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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

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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

$$ \eqalign{ \dfrac{dx}{dt} &= f x (1-x/k) - a x y/(c + x)\cr \dfrac{dy}{dt} &= - g y + b x y/(c+x) \cr} $$

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  • $\begingroup$ Why would the predator birth rate also be affected by saturation? If the predators are full, they should still be able to reproduce. If anything they'd have more offspring rather than less. $\endgroup$ Apr 5, 2020 at 3:10
  • $\begingroup$ Also. why did you decide c+x for saturation? Is c how full they are at the moment or how much they need to eat to be full? Would we not have to account for that with two separate variables? Sorry if this is a lot of questions, I am making sure that I understand your solution. $\endgroup$ Apr 5, 2020 at 3:11
  • $\begingroup$ They do reproduce more, but only to a point. $x/(c+x)$ goes to a limit of $1$ as $x \to \infty$. $\endgroup$ Apr 5, 2020 at 3:59

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