I'm reading an economics paper, and have stumbled upon this differential equation system:
$\dot x = -\beta xy^\alpha$
$\dot y = \beta xy^\alpha - \frac{1}{\sigma}y$
I haven't taken any classes on differential equations, but would like to read up on this system. Could someone offer me some pointers?
The paper is Financial crisis: An attempt of mathematical modelling by Andrei Korobeinikov
These kinds of models are called "compartmental" $-$ but other names exist $-$, because the variables $x,y$ represent separate compartments (chemical species, capitals of different entities, infected/recovered/dead people in epidemiology, etc.).
The basic idea is that the variation of the compartments is given by the derivatives of the variables. In the present case, $x$ and $y$ "interact", which results in the decreases of $x$ and the increase of $y$ (assuming positive $\alpha,\beta$); this interaction is modelled by the term $\beta xy^\alpha$, where $\alpha$ somewhat "tunes" the influence of $y$ ($\alpha=0$ means for example that the interaction is independent of the value of $y$). Finally, on top of that, the term $-\frac{1}{\sigma}y$ means that $y$ decreases independently of $x$ at a rate $\frac{1}{\sigma}$ (N.B. : it is not a linear rate, but an exponential rate).
Finally, it is to be noticed that the case where $\dot{x} = \dot{y} = 0$ corresponds to stationary solutions, which don't evolve over time.