Question

Consider a population whose growth over a given time period can be described by the exponential model: dN/dt = rN. Select the correct statement about this population.a. A population with an r of 0.1 will decrease over time.b. A population with an r of 0 will have no births or deaths during the time period under consideration.c. A population with a positive value of r will grow exponentially.

Answer 1

Answer:

b and c

Step-by-step explanation:

We are given that a population whose growth over a given time period can be described by the exponential model

$\frac{dN}{dt}=r N$

Let initial population =$N_0$ when time t=0

$\int\frac{dN}{N}=r\int_0^tdt$

After integrating

We get ln N=rt +C

Where C is integration constant

When t=0 then N=$N_0$

$ln N_0=C$

Substitute the value of C then we get

$ln N=rt +ln N_0$

$ln N-ln N_0=rt$

$ln\frac{N}{N_0}=rt$

$\frac{N}{N_0}=e^{rt}$

$N=N_0e^{rt}$

When r=0.1 then we get

$N=N_0e^{0.1t}$

Hence, the population increase not decrease.

When r= 0

Then we get

$N=N_0e^{0}$

$N=N_0$

Hence, the population do not increase or decrease.

So, a population with r of 0 will have no births or deaths during the time period under consideration.

If we take a positive value of r then the population will increase exponentially .

Hence, option b and c are both correct.