Best Solution in O(n) time:

import java.io.IOException;

public class MigratoryBirds{ public static void main(String[] args) throws IOException { System.out.println(migratoryBirds()); }

//Worst Case: O(n)
static int migratoryBirds() throws IOException {
    Reader reader = new Reader();
    int n = reader.readInt();

    final int k = 6;
    int[] counts = new int[k];
    int type = 0;
    int maxCount = 0;

    for (int i = 0; i < n; i++) {
        int birdType = reader.readInt();
        counts[birdType]++;

        if(counts[birdType] > maxCount ||
            (counts[birdType] == maxCount && birdType < type)) {
            type = birdType;
            maxCount = counts[birdType];
        }
    }
    return type;
}

}