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?


We need to consider all the possible cases.

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

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


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

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

  • $\begingroup$ Thanks, i have a quaestion, how did you know that we need to analize only the positive cases? $\endgroup$ – user557550 Jun 15 '18 at 16:41
  • $\begingroup$ @MariyaKav We can also analize all possible cases but they are a lot if we don't have any limitation. In any case we can assume $c>0$ for the harvesting but then we should consider the cases $b>0$ and $b<0$ and for each one $a>0$ and $a<0$. The cases become even more if we need also to consider the values $a=0$ or $b=0$. $\endgroup$ – gimusi Jun 15 '18 at 16:44
  • $\begingroup$ @MariyaKav Once we have fixed $a,b,c$ the analysis is simple. Do you have any limitation on thise values? $\endgroup$ – gimusi Jun 15 '18 at 16:45
  • $\begingroup$ No there are no limitations in an exercise, it is only written that analize depending on conctrant c.. So then we can assume that a,b are positive and analize when c is negative or positive?? Or? $\endgroup$ – user557550 Jun 15 '18 at 17:40
  • $\begingroup$ @MariyaKav As you can see in the fisrt example of the link given in the comment, the solution is obtained assuming $a,b,c>0$, thus I think that it is a reasonable assumption. $\endgroup$ – gimusi Jun 15 '18 at 18:02

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