Imagine two situations (prey and predator) with the logistic growth :

1. Prey, changes fast:

2. Predator, changes very slowly:

$K$ is prey carrying capacity
$\beta$ is capture rate of searching predators
$\delta$ is predator death rate
$\gamma$ is conversion rate - if predator eats one prey, it turns it into $\gamma$ offpsring
$r$ is prey growth rate

Since the number of predators changes slowly, we can sort of consider $P$ as a constant in formula 1. This gives us the prey "quasi-equilibrium" ${\hat{N}}_2=\frac{r-\beta P}{r}K$, if $r>\beta P$.

(using same method as Harvested Logistic Model)

For the predator function, if the prey can be assumed to always be close to its quasi-equilibrium, we can model the predator change over this slower time (years for example):

This last one looks like the logistic function again. So we know that we have the intrinsic growth rate $\gamma \beta K-\delta$ and the carrying capacity $\frac{r}{\gamma {\beta }^{2}K}(\gamma \beta K-\delta )$

This shows that the trivial equilibrium ${\hat{P}}_1=0$ is stable if $\delta >\gamma \beta K$, meaning the predator population will always trend to zero if this is met.

The predator is only viable if $\gamma \beta K>\delta$ - they need to be able to catch enough prey, produce enough offspring and prey needs a large enough carrying capacity to exceed their death rate.

Predator population model

left=-0.5
top=5
---
g = 1
B = 1
K = 3
d = 1.2
R = 0.8

y=(gBK-d) (1-x/((R/gB^2K)gBK-d))x