathrm{qq}\right)-\left(\overline{q}\overline{q}\right)$ production from the sea was introduced into the chain formation process with the same probability as for the $q\to \mathrm{qq}$ branching within the chain decay process. With these assumptions, multiplicity ratios and Feynman-$x$ distributions for strange particles in $h-h$ and multiplicity ratios in heavy ion collisions are reasonably well reproduced.