Existence and uniqueness of limit cycles on a cubic Kolmogorov differential system in the predator-prey relation (Q1118736)

The author analyses qualitatively a cubic differential system which has ecological significance (the predator-prey relation): \[ dx/dt = x(a_0 + a_1x - a_3 x^2 - a_4 y + a_5 xy),\quad dy/dt = y(x-1 )(1+by). \] The author's results can be shown by following tabel: \[ \vbox{\settabs 3 \columns \+case &condition &the number of\cr \+ &&limit cycle\cr \+ &&in 1st quadrant\cr \+\(a_5=0\) &1) \(p\geq0\), &0\cr \+ &2) \(p>0\), \(| p| << 1\); &1\cr \+\(a_5>0\) &1) \(q>0\), \(r<0\), &0\cr \+ &2) \(a_3 a_0^{-1} q^2 > r > 0\), \(| r| << 1\), &1\cr \+ &3) \(q>0\), \(r>a_3 a_0^{-1}q^2\); &0\cr \+\(a_5<0\) &1) \(q>0\), \(r\geq0\), &0\cr \+ &2) \(q>0\), \(r<0\), \(| r| << 1\) &1\cr} \] where \(p=a_0^{-1} (2a_3 - a_1)\), \(q=a_3^{-1}a_4-1\), \(r = a_0^{-1}(a_1+a_3) + a_4a_5^{-1}(1-2a_3a_0^{-1})+1.\)