I am reading the article in here. There are few things that I don't understand in it and can someone please help me to clarify these.
First the Lotka Volterra type predator prey is defined as below with $x_1$ being prey and $x_2$ being predators.
Then two functional forms are introduces as
$\phi(x_1)=x_1/(k+x_1) ----------(1)$ and
$\phi (x_1/x_2)=x_1/(kx_2+x_1)-----------(2)$
Then it is said as resource becomes scarce (the value of $x_1$ no longer dominates $k$) equation (2) reflects the intraspecific competition for resources. I don't understand why only equation (2) reflects this and why equation (1) doesn't? How is this decided?
Then it is said that if $b_i x_1x_2$ is replaced in the Lotka Volterra equation with $b_ix_1\phi(x_2)$ then intraspecific competition for prey is only indirectly accounted for through the increased rate at which the prey are depleted. I don't understand what is meant by this. How does it result in an increased rate at which the prey are depleted?
What is it meant by indirectly accounted? Doesn't this impact when $b_i x_1x_2$ is replaced with $b_ix_1\phi(x_2/x_1)$