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Question

Suppose that the population dynamics of a species obeys a modified version of the logistic differential equation having the following form: $\frac{d N}{d t}=r\left(1-\frac{N}{K}\right)^{2} N$ where $r \neq 0 \text { and } K>0$ Apply the local stability criterion to the equilibrium $\hat{N}=K.$ What do you think your answer means about the stability of this equilibrium?

Solution

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The derivative of the differential equation with respect to $N$ is

$g'(N)=2r \left( 1 - \frac { N } { K } \right) \left(-\frac{2N}{K}+1 \right)$

Now, $g'(K)=2r \left( 1 - \dfrac { K } { K } \right) \left(-\dfrac{2K}{K}+1 \right)=0$.

$\hat{N}$ is locally stable if $g'(\hat{N})<0$ , and $\hat{N}$ is unstable if $g'(\hat{y})>0$. Here $g'(K)=0$ which means that it cannot be said whether $\hat{N}=K$ is locally stable equilibrium or not.

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