Lotka-Volterra equations modified

Question

Okay, so I'm just learning about the Lotka-Volterra and the question I have regards the following model:

$dx/dt = x(1-y/2)$

$dy/dt = -y(1 - (x/0.8) + (x^2)/4)$

I need to state what term has been added to the standard Lotka-Volterra equations and the effect it has. Now $dx/dt$ is standard and I want to say $((x^2)/4)$ is the term that has been added to $dy/dt$. Surely this cannot be correct however as 0.8 is not a constant positive integer, which I thought was one of the conditions of the Lotka-Volterra.

I have $dy/dt = -y(1 - (x/4)(5 - x))$ but I'm not sure how this helps.

Many thanks in advance for any help/explanation.

Answer 1

0.8 is a constant, and yes, you are right, $x^2/4$ is the term that has been added to the standard model.

The standard model only contains a birth/death process, f.e. $dx/dt = x$ and mass-action law terms to model the predator-prey relationship.

Answer 2

Interestingly, in this class of differential systems, for every initial condition near the fixed point of smallest abscissa, the behaviour is cyclic, but for every initial condition not inside the closed curve around this fixed point and not on the curves intersecting at the fixed point with largest abscissa, $x(t)\to\infty$ and $y(t)\to0$, which corresponds to an extinction of the population of predators and an explosion of the population of preys.

Below, as an example, the system $$x'=x-xy\quad y'=yx(3-x)-2y$$ The fixed points are $(0,0)$ (saddle), $(1,1)$ (center) and $(2,1)$ (saddle). The level curves are $$C(h):\quad yx^2\exp(-y-3x+x^2/2)=h$$ and the critical (closed) level curve is the loop part of $C(4e^{-5})$.