@wnx I was (and am) totally calm! My tone wasn't aggressive, maybe I had to add some emoticon? :) This said, the OPs question lacks a lot of information and has indeed some mistakes in it! It is indeed quite indecipherable. But it's not a big deal, right? I was just trying to push the OPs to reformulate better his question, for his own advantage!

Following Lazarsfeld (the example you cited), for it is sufficient to have $r+s\ge 4$.

@YangMills thanks! I looked at the paper. I'll add something in my answer later or tomorrow! Thanks again to Jason, too!

Hi Jason! Now that you say this, I have some reminiscence of a conference in Luminy where McKernan gave a talk around this, but I cannot remember the precise statements... I’ll take a look, cheers!

@Samir, thai is as wrong as possibile, in general! Once again you should write better your original question! What do you have in mind? A projective manifold? Or merely a compact complex one?

In any case I really don't understand what the OP would like to know...

@abx what you say it's ok for projective manifolds. But this is merely compact complex. Even in the compact Kähler case "$K_X$ not nef implies presence of rational curves" is not known!

@abx thanks. But still, I don't immediately see why if $X$ satisfies 1) and 2), then this is true...

$K_X$ big does NOT imply $K_X$ nef, nor in general neither in this case.

I don't understand. The general fibre has genus one, indeed.