| Citation: |
Huilan Wang, Chunhua Ou, Binxiang Dai. THE EXISTENCE AND STABILITY OF ORDER-1 PERIODIC SOLUTIONS FOR AN IMPULSIVE KOLMOGOROV PREDATOR-PREY MODEL WITH NON-SELECTIVE HARVESTING[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1348-1370. doi: 10.11948/20200181
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THE EXISTENCE AND STABILITY OF ORDER-1 PERIODIC SOLUTIONS FOR AN IMPULSIVE KOLMOGOROV PREDATOR-PREY MODEL WITH NON-SELECTIVE HARVESTING
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Abstract
In this paper, we focus on a general state-dependent Kolmogorov predator-prey model subject to non-selective harvesting along with delivery. Certain criteria are established for the existence, non-existence and multiplicity of order-1 impulsive periodic solutions to the system. Based on the geometric phase analysis and the method of Poincaré map or successor function with Bendixson domain theory, three typical types of Bendixson domains (i.e., Parallel Domain, Sub-parallel Domain and Semi-ring Domain) are introduced to deal with the discontinuity of the Poincaré map or successor function. We incorporate two discriminants $ \Delta_1 $ and $ \Delta_2 $ to link with the existence, non-existence and multiplicity as well as the stability of order-1 periodic solutions. At the same time, we locate the order-1 periodic solutions with the help of three characteristic points and the parameters ratio of delivery over harvesting. The results show that there must exist at least one order-1 periodic solution when the trajectory, that is tangent to the mapping line, can hit the impulsive line. While the trajectory tangent to the mapping line cannot hit the impulsive line, there is not necessary the existence of an order-1 periodic solution, which means the impulsive control may be invalid after finite times stimulation or suppression. In conclusion, we reveal that the delivery can prevent the predator from extinction and stabilize the order-1 periodic solution.
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References
[1] G. Butler, S. Hsu and P. Waltman, Coexistence ofcompeting predators in achemastat. Journal of Mathematical Biology., 1983, 17(2), 133-151. doi: 10.1007/BF00305755 [2] D. Bainov and P. Simeonov, Impulsive differential equations: periodic solutions and applications, Longman Scientific & Technical, New York, 1993. [3] J. Chavez, D. Jungmann and S. Siegmund, A comparative study of integrated pest management strategies based on impulsive control, Journal of Biological Dynamics, 2018, 12(1), 318-341. doi: 10.1080/17513758.2018.1446551 [4] K. Chakraborty, S. Das and T. K. Kar, On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations, Applied Mathematics and Computation, 2013, 221(C), 581-597. [5] R. Hakl, M. Pinto and V. Tkachenko, Almost periodic evolution systems with impulse action at state-dependent moments, Journal of Mathematical Analysis and Application, 2017, 446(1), 1030-1045. doi: 10.1016/j.jmaa.2016.09.024 [6] G. Jiang, Q. Lu and L. Qian, Complex dynamics of a Holling type Ⅱ prey-predator system with state feedback control, Chaos, Solitions and Fractals, 2007, 31(2), 448-461. doi: 10.1016/j.chaos.2005.09.077 [7] T. Kar and S. Chattopadhyay, A focus on long-run sustainability of a havested prey predator system in the presence of alternative prey, Competes Rendus Biologies, 2010, 333(11-12), 841-849. doi: 10.1016/j.crvi.2010.09.001 [8] L. Nie, Z. Teng and B. Guo, A state dependent pulse control strategy for a SIRS epidemic system, Bulletin of Mathematical Biology, 2013, 75(10), 1697-1715. doi: 10.1007/s11538-013-9865-y [9] L. Nie, J. Shen and C. Yang, Dynamic behavior analysis of SIVS epidemic models with state-dependent pulse vaccination, Nonlinear analysis- Hybrid System, 2018, 27, 258-270. doi: 10.1016/j.nahs.2017.08.004 [10] P. Simeonov and D. Bainov, Orbital stability of periodic solutions of autonomous systems with impulse effect, International Journal of Systems Science, 1989, 19(12), 2561-2585. [11] R. Smith and E. Schwartz, Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: How often should you vaccinate and how strong should the vaccine be?, Mathematical Biosciences, 2008, 212(2), 180-187. doi: 10.1016/j.mbs.2008.02.001 [12] X. Tang and X. Fu, On period-k solution for a population system with state-dependent impulsive effect, Journal of Applied Analysis and Computation, 2017, 7(2), 439-454. doi: 10.11948/2017028 [13] S. Tang and R. