What is "the 'list of list'"? Could you give a example of the code to generate it? Or could you upload a worksheet?

In your Question, there is just a blank space where I think that you intended to paste a Matrix.

@J4James I'll write the code with the changeable bounds and post it as a separate Answer. In the above the, the 2,2,2,2 are not the upper bounds---they are the numbers of increments. I'll write it so that for each parameter you'll specify the upper and lower bounds and the number of increments, rather than specifying the size of the increments.

@J4James Do you mean change the 8 that is the upper bound for eta to 12?

@J4James No I don't mean that. If you want to change the lower bound from 0, that's fine. But you didn't say that before. You said from 0 to 2 with increment 0.01. The 0, the 2, and the 0.01 are all easy to change.

@Preben Alsholm What made you think of replacing (D@@2)(f)(0) with a parameter?

@adel-00 What do mean by t? Do you mean the value of t such that N'(t) = 0? In other words, given a value of beta, how do we choose a t to match that beta?

@J4James That's 200 values for each of the 4 parameters. That's 200^4 = 1.6 billion combinations of parameters. That's too much to do in a reasonable time.

As for the time, each change of parameters requires a separate call to dsolve. There's no parameters option for BVPs.

@adel-00 Do you mean beta versus the value of t such that N'(t) = 0? Won't that encounter the same problem as Preben mentioned? It's still an autonomous IVP.

Your examples just appear as blank spaces in your Question.

@Carl Love My results so far are that y' = 0 only at t = +infinity regardless of epsilon.

@adel-00 Okay, that makes sense to me now. I'll be thinking about it. Yes, I think that there's an easy way to do it.

Plotting epsilon against t makes no sense to me because they are both independent variables.

@J4James Do you mean 0 to 2 times the increment?

@J4James I don't know a way. There is no help page for that error message. Someone will probably chime in here soon with an analysis of the error. They might find a way around.