Question

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation

\(\frac{{dP}}{{dt}} = {r_0}\left( {1 - \frac{{P\left( t \right)}}{{{P_c}}}} \right)P\left( t \right) - \beta P\left( t \right)\)

Where \({r_0}\) is the birth rate of the fish, \({P_c}\) is the maximum population that the pond can sustain (called the carrying capacity), and \(\beta \) is the percentage of the population that is harvested.

a) What value of \(\frac{{dP}}{{dt}}\) corresponds to a stable population?

b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level.

c) What happens if \(\beta \) is raised to 5%.

Short answer

a)When \(\frac{{dP}}{{dt}} = 0\) then, the population remains stable (which is constant).

b)The stable population level is \(P = 2000\).

c) If \(\beta \) is raised to 5%, then the stable population does not exist.

Step-by-step solution

Determine the value of \(\frac{{dP}}{{dt}}\) corresponds to a stable population

a)

When \(\frac{{dP}}{{dt}} = 0\) then, the population remains stable (which is constant).

Determine the stable population level

b)

A model for the rate of change of the fish population is \(\frac{{dP}}{{dt}} = {r_0}\left( {1 - \frac{{P\left( t \right)}}{{{P_c}}}} \right)P\left( t \right) - \beta P\left( t \right)\).

Evaluate the stable population level as shown below:

\(\begin{aligned}\frac{{dP}}{{dt}} & = 0\\{r_0}\left( {1 - \frac{P}{{{P_c}}}} \right)P - \beta P & = 0\\\beta P & = {r_0}\left( {1 - \frac{P}{{{P_c}}}} \right)P\\\frac{\beta }{{{r_0}}} & = 1 - \frac{P}{{{P_c}}}\\\frac{P}{{{P_c}}} & = 1 - \frac{\beta }{{{r_0}}}\\P = {P_c}\left( {1 - \frac{\beta }{{{r_0}}}} \right)\end{aligned}\)

Take \({P_c} = 10,000,{r_0} = 5\% = 0.05,\) and \(\beta = 4\% = 0.04\) then the stable population level as shown below:

\(\begin{aligned}P & = 10,000\left( {1 - \frac{4}{5}} \right)\\ & = \left( {\frac{{10,000}}{5}} \right)\\ & = 2000\end{aligned}\)

Thus, the stable population level is \(P = 2000\).

Explain what happens if \(\beta \) is raised to 5%

c)

When \(\beta = 0.05\) then

\(\begin{aligned}P & = 10,000\left( {1 - \frac{5}{5}} \right)\\ & = 10,000\left( 0 \right)\\ & = 0\end{aligned}\)

The population is not stable.

Thus, if \(\beta \) is raised to 5% then the stable population does not exist.