Answer to Question #163071 in Differential Equations for Ajay

Question

A park has a stable population of birds. Prior to this situation, the birds’ population

increased from an initial low level. When the population of birds was 1000, the

proportionate birth rate was 40% per year and the proportionate death rate was 5% per

year. When the population was 3,000, the proportionate birth rate was 30% and the

proportionate death rate was 10%. Consider the population model under the following

assumptions:

(i) There is no migration and no exploitation.

(ii) The proportionate birth rate is a decreasing linear function of the population.

(iii) The proportionate death rate is an increasing linear function of the population.

Show that

The population grows according to the logistic model.

Find the stable population size.

If the shooting of birds is allowed at the rate of 15% of the population per year, find the

new equilibrium population.

Answer 1

$1)\ \frac{dN}{dt}=(\beta-\delta)N\\ N(t)\text{ --- population size}\\ \beta(N)\text{ --- birth rate}\\ \delta(N)\text{ --- death rate}\\ N=1000\Longrightarrow \beta=40\\ N=3000\Longrightarrow \beta=30\\ \beta(N)=-0.005N+45\\ N=1000\Longrightarrow \delta=5\\ N=3000\Longrightarrow \delta=10\\ \delta(N)=-0.0025N+2.5\\ \beta-\delta=-0.0075N+42.5\\ \frac{dN}{dt}=(-0.0075N+42.5)N\\ \frac{dN}{dt}=0\\ (-0.0075N+42.5)N=0\\ \text{Equilibrium points}:\\ N=0,\ N\approx 5667\\ 2)\ \frac{dN}{dt}=(-0.0075N+42.5-15)N\\ \frac{dN}{dt}=(-0.0075N+27.5)N\\ (-0.0075N+27.5)N=0\\ \text{Equilibrium points}:\\ N=0,\ N\approx 3667$