You should start by entering the 5 equations, 5 solutions, and the intermediate variable definitions into Maple. Then, it should be very easy to verify that it is a solution. It may be a little more difficult to verify that it's the only solution.

You should change I and D to different variables because those are otherwise used by Maple.

@Kitonum The OP is referring to the kind of plot produced by Statistics:-ColumnGraph.

@Kitonum Based on the picture in the first version of this Question, I know that the intention is that k is a fixed number between 0 and n, inclusive.

@daljit97 I see that you've attempted to correct the Question. But "multiplication modulo 42" is not the same thing as "modular multiplication by 42". The former is the correct group operation. The latter isn't even a binary operation, let alone a group operation. 

"Modular division by 42" is not a well-defined group operation. A group operation has two operands from the group. You may be thinking of either "addition modulo 42" or "multiplication modulo 42", but those are very different groups.

@jgray07m I think that all that you need to do is keep the sliders for birth and death rates at 0. I don't think that dN/dt = 0 will be a problem.

@Joe Riel Thanks, Joe, for the Tally-once idea and for the profiling. I haven't looked at your code yet, but I'm sure that it will be enlightening. I wouldn't have guessed that Tally was the culprit, but it makes sense now. The reason that I count nonzeros rather than zeros is because that's all that the too-vaguely-named ArrayTools:-SearchArray does.

@Joe Riel Using expand will do more than using simplify in this and several related cases.

@Kitonum Yes, I like your interpretation.

Having taught 3-D Calculus with Analytic Geometry, my mental imagery of the problem involves lines and planes. Indeed, I was visualizing it as I fell asleep last night, and I put it into Maple when I awoke.

@radaar To be even more specific topologically, do you mean embeddings^[*1] of [0,1]x[0,1] into R³? So, for example, a mountainous region of the Earth's surface, but not the whole Earth itself?

[*1] An embedding f:A->B is a bicontinuous injection, i.e., a homeomorphism between A and its image in B, (f(A)).

@Mariusz Iwaniuk 

The final condition is D(f)(infinity) = 1, not f(infinity) = 1.

When you put comments in worksheets, they should be marked as comments so that the syntax checker does not interpret them as code, which'll surely produce error messages. There are a few ways to do this. The easiest is to begin the comment with #. Then everything to the end of that line will be ignored by the syntax checker.

It's also possible to set aside whole execution groups (which are somewhat analogous to paragraphs) as fully formatted text, similar to that produced by a word processor. 

@Zeineb 

X_k is the k^th column of A^-1.

@jgray07m So, populations are relative, as in percentages, rather than actual population numbers? Yes, it seems reasonable. I guess then that N is total (alive) population? 

And I need the 5 parameters. Obviously, the v slider should go from 0 to 1.

@jgray07m You can read the help page ?Explore.

Can you come up with some numeric values?