The model that we are given is a Lotka-Volterra system where the predator $(y)$ and prey $(x)$ follow the logistic growth:
$$\frac{dx}{dt}=fx(1-x/k)-axy,$$ $$\frac{dy}{dt} =-gy+bxy$$
I have to create a model where the predator gets full, so after a certain amount of prey they aren't hungry. I am completely stumped. The models that I have found online are far too complicated than the prof likely expects from us (i.e. Rosenzweig-MacArthur Model, functional growth, etc.), and they account for things like predator handling time?
I thought of a potential equation for the prey model but I am having trouble coming up with a predator equation (let me know if prey model is incorrect or doesn't make sense):
$$\frac{dx}{dt} =fx(1-x/k)-axy(1-c/f), $$ where $c=$prey eaten per predator (average across predator population) and $f=$amount of prey eaten before full
In the original model, the terms $-axy$ and $bxy$ represent the death of prey from being eaten by predators and the resulting births of predators, respectively. So each predator eats prey at a rate proportional to the number of prey ($x$). You want to replace that $x$ by something that will saturate: I would try $x/(c + x)$ where $c$ is a positive constant (don't use $f$ because you already have an $f$ in your equations).
So the equations could become
$$ \eqalign{ \dfrac{dx}{dt} &= f x (1-x/k) - a x y/(c + x)\cr \dfrac{dy}{dt} &= - g y + b x y/(c+x) \cr} $$