Bacteria population growth confusion
0
In one of my course, it is said that a bacteria population growth can be modelled by the differential equation $P(t)'=rP(t)$ where $r$ is the growth rate. The solution of the above equation is $P_{0} e^{rt}$ which bugs me quite a lot. I looked up the Internet and found this article which explains that bacteria growth is characterised by the function $P_{0} 2^t$. Can someone help me to clear this up?
differential-equations
asked Sep 20 '16 at 13:08
Dory
627
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show 3 more comments
0
In one of my course, it is said that a bacteria population growth can be modelled by the differential equation $P(t)'=rP(t)$ where $r$ is the growth rate. The solution of the above equation is $P_{0} e^{rt}$ which bugs me quite a lot. I looked up the Internet and found this article which explains that bacteria growth is characterised by the function $P_{0} 2^t$. Can someone help me to clear this up?
differential-equations
asked Sep 20 '16 at 13:08
Dory
627
My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15
|
show 3 more comments
0
0
0
In one of my course, it is said that a bacteria population growth can be modelled by the differential equation $P(t)'=rP(t)$ where $r$ is the growth rate. The solution of the above equation is $P_{0} e^{rt}$ which bugs me quite a lot. I looked up the Internet and found this article which explains that bacteria growth is characterised by the function $P_{0} 2^t$. Can someone help me to clear this up?
differential-equations
asked Sep 20 '16 at 13:08
Dory
627
In one of my course, it is said that a bacteria population growth can be modelled by the differential equation $P(t)'=rP(t)$ where $r$ is the growth rate. The solution of the above equation is $P_{0} e^{rt}$ which bugs me quite a lot. I looked up the Internet and found this article which explains that bacteria growth is characterised by the function $P_{0} 2^t$. Can someone help me to clear this up?
differential-equations
differential-equations
asked Sep 20 '16 at 13:08
Dory
627
asked Sep 20 '16 at 13:08
Dory
627
edited Sep 20 '16 at 13:33
asked Sep 20 '16 at 13:08
Dory
627
asked Sep 20 '16 at 13:08
Dory
627
asked Sep 20 '16 at 13:08
Dory
627
627
My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15
|
show 3 more comments
My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15
My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15
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My problem with this reasoning is that r is supposed is suppose to represent a growth rate of 100% as in the article. How do you know in advance that r in the differential equation will be different? This DE is derived in the course from the fact that $P(t+Delta t) - P(t) = rP(t)Delta t$ which is a priori found by simple book-keeping. Intuitively for me, this r of 100% should lead to a function such as $2^t$ and not $e^t$. That is what bugs me.
– Dory
Sep 20 '16 at 13:29
That web page is wrong about the bacteria. You do not start at time $0$ with a bunch of "Mr. Blue" bacteria of which every one is about to start spawning a new "Mr. Green." An actual colony of bacteria has bacteria in every stage of reproduction at any given time, so you are (almost) continuously getting new bacteria. It actually works more like the compound-interest-on-money example that is shown after the bacteria.
– David K
Sep 20 '16 at 13:38
@DavidK But isn't it true that a not completely born/formed bacteria cannot start to divide into two bacterial?
– Dory
Sep 20 '16 at 13:43
It's also important to realise that these equations are only simple models of what is happening and won't necessarily be exact in the real world. Also note that having $r$ as an unknown still includes the possibility that $r$ relates to 100%, but it also includes the possibility that it has another value.
– EHH
Sep 20 '16 at 14:19
@Dory it is not the incompletely formed bacteria that divide. My point is that if the bacteria divide once every 60 seconds, then at any instant that you look at the colony there will be some bacteria that last divided 59 seconds ago and have just 1 second left before they finish dividing again, some that divided 58 seconds, ago, and so forth. By the way, there will be almost twice as many that divided 1 second ago as the number that divided 59 seconds ago.
– David K
Sep 20 '16 at 15:15