# Coming up with a predator-prey model that accounts for predator satiation

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).
\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}
• They do reproduce more, but only to a point. $x/(c+x)$ goes to a limit of $1$ as $x \to \infty$. Apr 5, 2020 at 3:59