NSolve Unreliable

Question

NSolve is unpredictably unreliable. I change a parameter that has no effect on the solution and get one of four outcomes: 1) both correct answers, 2) one correct answer, 3) it just returns the command back, or 4) it hangs for a long time.

Do[
  Print[r, " ",
    NSolve[
      {n1 == E^(r (1 - n1 - 0.6 n2)) n1, n2 == E^(r (1 - 1.2 (0.6 n1 + n2))) n2}
    , {n1, n2}]
  ]
, {r, 1.0, 2.0, 0.1}]

gives

1. {{n1->1.,n2->0.},{n1->0.78125,n2->0.364583}}
1.1 NSolve[{n1==E^(1.1 (1-n1-0.6 n2)) n1,n2==E^(1.1 (1-1.2 (0.6 n1+n2))) n2},{n1,n2}]
1.2 {{n1->0.78125,n2->0.364583}}
1.3 {{n1->0.78125,n2->0.364583}}
1.4 {{n1->0.78125,n2->0.364583}}
1.5 {{n1->1.,n2->0.},{n1->0.78125,n2->0.364583}}
1.6 {{n1->0.78125,n2->0.364583}}
1.7 {{n1->0.78125,n2->0.364583}}
1.8 {{n1->0.78125,n2->0.364583}}
1.9 NSolve[{n1==E^(1.9 (1-n1-0.6 n2)) n1,n2==E^(1.9 (1-1.2 (0.6 n1+n2))) n2},{n1,n2}]

and hangs on r=2.0 when I run it on Mathematica 10.4.1 on MacOS 10.11.6. I tried all of the NSolve Methods from this answer but they make no difference.

Bug? Possible workaround (not for this particular problem, but this phenomenon in general)?

Answer 1

Rationalize the equations and use rational values of r (i.e., use exact numbers) and restrict the domain to nonnegative Reals. Although this is quite slow, it works.

$Version

(*  "10.4.1 for Mac OS X x86 (64-bit) (April 11, 2016)"  *)

eqns = {
   n1 == E^(r (1 - n1 - 3/5 n2)) n1,
   n2 == E^(r (1 - 6/5 (3/5 n1 + n2))) n2,
   n1 >= 0, n2 >= 0};

Do[Print[r // N, " ",
  soln = NSolve[eqns, {n1, n2}, Reals],
  " ", And @@ (And @@ eqns /. soln)],
 {r, 1, 2, 1/10}]

(*  1. {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.1 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.2 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.3 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.4 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.5 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.6 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.7 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.8 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

1.9 {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True

2. {{n1->0,n2->0},{n1->0,n2->0.833333},{n1->0.78125,n2->0.364583},{n1->1.,n2->0}} True  *)