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I am reading Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, 2nd Ed. (Strogatz 2015). I came across a problem involving the Allee Effect which I have not been able to solve. Problem: Strogatz problem

I tried comparing terms with the Allee effect as stated on Wikipedia, allee effect equation but it is still not clear to me how to solve this. Can someone please help?

Since my last post, I have tried plotting several values for the parameters, and I solved for the zeros of this equation. My guess, is that I am supposed to qualitatively analyze this plot and see that it is maximized for intermediate values of N. I am including a picture from Mathematica of my work. Please let me know if I am on the wrong track. enter image description here

Anne
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You're not interpreting that Wikipedia page correctly. That equation provides an example of the Allee effect, but is not exclusive.

The Allee effect occurs when the non-zero steady state of a system is not globally attractive, so that such populations may be stable at non-zero population levels for a long time, but are unable to recover once the population density drops below a certain threshold.

In terms of your problem, the tricky bit (conditions for global attraction) is done for you as it instructs you to find parameters so that $\frac{\dot{N}}{N}$ is maximum for intermediate (positive) $N$.

So when does $f(N) = r - a(N-b)^2$ have a maximum admitted at some positive $N$? This is a quadratic, so simple mathematics is enough.

  • $a>0$ is needed to ensure the parabola admits a maximum
  • $b>0$ is needed for the maximum is attained at positive $N$.
  • $r>0$ is also needed to ensure the maximum is positive. This is not clear from the information in the question, but the non-trivial steady states involve $\sqrt{\frac{r}{a}}$ and so $r > 0$ for these steady states to even exist.

So overall quite boring conditions and not a very satisfying question.

In general, a population $x$ with reproductive rate $b(x)$ and death rate $d(x)$ with dynamics $$\frac{dx}{dt} = x(b(x) - d(x))$$ displays the Allee effect if $b(0) < d(0)$. This is easy to prove with a basic grasp of ODE stability.

PhysicsMathsLove
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