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 $r \neq 0$ and $K>0$ (a) Show that $\hat{N}=0$ and $\hat{N}=K$ are equilibria. (b) For which values of $r$ is the equilibrium $\hat{N}=0$ unstable? (b) For which values of $r$ is the equilibrium $\hat{N}=0$ unstable? (c) Apply the local stability criterion to the equilibrium $\hat{N}=K .$ What do you think your answer means about the stability of this equilibrium? (Note: This is an example in which the local stability criterion is inconclusive.) (d) Construct two phase plots, one for the case where $r>0$ and the other for $r<0,$ and determine the stability of $\hat{N}=K$ in each case. Does the answer match your reasoning in part $(c) ?$

          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 $r \neq 0$ and $K>0$
(a) Show that $\hat{N}=0$ and $\hat{N}=K$ are equilibria.
(b) For which values of $r$ is the equilibrium $\hat{N}=0$ unstable?
(b) For which values of $r$ is the equilibrium $\hat{N}=0$ unstable?
(c) Apply the local stability criterion to the equilibrium $\hat{N}=K .$ What do you think your answer means about the stability of this equilibrium? (Note: This is an example in which the local stability criterion is inconclusive.)
(d) Construct two phase plots, one for the case where $r>0$ and the other for $r<0,$ and determine the stability of $\hat{N}=K$ in each case. Does the answer match your reasoning in part $(c) ?$

Added by Joann R.

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