In my following question, I am referring to an article dealing with reaction-diffusion equations. I'll give screenshots of the parts of the text I am dealing with.
The setup is the following:
My question is how to see these described "jumps" within the dynamics: How do one see these jumps from (3.2)? For example: How can one see that, starting with initial values $(u_0,v_0)$ with $v_0\geq S(u_0)$, the singular flow jumps instantaneously to the point $(u_1,v_0)$ where $v_0=f(u_1), u_1\leq u_{\text{min}}$? Or how does one see from (3.2) that from point $(u_{\text{max}},v_{\text{max}})$ we jump instantaneously to the point $(u_2,v_{\text{max}})$ where $v_{\text{max}}=f(u_2), u_2>0$?
I think, here, the case $\varepsilon=0$ is considered. Thus, (3.2) is $$ 0=F(u,v)=u(u-a)(1-u)-v,~~\frac{dv}{dt}=G(u,v). $$ Now, $0=F(u,v)=u(u-a)(1-u)-v$ means that $v=u(u-a)(1-u)=f(u)$, so the dynamics only takes place on $f(u)$. But thats all I can say.
First of all take $\varepsilon=0$ (and don't think in terms of the limit behavior in $\varepsilon$).
The reason for the jumps is that the first equation now takes the form $F(u,v)=0$. This causes that if we don't start in the solution of this equation, we must jumpt to them (this is basically the last sentence in your question).