Expert: Scott A Wilson - 6/5/2009

Question
As a vet I'm constantly trying to explain the population growth rates for fleas reproducing under ideal environmental conditions (85% humidity and 26C). The life cycle: an adult female will produce 40 eggs a day, laying 500 in a lifetime, and will live for 4 weeks. 50% are offspring are female, and the egg/larval/pupal stages take 28 days to reach maturity and start laying the next generation. If a single adult female flea is introduced on day 0, on day 29 there is 41 adults and 460 eggs; day30 - 80 adults(mom died), 420+800 eggs (only half are female and lay eggs); day 30 120 adults, 380+800+1600 eggs, etc etc. Does this make sense? Would be great to be able to input the number of days after introduction of a single adult female, and get a figure of entire adult and egg populations.  I'd be interested in a copy of the calulations if possible. Thanks for your help.

Answer
I designed a program for you in C++.

It assumes that there are 2 fleas to start with
(doesn't it take 1 male and 1 female?).

Every time u is pushed it updates them by 1 year.
Note that for the first 27 numbers listed out,
the new fleas are still developing.

Since they only lay about 500 eggs at 40 per day, I have them laying 40 for the first 12 days and 20 on the 13th day.

The first set produces the second generation, and then holds for awhile (a few days) until that generation starts production.

After the third generation, production has spread out enough that new fleas appear evevery day.  The fleas are kept in the computer, so there is no need to start itching.

Don't worry - when the program ends, all the fleas are killed.
Yes, I'm sure of it.

After 100 days, there are over 2,000,000,000 fleas.
Unfortunately, on the computer, this is almost too many, for on the next day the population seems to have died out.

Now if you only wanted an approximate number of fleas, I could make that into a real number, but that would round when it got that high.

I could make adjustments in the program to possible apply some medicine to kill of most of the fleas at a later time or sonething.

It works great, but I'm not sure how to send an execuatable program without your email address...

After 10 days, I have 402.
After 20 days, there are 442.
After 30 days, there are 2,842.
After 40 days, there are 62,040.
After 50 days, there are 106,040.
After 60 days, there are 426,040.
After 70 days, there are 8,777,960.
After 80 days, there are 22,809,600.
After 90 days, there are 65,769,600.
After 100 days, there are 1,209,761,200.
After a few more days, there were too many and they all died.

Not really, but the numbers were too big for C++ in long int.

I'm not sure if the program I wrote was entirely correct.

Here is the copy of it you could highlight and put into C++
{ if you have C++ }.  Note that there are three lines near the bottom that wrapped around as I was sending them.   They may or may not need to be fixed on arrival.

/*
females produce 40 eggs a day for 12 days and 20 egss on 13th day
half of the eggs are female

eggs take 28 days to reach maturity

life after birth is 4 weeks
*/
// population.cpp

#include <stdio.h>
#include <stdlib.h>
#include <conio.h>

#define MAX 68

void intro() {
 printf ("This program will list out the number of fleas that there are.\n");
 printf ("They start production on day 28 and go to day 39.\n");
 printf ("At that point, they have produced 500 eggs.\n");
}

void kill (long int *p) {
 int i;   // used in for loop
//  for (i=0; i<MAX; i++)
//printf ("%9.4f", rand())
//    p[i] = rand()*p[i];
}

int upper (char ch) {
 if (ch >= 'a' && ch <= 'z') ch = ch - 32;
 return (ch);
}

void help () {
 printf ("\nAssistance for the program\n\n");
 printf (" I list individuals\n");
 printf (" P print out\n");
 printf (" U update\n");
 printf (" Q quit\n\n");
 printf (" Choice? ");
}

void list (long int *p) {
 int i;    // used in for loop

 for (i=0; i<MAX; i++) {
   printf (" %3d:%9ld", i, 2*p[i]);
   if ((i + 1) % 5 == 0)
     printf ("\n");
  }
 printf ("\n");
}

long int out (long int *p) {
 int i;   // used in for loop
 long int sum=0;    // population sum

 for (i=0; i<MAX; i++)  sum += p[i];

 return (sum);
}

long int update (long int *p) {
 int i;
 long int sum=0;

 for (i=MAX-1; i>0; i--) {
   p[i] = p[i-1];
   sum += p[i];
  }

 p[0] = 0;   // account for new born fleas
 for (i=28; i<40; i++)
   p[0] += 20*p[i];
 sum += p[0];
 return (sum);
}

void main() {
 int i;      // used in for loop
 int ch;      // command for program to do
 int d=0;      // number of days
 long int p[MAX];      // used for population
 long int pop;      // the population

 intro ();
 // initialize the population
 for (i=0; i<MAX; i++)  p[i] = 0;
 p[28] = 1;   // start with one mature
 pop = 1;

 help();
 ch =upper(getch());
 if (ch != 'Q')
   do {
     printf ("ch %c\n", ch);

     if (ch == 'U') {
       d++;    // add one to number of days
       printf ("The number of days is %d.\n", d);
      }

     if (ch == 'H') help();
     else if (ch == 'I') list (p);
     else if (ch == 'P') printf ("The total is %ld\n", pop = 2*out(p));
     else if (ch == 'O') kill (p);
     else if (ch == 'U') printf ("Another day passed.  Now there are %ld fleas.\n", pop = 2*update(p));
     else if (ch != 'Q') help();

     if (pop < 0)  for (i=0; i<MAX; i++)  p[i] = 0;   // if population goes negative, eliminate all

     help();
     ch = upper(getch());
    }
   while (ch != 'Q');
 else printf ("Boy, you killed them before they got started!\n\n");

 if (d > 0) printf ("And that exterminated the fleas ...\n\n");
}