Can anyone explain me please what this terms mean one by one and more explicitly like (1- x/K) and mx/b+x2

Question

Actually it's a prey-predator model (for specifically, Bogdanov–Takens bifurcation of a Holling IV prey–predator model with constant-effort harvesting). Please help me to understand these terms so that I can explain it and solve it more easily. I really want to get where these terms come from. Or at least understanding (1- x/K) and mx/b+x2 terms will be enough for me . I searched a lot but couldn't find the explicit explanation. What does the (1-x/K) - (mx)/(b+x^2) mean in the equation: x'= r1x(1-x/K) - (mx)/(b+x^2)- c1x . I know that it's a rate of change of x during the time period (x as a prey in this particular equation) and r1x is the initial population of prey and the rest of the terms effect to change of x. But I'm trying to figure out what are they and how did we write it. Where did they come from. Just wanted to know the meaning of them. That's all

Thanks in advance Prey-predator model with Leslie-Gower and Holling IV schemes with constant-effort harvesting

Answer 1

I am not familiar with this particular model so I won't be able to give you a perfectly detailed answer, but here is some insight.

It is helpful to understand the Logistic function in the context of population growth. It is of the form

$\frac{dP}{dt} = rP(1-\frac{P}{K})$.

This is a model for a population, $P$, with growth at a rate $r$ but with a maximum population $K$.

So we can see, in your model, that each equation has a term like this. $r_1$ and $r_2$ are the prey and predator population growth rates, the prey has a carrying capacity of $K$ while the predator has one of $sx$ (i.e. the more prey that exist, the more food there is to support a larger predator population).

The term $\frac{mx}{b+x^2}y$ causes the prey population to decrease and seems to take into account the fact that prey are hunted and killed off by predators with each predator killing at a rate of $\frac{mx}{b+x^2}$.

The last two terms decrease each population at a rate equal to a constant times their population. This probably accounts from death due to old age or being hunted by some third group.