Question

The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community it is assumed that the rate at which the population changes is a net rate-that is the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t>0?

Short answer

$\frac{dP}{dt}=(b-d)P$

Step-by-step solution

Definition of differential equation and formula

A system of ordinary differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable.

$dy/dx=f\left(x\right)$

Where {rate of births – rate of deaths}

Determination of birth and death rate

The new population model is that difference between the rate of births and the rate of deaths

$\frac{dP}{dt}=\left\{\text{rate of births}\right\}\text{-}\left\{\text{rate of deaths}\right\}$

Consider, bP=rate of births

dP=rate of deaths

Here, the given statement says that the both the rate of births and deaths rates are proportional to the current population

So, we take that the births as b and deaths rate as d

Step 3: Determination of a model for the population

The formula will be

$$\begin{gathered} \frac{dP}{dt}=\left\{\text{rate of births}\right\}\text{-}\left\{\text{rate of deaths}\right\} \\ \frac{dP}{dt}=bP-dP \\ \frac{dP}{dt}=(b-d)P \end{gathered}$$

Hence, the population model of proportional birth and death rates will be$\frac{dP}{dt}=(b-d)P$