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, Mathematical Biology, 2015, 50(3), 257-292. [14] S. Tang, W. Pang and R. Cheke, Global dynamics of a state-dependent feedback control system, Advances in Difference Equations, 2015. DOI: 10.1186/s13662-015-0661-x. [15] S. Tang, X. Tan and J. Yang, Periodic solution bifurcation and spiking dynamics of impacting predator-prey dynamical model, International Journal of Bifurcation and Chaos, 2018, 28(12), 1850147. doi: 10.1142/S021812741850147X [16] J. Wang, H. Chen, Y. Li and X. Zhang, The geometrical analysis of a predator-prey model with multi-state dependent impulses, Journal of Applied Analysis and Computation, 2018, 8(2), 427-442. doi: 10.11948/2018.427 [17] H. Wang, B. Dai and Q. Xiao, Existence of order-1 periodic solutions for a viral infection model with state-dependent impulsive control, Advances in Difference Equations, 2019. DOI: 10.1186/s13662-019-1967-x. [18] A. Wang, Y. Xiao and R. Smith, Using non-smooth models to determine thresholds for microbial pest management, Journal of Mathematical Biology, 2019, 78(5), 1389-1424. doi: 10.1007/s00285-018-1313-z [19] Q. Xiao, B. Dai, B. Xu and L. Bao, Homoclinic bifurcation for a general state-dependent Kolmogorov type predator-prey model with harvesting, Nonlinear Analysis: Real World Applications, 2015, 26, 263-273. doi: 10.1016/j.nonrwa.2015.05.012 [20] Q. Xiao and B. Dai, Dynamics of an impulsive predator-prey logistic population model with state-dependent, Applied Mathematics and Computation, 2015, 259, 220-230. doi: 10.1016/j.amc.2015.02.061 [21] G. Zeng, L. Chen and L. Sun, Existence of periodic solution of order one of planar impulsive autonomous system, Journal of Computational and Applied Mathematics, 2006, 186(2), 466-481. doi: 10.1016/j.cam.2005.03.003 [22] Q. Zhang, B. Tang and S. Tang, Vaccination threshold size and backward bifurcation of SIR model with state-dependent pulse control, Journal of Theoretical Biology, 2018, 455, 75-85. doi: 10.1016/j.jtbi.2018.07.010 -
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Huilan Wang, Chunhua Ou, Binxiang Dai. THE EXISTENCE AND STABILITY OF ORDER-1 PERIODIC SOLUTIONS FOR AN IMPULSIVE KOLMOGOROV PREDATOR-PREY MODEL WITH NON-SELECTIVE HARVESTING[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1348-1370. doi: 10.11948/20200181
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Figure 1. The positive invariant set
$ \Omega $ and four parts of it, and the direction of the manifolds, where$ c_1: F(x, y) = 0 $ or$ y = \varphi_1(x) $ ,$ c_2:G(x, y) = 0 $ or$ y = \varphi_2(x) $ and$ l_1: x = K $ and$ l_2: y = \sigma $ . -
Figure 2. The location of the three characteristic points
$ T, W, R $ . -
Figure 3. (a) There is no order-1 periodic solution in
$ D $ even if$ S(Y_{1})\cdot S(Y_{2})<0 $ holds (b) There is an order-1 periodic solution in$ D $ in spite of$ S(Y_1)\cdot S(Y_2)>0 $ (c) There may exist two order-1 periodic solutions in$ D $ in view of$ S(Y_1)\cdot S(Y_2)<0 $ and$ S(Y_{1})\cdot S(Y_{3})<0 $ . -
Figure 4. The illustration for three categories of Bendexion domain.
$ N $ is a tangent line of the orbit$ L_2 $ at$ B $ in (b), and$ M $ is a tangent line to the orbit$ L_2 $ at$ \bar{B} $ in (c). -
Figure 5. If
$ \bar(Y) = q_{0} $ , then$ I(\bar{Y}) = \bar{Y} $ ; If$ \bar{Y}>q_{0} $ , then$ q_{0}<I(\bar{Y})<\bar{Y} $ ; If$ \bar{Y}<q_{0} $ , then$ \bar{Y}< I(\bar{Y})<q_{0} $ , where$ q_0 = \frac{\tau }{q} $ . -
Figure 6. The illustration for the location of
$ W, W^{+}, W^{-} $ , and$ \overline{W^{+}} $ when$ O^{-}(W) $ intersects with$ N $ at least two points. -
Figure 7. Two possible cases of that the trajectory
$ O^{-}(W) $ intersects with$ N $ .