Multiple positive periodic solutions for a food-limited two-species ratio-dependent predator-prey patch system with delay and harvesting

By using Mawhin's coincidence degree theory, this paper establishes some sufficient conditions on the existence of four positive periodic solutions for a food-limited two-species ratio-dependent predator-prey patch system with delay and harvesting. Some novel techniques are employed to obtain the appropriate a priori estimates. An example is given to illustrate our results.