Question

Modify the model in problem 3 for net rate at which the population P(t) of a certain kind of fish changes by also assuming that the fish harvested at a constant rate h>0.

Short answer

$\frac{dP}{dt}=bP-d{P}^2-h$

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 harvested fish

The population model from 3, where

b = proportionality constant of birth rate

d = proportionality constant of death rate

The function will be:

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

Consider,

$\begin{array}{l}bP=\text{rateofbirths}\\ d{P}^2=\text{rateofdeaths}\end{array}$

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

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

$\begin{array}{l}bP=\text{rateofbirths}\\ d{P}^2=\text{rateofdeaths}\end{array}$

in above function

$\frac{dP}{dt}=bP-d{P}^2$

Here, the given statement says that the fish is harvested at rate of h>0,

It means $\frac{dP}{dt}$has been decreased by h

Where,

$\frac{dP}{dt}=bP-d{P}^2-h$

Therefore, the population model proportional of fish harvested at constant rate h>0 will be

$\frac{dP}{dt}=bP-d{P}^2-h$