Sometimes the Mapleprimes worksheet display is broken for a time, and the files display incorrectly as you saw. But sometimes there is just something slightly different about a file that leads to problems, as is the case here. I opened your file and had the same poblem uploading, then deleted the last two plots and then redid them. Now it seems identical but can upload. 

Download predator_prey2.mw

In your first section of code, after the first solve, you assign x[2]^2/3 to x[1]. So now fx is (x[2]^2/3)^2 - 3*x[2] = 0, so a quartic equation in x[2]. When you solve it you get the 4 roots of this quartic. So this is correct.

A better approach is to just solve the two equations simultaneously, i.e., let Maple do the work for you.

sol := solve({fx, fy},{x[1],x[2]})

you could then use assign to set the values of x[1] and x[2], or better, use the solution you want with eval to work out other things involving x[1] and x[2].
(For future reference, please upload your worksheet contents and leave the link, so we can directly use it without retyping it in.)

Download solve.mw

I assume you are OK with the standard mathematical way of writing the limit. Then the following works.

InertForm:-Display(`%>`(%limit(X[i] %^ (rho = 0), alpha = infinity), 0), inert = false)

Maple calculates THE sqrt (sqrt is a function); sqrt(4) does not give +/-2. From the help page "sqrt(x) represents the "principal square root", defined by the formula sqrt(x) = exp(1/2 * ln(x))". 

coeffs finds all the coefficients, and then they can be used in solve (by default solve assumes they are equal to zero). But solve(identity(,...) automates this process. There is no solution for eq4, but for eq1 if you do not specify q[0], then there is a trivial solution.

Solve_for_coefficients2.mw

I'd say it is a bug. A workaround if to use evalf

sol:=[-1/4*x^2, (-1/2-1/2*(-3)^(1/2))^2+(-1/2-1/2*(-3)^(1/2))*x, (-1/2+1/2*(-3)^(1/2))^2+(-1/2+1/2*(-3)^(1/2))*x];
plot(evalf(sol),legend=sol)

I've added missing information with random values. This gives a magnitude plot colored by phase.

ps - if providing code, please don't include the output.

Download complexplot3d.mw

If you try

for y in [0, 0.5, 1, 5, 10] do
  p := plot(y);
  plottools:-getdata(p, rangesonly);
end do;

Then the y range is always reported just y..y, but the actual ranges are

y=0: -1..1

other y:   y-y/2..y+y/2

so perhaps that is a general rule.

The plot,ranges help just says "If the vertical range is unspecified, a suitable range will be deduced from the values computed" :-(

dsolve works here, and finds a simple correct solution. I'm guessing something about the problem is not correctly formulated here.

Download dsolve.mw

You have a large polynomial system that is going to be slow. Among other changes, removing the explicit option saves about 25%. I have also added general suggestions that may be useful in the future.

new1.mw

Your worksheet does not run so I copied the output.

Download choosecase.mw

The problem is the O(h^7) term. If q is your expand(SUMY-T) then

coeff(convert(q, polynom), h, 6)

(actually you should remove it from SUMY)

I don't see any way in the general case. But Z² is simple enough to do. The comments suggest why it will be difficult in the general case. Zp is the cyclic group of order p.

Download CayleyGraphZ2.mw

Try

showstat(Units:-TestDimensions);

invlaplace very often needs assumptions to make progress. Even though s is complex, assuming everything is positive often seems to work. I just worked out lap3, since I am out of time, but this method is probably enough to get the rest done.

Download lap3.mw