The authors study the following non-autonomous predator-prey model with harvest efforts for both the prey and the predator population and a time delay due to the gestation of the predator \[ \begin{aligned} \dot x(t)=&x(t)(r(t)-a(t)x(t))-\frac{b(t)x(t)}{m+x(t)}y(t)-h_1(t),\\ \dot y(t)=&y(t)(\frac{c(t)x(t-\tau(t))}{m+x(t-\tau(t))}-d(t))-h_2(t), \end{aligned}\tag{1} \] where \(x(t)\) and \(y(t)\) represent the densities of the prey and the predator populations, respectively; \( a, b, c, d, r, \tau, h_1\) and \(h_2\) are all nonnegative continuously periodic functions with period \(\omega>0\), \(m>0\) is a constant. Here, the prey population \(x\) grows logistically with the intrinsic growth rate \(r(t)\) and carrying capacity \(r(t)/a(t)\), \(d(t)\) is the death rate of the predator, \(b(t)\) is the capturing rate of the predator, \(c(t)/b(t)\) is the conversion rate, \(m>0\) is the half saturation rate of the predator.
Sufficient conditions are derived for the existence of at least two positive periodic solutions to system (1) by using the generalized continuation theorem developed by Gaines and Mawhin [R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag. (1977; Zbl 0339.47031)].