Question

Using the concept of net rate introduced in problem 2,determine a model for a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t.

Short answer

$\frac{dP}{dt}=k_{1}P-k_2{P}^2$

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,

$\begin{array}{l}{k}_{1}P=\text{rateofbirths}\\ k_2{P}^2=\text{rateofdeaths}\end{array}$

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 asK₁ and deaths rate as K₂

Step3: Determination of population model  

The formula will be

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

The population model is:

Applying

$\begin{array}{l}{k}_{1}P=\text{rateofbirths}\\ k_2{P}^2=\text{rateofdeaths}\end{array}$in above function

$\frac{dP}{dt}=k_{1}P-k_2{P}^2$

The population model is proportional to birth and the population model is square to the proportional of death rate.Hence the solution is $\frac{dP}{dt}=k_{1}P-k_2{P}^2$