Question

Consider the following first-order ODE formulations
0 0
{ ( )}, ( )
( )
a L n t n t n
dt
dn t
= − =
Associate the physical meaning to the variables {t, n(t)} and the parameters {a, L} so that the
above formulation becomes a mathematical model for population changes.

Answer 1

By the condition of the problem, this equation

$$\frac{dn}{dt}=a\left(L-n(t)\right)$$

should be a mathematical model to describe population changes.

Therefore, we can immediately give physical meaning to some variables and constants

$$t-\text{time}\\[0.3cm] t_0-\text{initial time}\\[0.3cm] n(t)-\text{population at time}\,\,\,t\\[0.3cm] n_0-\text{initial population}\\[0.3cm]$$

For further analysis of the model and interpretation of constants, we solve this differential equation:

$$\displaystyle\frac{dn}{dt}=a\left(L-n(t)\right)\rightarrow\displaystyle\frac{dn}{L-n(t)}=adt\\[0.3cm] \int\displaystyle\frac{dn}{L-n(t)}=\int adt\rightarrow-\ln|L-n(t)|=at-\ln|C|\\[0.3cm] \ln|L-n(t)|=-at+\ln|C|\rightarrow L-n(t)=C\cdot e^{-at}\\[0.3cm] \boxed{n(t)=L-C\cdot e^{-at}}$$

To determine the constant, we use the initial condition

$$n(t_0)=n_0=L-C\cdot e^{-at_0}\rightarrow C\cdot e^{-at_0}=L-n_0\\[0.3cm] \boxed{C=e^{at_0}\cdot(L-n_0)}$$

Conclusion,

$$n(t)=L-e^{at_0}\cdot(L-n_0)\cdot e^{-at}\rightarrow\\[0.3cm] \boxed{n(t)=n_0\cdot e^{-a(t-t_0)}+L\cdot\left(1-e^{-a(t-t_0)}\right)}\\[0.3cm] \lim\limits_{n\to\infty}n(t)=\lim\limits_{n\to\infty}\left(n_0\cdot e^{-a(t-t_0)}+L\cdot\left(1-e^{-a(t-t_0)}\right)\right)=L$$

Now we can explain the physical meaning of constants $a$ and $L$ :

$$L-\text{maximum possible population}\\[0.3cm] a-\text{characteristic time at which the difference decreases e times}\\[0.3cm]$$

ANSWER

$$n(t)=n_0\cdot e^{-a(t-t_0)}+L\cdot\left(1-e^{-a(t-t_0)}\right)\\[0.3cm] t-\text{time}\\[0.3cm] t_0-\text{initial time}\\[0.3cm] n(t)-\text{population at time}\,\,\,t\\[0.3cm] n_0-\text{initial population}\\[0.3cm] L-\text{maximum possible population}\\[0.3cm] a-\text{characteristic time at which the difference decreases e times}$$