Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

Cai, Zhiping; Huang, Lihong; Zhang, Lingling; Hu, Xiaoyun

doi:10.1002/mma.3379

Cited by 8 publications

(7 citation statements)

References 42 publications

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“…We adopt a reasonable definition from Filippov (see previous studies()) by considering the following delayed differential equation with discontinuous right‐hand side

z^{'} false(t false) = f false(t, z false(t - τ false), z false(t false) false) .

…”

Section: Preliminaries and Lemmasmentioning

confidence: 99%

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

Huang

2019

Math Methods in App Sciences

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This paper investigates the periodic solution of a delayed Beddington-DeAngelis (BD) type predator-prey model with discontinuous control strategy. Firstly, the regularity and visibility analysis of the delayed predator-prey model is carried out by using the principle of differential inclusion. Secondly, the positiveness and boundeness of the solution is discussed by employing the comparison theorem. Based on the boundary conditions of the model and the Mawhin-like coincidence theorem, it is shown that the solution of the delayed BD system is asymptotically stable in finite time. Furthermore, it is found that there exists at least one periodic solution of the nonautonomous delayed predator-prey model by using the principle of topological degree and set value mapping. Specially, when the nonautonomous delayed BD system degenerates into an autonomous system, some criteria are obtained to guarantee the convergence behavior of the harvesting solutions for the corresponding autonomous delayed BD system.Finally, numerical examples are given to demonstrate the applicability and effectiveness of main results. It is worthy to point out that the discontinuous control strategy is superior to the continuous harvesting policies adopted in existing literature. KEYWORDSdelayed predator-prey systems, discontinuous control strategy, functional differential inclusion, global asymptotic stability, periodic solution 4498

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“…We adopt a reasonable definition from Filippov (see previous studies()) by considering the following delayed differential equation with discontinuous right‐hand side

z^{'} false(t false) = f false(t, z false(t - τ false), z false(t false) false) .

…”

Section: Preliminaries and Lemmasmentioning

confidence: 99%

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

Huang

2019

Math Methods in App Sciences

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“…Next, we specify what is meant by a solution of delayed differential Equation (1.3) with a discontinuous right-hand side. For more details, please see the books [28,29] and the papers [17,[30][31][32].…”

Section: Definition 21mentioning

confidence: 99%

“…Based on the properties of upper semi-continuous set-valued map with nonempty compact convex values and the continuation theorem of differential equation with discontinuous right-hand side, we obtain the positivity and viability of solution for discontinuous biological system with positive initial conditions. It should be pointed out that the main idea of the positivity and viability of solution for discontinuous biological system arises from Wang, Huang, Guo, and Cai (see [17,[30][31][32]35] and their references).…”

Section: Remarkmentioning

confidence: 99%

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

Luo

Wang

2016

Math Methods in App Sciences

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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...

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“…Generally speaking, there are two types of discontinuous predator-prey models, one is a delayed predator-prey model with control strategy, and the other is a nonautonomous system with control strategy ([4, 32, 9, 42, 27, 14, 17, 20, 36]). Cai et al [2,3] studied the periodic dynamical behavior of a class of delayed Filippov system under the framework of differential inclusion in set-valued analysis. Duan et al [9,7] considered a delay Lasota-Wazewska model with discontinuous harvesting policy.…”

mentioning

confidence: 99%

“…Although the periodic solution of a predator-prey model ( [2,3,33,32,37,13,15,18,40,23,24,16]) was extensively studied, to the best of our knowledge, the general delayed predator-prey model with discontinuous prey harvesting control has not yet been considered. Due to the characteristic of discontinuous harvesting control strategy, the existing results obtained for the optimal continuous harvesting policy, threshold policy and weighted escapement policy cannot be directly applied to the general delayed predator-prey model with discontinuous harvesting control strategy.…”

mentioning

confidence: 99%

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

Li¹,

Huang²,

Ji³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This paper studies the solution behaviour of a general delayed predator-prey model with discontinuous prey control strategy. The positiveness and boundeness of the solution of the system is firstly investigated using the comparison theorem. Then the sufficient conditions are derived for the existence of positive periodic solutions using the differential inclusion theory and the topological degree theory. Furthermore, the positive periodic solution is proved to be globally exponentially stable by employing the generalized Lyapunov approach. The global finite-time convergence is also discussed for the system state. Finally, the numerical simulations of four examples are given to validate the correctness of the theoretical results.

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Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

Cited by 8 publications

References 42 publications

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

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