Harvesting in logistic equation

Question

So the question is:

Analyze the equation depending on c : $$y'=y(a-by)-c.$$ When does the population die out in time?

My attemp for solution

So to draw phase portrait we need to find first the zeros of the equation(stationary points), so lets do that.

$$ya-by^2 - c=0 \Rightarrow by^2-ay+c =0 \Rightarrow D = a^2-4bc$$

Hence our roots looks as follows :

$$\lambda_{1,2} = \frac{a\pm \sqrt{a^2-4bc}}{2b}.$$

Now we need to analyze the solutions.

Question However i do not know how to proceed from now, do we know anything about a,b? And should we analyze the case when $\sqrt{a^2-4bc} > a$ and $<a$? Or how should the answers vary?

Answer 1

We need to consider all the possible cases.

For example for $a,b,c>0$ and $a^2-4bc\ge 0$ we have

• $\lambda_{1} = \frac{a - \sqrt{a^2-4bc}}{2b}>0$
• $\lambda_{2} = \frac{a + \sqrt{a^2-4bc}}{2b}>\lambda_{1}$

with

• $y'=-by^2+ya-c<0$ for $y(0)<\lambda_1$

thus in this case the population dies out in time when $y(0)<\lambda_1$.