BIOL 103 Home

Biomorph Challenge 1: Key

1. "You infinitely insignificant Terrestrial creatures have nothing to say about it, so relax--Biomorphs produce only two offspring per year, so in a hundred years you'll see only two hundred of us.

Actually, if each Biomorph produces two offspring (converting the parent like bacteria), then after 100 years there will be:

N₀2 ^t/d =

N₀2 ^{100 years/1 year} =

N₀2 ¹⁰⁰ = 1 biomorph x 1.27 x 10³⁰ (a number with thirty zeros). That's considerably larger than our own human population.

If the Biomorphs are immortal after making offspring, it gets even worse:

N₀3¹⁰⁰ = 5.15 x 10⁴⁷ times N₀ biomorphs.

2. Tribbles actually don't age, so they live forever after producing each litter (unless they eat poisoned grain.) With this new assumption:

(a) Uhura brought one tribble onto the Enterprise. Each tribble produced a litter of 10 every 12 hours, calculate how many tribbles there should be in 3 days.

After three days there will be:

N_t = N₀11^t/d = 1tribble x 11^{(3 days*(24 hours/day)/12 hours)} = 11⁶ tribbles = 1,771,561 tribbles

(b) Set up an Excel file to calculate the number of tribbles produced over 5 days. In Excel, plot your results (number of tribbles over time). Print a copy of the plot and the data table.

For each day, there will be 11^{t (days)/0.5 days} tribbles.

3. Spock estimates that there would be 1,771,561 tribbles on board (see above question) after 3 days. Which assumption did Spock use?
Spock must have assumed that the tribbles continue living after producing each litter, and that they have infinite resources.

What actually became of the tribble population on the Enterprise, and why?
Various answers are accepted. In the story, the tribbles in the space station storage compartment were killed by poison. Those in the Enterprise probably ran out of food (as the doctor suggested). At the end of the story (not shown in class), engineer Scotty beams them all onto the Klingon ship.

4. What is the doubling time of the tribble population?
We are solving (N_t = N₀2^{t days/d days )}) for the doubling time, d.

The number N_t has to be the same whether the exponential base is 2 or 11. So you can calculate the doubling time (Q=2) from the generation time (Q=11, after 12 hours=0.5 days).

N₀11^{t days/0.5 days} = N₀2 ^{t days/d days}

The equation will be true for any N₀ and any total time t, so set N₀=1, t=1.

11^1/0.5 = 2 ^1/d

log 11² = log 2 ^1/d

2 log 11 = (1/d) log 2

d = log2/(2log11) = 0.144 days = 3.47 hours

During the poisoning, the population of tribbles drops to 500 during the poisoning. Assume those tribbles are still producing 10 offspring per litter, but only half survive the poison, and the parent dies after one litter.
What is their new doubling time?

Generation size is now decreased from 11 to 10/2 = 5.

log 5² = log 2 ^1/d

d = log2/(2log5) = 0.215 days = 5.17 hours

5. In an environment like the Enterprise:
a) How many tribbles would you need initially to produce 1 billion tribbles in 2 days?

N_t = N₀2^{t days/d days )} Solve for N_t. Assume d=0.144 days [or you may assume d=0.217 days]

10⁹ tribbles = N₀2^{(2 days/0.144 days)}

N₀ = 10⁹ tribbles/2^(2/0.144) = 65,922 tribbles after 2 days

b) How many hours would it take for 1 tribble to produce one billion tribbles?

10⁹ tribbles = N₀2^{(t hours/3.47 hours)} = 1 x 2^{(t hours/3.47 hours)} Solve for t = total time in hours.

log 10⁹ = log 2^{(t hours/3.47 hours)}

9 = (t hours/3.47 hours) x log 2

t hours = 9 x 3.47 hours / log 2 = 104 hours

6. In actual natural environments, what phenomena limit the size of growing populations?

In actual environments, population growth is limited by predation of individuals, diseases that kill or decrease fertility, starvation for resources, and overproduction of toxic wastes.

7. If there are 1,200,000 tribbles on the Enterprise and Cyrano Jones takes 12 years to pick them up:
Many answers are possible; you need to justify your answer and show your calculations.

a. How many seconds does it take for him to pick up one tribble?
Assuming he does nothing else:
(12 years)(365.25 days/year)(24 hours/day)(3600 sec/hour) / 1,2 x 10⁶ tribbles
= 26 sec per tribble [Note: Be sure to divide in the right direction; check units]

b. Is this answer reasonable? Explain what assumptions you made in finding your answer.
The answer is not reasonable because it leaves no time to sleep or eat. It also assumes the tribbles are all dead and inert for twelve years.

c. How many seconds do you think it would take you to pick up a tribble and stuff it into a bag?
I can pick up a tribble in 5 seconds, including the time to walk across the room for the next one and place it in a bag.

d. Neglecting the time it takes to store the full bags and get empty bags, how many tribbles could you pick up in 12 years? Allow yourself some time to eat and sleep.
Assume 10 hours of tribble picking per day:
(12 years)(365.25 days/year)(10 hours/day)(3600 sec/hour) / 5 sec / tribble
= 31,557,600 tribbles after 12 years

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