The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator-prey model with mutual interference and Beddington-DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set-valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system.where s is the threshold that should be appropriately chosen, depending on the problem to be solved. That is, if abundance is below the threshold level, there is no harvest; above the threshold, a constant harvest rate is applied. In order to effectively manage many fields of renewable resource, such as fisheries, grazing (named stock removal), control of non-native predators, species conservation and conflict uses of aquatic vegetation, managers often use TP. As an on-off control or as a special and simple case of variable structure control in the control papers, TP is also applied [13][14][15].However, Leard and Rebaza [16] pointed out that TP is impractical in real-world application. In fact, it would be difficult for managers to implement TP because of time delays and capital constraints. To maintain the sustainable development of the biological system as well as keeping the economic interest of harvesting at an ideal level, Cai and Huang [17] introduced a more practical harvesting management policy called discontinuous harvesting policy (DHP). And the DHP can be defined as a discontinuous function h.x/, which satisfies the following assumption (D1):h is piecewise continuous, that is, h is continuous on R except on a countable set of isolate points f k g, where there exist finite right limits lim s! C k h.s/ , h. C k / and left limits lim s! k h.s/ , h. k /, respectively. Moreover, h has only a finite number of discontinuities in each compact set of R. h is monotonically non-decreasing in R, that is, for any s 1 , s 2 2 R such that s 1 < s 2 and h is continuous at s 1 and s 2 , it results h.From the economic and managerial point of view, the DHP provides a more practical form of harvesting management policy. According to DHP, the managers are allowed to increase the harvesting rate by discontinu...

In this paper, a delayed prey–predator model with discontinuous harvesting policies is investigated, in which the predator population consumes the prey according to Hassell–Varley-type functional response. Under some reasonable assumptions on the discontinuous harvesting functions, the local existence and global existence of positive solution in sense of Filippov to the system are obtained. Based on the functional differential inclusions theory and the topological degree theory of set-valued analysis, a series of useful criteria on existence and non-existence of the positive periodic solution is established for the system. Finally, some numerical examples are given to show the applicability and effectiveness of the obtained results. It is worthy to point out that the discontinuous harvesting policies are superior to continuous harvesting policies, which are usually adopted in previous papers.

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In this paper we prove that the linear transformation y n = n i=0 n i x i , n = 0, 1, 2,. .. preserves Log convexity. As an application, we investigate the sequence S n = n i=0 n i .