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Firstly, we have these two Lotka-Volterra equations for prey and predator, 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 with no interaction between species of the same condition. If there were competition between the preys and the 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 do I have to do to change these equations if I would to introduce more conditions in addition to competence between species such as, for expample, life expectancy of both species, parasitism, diseases, lack of food depending on the season, etc. ?

Thank you for the help!

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I will provide examples in each case. life expectancy of both species, parasitism, diseases, lack of food depending on the season

(1) Life expectancy: this is usually denotes by a death term. For example, $x' = bx-dx$. In this case, the life expectancy of the $x$ population is $\frac{1}{d}$, where $d$ is the (exponential) death rate. To incorporate this into your population, just add the death terms in both populations.

(2) Parasitism: you can use another compartment $z(t)$, which represents the parasitic species. The specific interactions between the parasite and its host will depend. For example, you can have $azy$ where a is the rate at which the parasite consumes nutrition from its host. Then $z'(t) = azy$ and add $-azy$ into the $y'(t)$ expression.

(3) Disease: you can either try to do this using the epidemiology-type model (SIR model) or implicitly (by having disease-effect term).

(4) Lack of food depending on the season: this can be modeled explicitly by having a compartment $n(t)$ that represents the available nutrition. Then model the growth of $x(t)$ and/or $y(t)$ with respect to the available nutrition. A reasonable approach is a cell-quota model. To incorporate the seasonal effect, you can use something like $n'(t) = sin(at)$, where the periodicity of the sine function is used to represents seasonality.

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