An impulsive two-prey one-predator system with seasonal effects
The author considers the following impulsive system \[ \begin{aligned} x_1' &= x_1(t)\Biggl(a_1+ \lambda_1\sin(\omega_1 t)- b_1 x_1(t)- \gamma_1 x_2(t)- {e_1 y(t)\over 1+ c_1 x^2_1(t)}\Biggr),\\ x_2' &= x_2(t) \Biggl(a_2+ \lambda_2\sin(\omega_2 t)- b_2 x_2(t)- \gamma_2 x_1(t)- {e_2 y(t)\over 1+ c_2 x^2_2(t)}\Biggr),\\ y_1'(t) &= y(t)\Biggl(- a_3+ {\beta_1 x_1(t)\over 1+ c_1 x^2_1(t)}+ {\beta_2 x_2(t)\over 1+ c_2 x^2_2(t)}\Biggr),\end{aligned} \] \[ t\neq nT,\quad t\neq (n+\tau- 1)T,\tag{1} \] \[ \Delta x_1(t) = -p_1 x_1(t),\quad \Delta x_2(t) = -p_2 x_2(t),\quad t= (n+\tau- 1)T, \] \[ \Delta y(t)= -p_3 y(t),\quad \Delta x_1(t)= 0,\quad \Delta x_2(t)= 0,\quad t= nT, \] \[ \Delta y(t)= q, \] \[ (x_1(0^+), x_2(0^+), y(0^+))= (x_{01}, x_{02}, y_0). \] Sufficient conditions for the existence of a two-prey-free periodic solution of (1) and its global asymptotical stability are derived.
Holling-type IV two competitive-prey one-predator system
scientific article; zbMATH DE number 5637394
scientific article