Positive periodic solution for a two-species ratio-dependent predator-prey system with time delay and impulse
The paper deals with a predator-prey model with delay and impulses of the form \[ \begin{aligned} \dot{x}(t)&=x(t)\left(b_1(t)-a_1(t)x(t)-{{c(t)y(t)}\over{m_1(t)y(t)+x(t)}}\right),\\ \dot{y}(t)&=y(t)\left(-b_2(t)+{{a_2(t)x(t-\tau)}\over{m_2(t)y(t-\tau)+x(t-\tau)}}\right), \quad\text{for }t\neq t_k,\end{aligned} \] \[ \begin{aligned} x(t^+_k)-x(t^-_k)& =c_kx(t_k), \qquad y(t^+_k)-y(t^-_k)= d_ky(t_k), \quad (x(0+), y(0+))=(x_0,y_0),\\ (x(t), y(t))&=(\varphi_1(t), \varphi_2(t)),\text{ for }-\tau\leq t\leq 0.\end{aligned} \] The authors apply the continuation fixed-point theorem of coincindence degree theory to provide sufficient conditions for the existence of a periodic solution of the problem.
delay differential equations with impulses
scientific article; zbMATH DE number 5078960