I'm stuck with a little exercise and cannot find out where I'm wrong. Maybe you can help me.
So we have a differential equation (modified logistic growth):
$\frac{dN}{dt}=k \ N \left(1-\frac{N}{B}\right) -aN$
and the questions "What are the steady states $x^*$ and their linear stabilities?"
I can find the steady states; we set the RHS zero and solve: $N=0$ or $\left( 1-\frac{N}{B}-a\right)=0 \Leftrightarrow N=B(1-a).$ But what is linear stability? I presume that it does not mean continuity, i.e. a small change in $t$ leads to a small change in $N$.
I think it means that a small change in $N$ means that the change in $N$ as $t\rightarrow \infty$ will be zero? But how do I check this rigorously? And what does linearity have to do with this?
-Marie!