Dynamics of a Competitive Lotka–Volterra Systems in $\mathbb{R}^{3}$

Abstract

We describe the dynamics of the 3-dimensional competitive Lotka–Volterra systems $$ \dot{x}=x(a-x-y-z),\quad \dot{y}=y(b-x-y-z),\quad \dot{z}=z(c-x-y-z), $$ providing the phase portraits for all the values of the parameters $a$ , $b$ and $c$ with $0< a< b< c$ in the positive octant of the Poincaré ball.