Periodic solutions of a periodic delay predator-prey system

The existence of a positive periodic solution for { d H ( t ) d t = r ( t ) H ( t ) [ 1 − H ( t − τ ( t ) ) K ( t ) ] − α ( t ) H ( t ) P ( t ) ,   d P ( t ) d t = − b ( t ) P ( t ) + β ( t ) P ( t ) H ( t − σ ( t ) ) \begin{equation*} \begin {cases} \frac {\mathrm {d}H(t)}{\mathrm {d}t}=r(t)H(t) \left [1-\frac {H(t-\tau (t))}{K(t)}\right ] -\alpha (t)H(t) P(t),\ \frac {\mathrm {d}P(t)}{\mathrm {d}t}=-b(t)P(t)+\beta (t)P(t)H(t-\sigma (t)) \end{cases} \end{equation*} is established, where r r , K K , α \alpha , b b , β \beta are positive periodic continuous functions with period ω > 0 \omega >0 , and τ \tau , σ \sigma are periodic continuous functions with period ω \omega .