Applied Mathematics & Optimization

, Volume 75, Issue 2, pp 229–251

# Optimal Harvesting in a Periodic Food Chain Model with Size Structures in Predators

Article

## Abstract

In this paper, we investigate a periodic food chain model with harvesting, where the predators have size structures and are described by first-order partial differential equations. First, we establish the existence of a unique non-negative solution by using the Banach fixed point theorem. Then, we provide optimality conditions by means of normal cone and adjoint system. Finally, we derive the existence of an optimal strategy by means of Ekeland’s variational principle. Here the objective functional represents the net economic benefit yielded from harvesting.

### Keywords

Size-structure Predator–prey model Optimal harvesting

### Mathematics Subject Classification

49K20 49K15 35F50 92D25

## Notes

### Acknowledgments

The authors would like to thank the anonymous reviewer for her/his constructive comments which help to improve the presentation of this paper. This work was supported partially by the National Natural Science Foundation of China (No. 11371313) and by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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