A question regarding Logistic population model

[issacnewton]

Hi
I am going through Edx course on Introduction to Differential Equations by Paul Blanchard (BUx: Math226.1x). At one point, he is explaining the Logistic population model.$$\frac{dp}{dt} = kp\left(1- \frac{p}{N}\right) $$ After this, he says that since the right hand side does not involve $t$, if $p(0) = 0$ then $\frac{dp}{dt} = 0$ for all $t$. I don't quite get his logic here. Can anyone explain this please ?

Thanks

 

[jim mcnamara]

If you can answer a question like the one below you get the gist of what is going on.
Example: How many people does it take to start reproducing more people? Or mammals or bacteria?

 

[issacnewton]

Jim, it will take 2 persons or in case of bacterias, it will just take one.

 

[issacnewton]

Can anybody give more hints ?

 

[mathman]

If there are none to start with, how can you reproduce?

 

[issacnewton]

Ok that makes sense. But if we have $\frac{dp}{dt}$ function of $t$, then does $p(0) = 0$ still lead to $\frac{dp}{dt} = 0$ for all $t$ ?

 

[Stephen Tashi]

After this, he says that since the right hand side does not involve $t$, if $p(0) = 0$ then $\frac{dp}{dt} = 0$ for all $t$. I don't quite get his logic here.

The derivative of a constant function is zero -meaning the derivative of a constant function is the "zero function". For example, if $f(x) = 3x + 15$ when you compute $\frac{df}{dx}$ as the derivative of $3x$ plus the derivative of $15$ what do you get for the answer when you differentiate the $15$?

Note that $15$ must be taken to denote the constant function $g(x) = 15$ in order to differentiate it because we differentiate functions, not single numbers.