array (
  0 => '1',
  1 => '2',
  2 => '3',
  3 => '4',
  4 => '5',
  5 => '6',
  6 => '7',
  7 => '8',
  8 => '9',
  9 => '11',
  10 => '12',
  11 => '13',
)

$arr = [1, 2, 5, 6, 7, 9, 11];
    $missedNumbers = [];

    foreach ($arr as $id => $number) {
        if (!isset($arr[$id + 1])) {
            continue;
        }

        if ($arr[$id + 1] - $number !== 1) {
            for ($i = 1; $i < $arr[$id + 1] - $number; $i++) {
                $missedNumbers[] = $arr[$id + 1] - $i;
            }
        }
    }

Array
(
    [0] => 4
    [1] => 3
    [2] => 8
    [3] => 10
)