First, we have these two Lotka-Volterra equations for prey and predators, respectively:

$$ frac {dx} {dt} = r_ {x} x (1- alpha y) $$

$$ frac {dy} {dt} = r_ {y} y ( beta x -1) $$

$$ r_ {x}, r_ {y}, alpha, beta gt 0 $$

These equations mean the predator-prey model without interaction between species of the same condition. If there were competition between prey and predators, the equations would be:

$$ frac {dx} {dt} = r_ {x} x (1-x- alpha y) $$

$$ frac {dy} {dt} = r_ {y} y ( beta x + gamma y-1) $$

$$ r_ {x}, r_ {y}, alpha, beta gt 0, gamma in R $$

My question is: what should I do to change these equations if I wanted to introduce more conditions in addition to competition between species such as, for example, the life expectancy of both species, parasitism, diseases, lack of food according to the season? etc.?

Thanks for the help!