Ecological interpretation of Lotka-Volterra model

Question

I am studying Lotka-Volterra models, specifically one of prey-predator with intraspecific competition. The equation system is:

$$\frac{dx}{dt}= r_x x(1-x-\alpha y)$$ $$\frac{dt}{dt}= r_y y(\beta x + \gamma y -1)$$ where $x$ correspond to preys and $y$ to predators, $$r_x,r_y,\alpha,\beta >0$$ and $$\gamma \in \mathbb{R}$$

I know $r_x$ is the growth tax for preys, $r_y$ is the death tax for predators and $\alpha$, $\beta$ are their interaction taxes, all of them are real positive numbers.

$\gamma$ is the logistic term for predators, but I don't understand why it can be positive or negative. If $\gamma$ is negative, it means competition between predators, but which ecological interpretation has a positive $\gamma$ value?

Thanks in advance.

Answer 1

With negative $\gamma$, this term stands for the reaction $$ 2Y\to Y, $$ environmental pressure leads to a reduction in the population.

With $\gamma$ positive, the term stands for the reaction $$ 2Y\to 3Y, $$ which could for instance mean that any encounter of $2Y$ leads to (in average) an increase in the population

Answer 2

$\gamma y$ , according to Wikipedia, is the "loss rate of the predators due to either natural death or emigration."

Hence, a positive $\gamma$ would correspond to high immigration rates of the predator. (High, as in immigration > natural death)