| Citation: |
Yue Yang, Yancong Xu, Libin Rong. GLOBAL BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH HARVESTING AND STOCKING[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 788-816. doi: 10.11948/20250139
|
GLOBAL BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH HARVESTING AND STOCKING
-
Abstract
Predator-prey interactions play a fundamental role in ecological dynamics and are often influenced by external factors such as harvesting and stocking. Understanding how these interventions impact population stability and oscillatory behavior is crucial for effective resource management and conservation efforts. In this paper, we investigate a two-species Leslie predator-prey model incorporating constant-yield harvesting and stocking terms for both species. Through a detailed bifurcation analysis, we unveil the mechanisms behind the suppression or persistence of multiple oscillations in the system. We demonstrate that the model, under harvesting or stocking, exhibits cusp bifurcations of codimension 2 and 3, leading to a rich sequence of bifurcations, including saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension 2 and 3, and Hopf bifurcation of codimension at least 3. In particular, we focus on the role of the conversion rate and analyze its influence on the system's transition between oscillatory and non-oscillatory coexistence states. Our findings suggest that indiscriminate stocking or harvesting of both species is not necessarily beneficial for population persistence; rather, appropriate management strategies should be designed according to specific parameter conditions. Finally, numerical simulations, including bifurcation diagrams and phase portraits, are presented, which provide valuable insights and potential applications in ecological management and conservation biology.
-
-
References
[1] R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funct. Anal. Appl., 1975, 9(2), 144–145. doi: 10.1007/BF01075453 [2] R. I. Bogdanov, Bifurcation of the limit cycle of a family of plane vector field, Sel. Math. Sov., 1984, 1, 373–387. [3] F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 1979, 7(4), 319–337. doi: 10.1007/BF00275152 [4] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 1979, 8(1), 55–71. doi: 10.1007/BF00280586 [5] F. Brauer and A. C. Soudack, Constant-rate stocking of predator-prey systems, J. Math. Biol., 1981, 11(1), 1–14. [6] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 1982, 12(1), 101–114. doi: 10.1007/BF00275206 [7] S. Chen, J. Wei and J. Zhang, Dynamics of a diffusive predator-prey model: The effect of conversion rate, J. Dyn. Differ. Equ., 2018, 30(4), 1683–1701. doi: 10.1007/s10884-017-9618-y [8] G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 1998, 58(1), 193–210. doi: 10.1137/S0036139994275799 [9] E. J. Doedel, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. A. Kuznetsov, B. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, 2007. [10] R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type Ⅲ, J. Diff. Eqs., 2010, 249(9), 2316–2356. doi: 10.1016/j.jde.2010.06.021 [11] W. L. Hogarth, J. Norbury, I. Cunning and K. Sommers, Stability of a predator-prey model with harvesting, Ecological Modelling, 1992, 62(1–3), 83–106. doi: 10.1016/0304-3800(92)90083-Q [12] S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 1978, 39(1–2), 1–10. [13] Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalised Holling type Ⅲ functional response, J. Dyn. Differ. Equ., 2008, 20(3), 535–571. doi: 10.1007/s10884-008-9102-9 [14] S. Laurin and C. Rousseau, Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type Ⅲ, J. Diff. Eqs., 2011, 251(10), 2980–2986. doi: 10.1016/j.jde.2011.04.017 [15] C. Li, J. Li and Z. Ma, Codimension 3 BT bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 2015, 20(4), 1107–1116. doi: 10.3934/dcdsb.2015.20.1107 [16] C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Diff. Eqs., 1989, 79(1), 132–167. doi: 10.1016/0022-0396(89)90117-4 [17] Y. Liu, X. Feng, S. Ruan and J. Yu, Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting, Ⅱ: Existence of two periodic solutions, J. Diff. Eqs., 2024, 388, 253–285. doi: 10.1016/j.jde.2024.01.004 [18] R. M. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 1979, 205(4403), 267–277. doi: 10.1126/science.205.4403.267 [19] M. R. Myerscough, B. F. Gray, W. L. Hogarth and J. Norbury, An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking, J. Math. Biol., 1992, 30(4), 389–411. [20] G. Peng, Y. Jiang and C. Li, Bifurcations of a Holling-type Ⅱ predator-prey system with constant rate harvesting, Int. J. Bifurc. Chaos, 2009, 19(08), 2499–2514. doi: 10.1142/S021812740902427X [21] L. Perko, Differential Equations and Dynamical Systems, Springer Science & Business Media, 2013. [22] N. Sarif and S. Sarwardi, Analysis of Bogdanov-Takens bifurcation of codimension-2 in a Gause-type model with constant harvesting of both species and delay effect, J. Biol. Syst., 2021, 29(03), 741–771. doi: 10.1142/S0218339021500169 [23] F. Takens, Forced Oscillations and Bifurcations, Springer, 1974. [24] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 2005, 65(3), 737–753. doi: 10.1137/S0036139903428719 [25] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 1999, 21, 493–506. [26] Y. Xu, Y. Yang, F. Meng and S. Ruan, Degenerate codimension-2 cusp of limit cycles in a Holling-Tanner model with harvesting and anti-predator behavior, Nonlinear Anal. Real World Appl., 2024, 76, 103995. doi: 10.1016/j.nonrwa.2023.103995 [27] Y. Yang, F. Meng, Y. Xu and L. Rong, Complex dynamics induced by additive Allee effect in a Leslie-Gower predator-prey model, Discrete Contin. Dyn. Syst. Ser. B, 2024, 29(8), 3471–3511. doi: 10.3934/dcdsb.2024011 [28] Y. Yang, Y. Xu, L. Rong and S. Ruan, Bifurcations and global dynamics of a predator-prey mite model of Leslie type, Stud. Appl. Math., 2024, 152(4), 1–54. [29] H. Zhang and M. Zhu, Population dynamics of a logistic model incorporating harvesting pulses on a growing domain, Commun. Nonlinear Sci. Numer. Simul., 2025, 146, 108768. doi: 10.1016/j.cnsns.2025.108768 [30] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., 1992. [31] C. Zhu and L. Kong, Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting, Discrete Contin. Dyn. Syst. Ser. S, 2017, 10(5), 1187–1206. doi: 10.3934/dcdss.2017065 -
-
Figures(9) / Tables(4)
Export File
Citation
Yue Yang, Yancong Xu, Libin Rong. GLOBAL BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH HARVESTING AND STOCKING[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 788-816. doi: 10.11948/20250139
Format
Content
DownLoad:
-
Figure 1.
One-parameter bifurcation diagram of system (1.2) with respect to the conversion rate (m). (a) m vs. x. (b) Zoomed-in section of (a).
-
Figure 2.
Three limit cycles and their corresponding time evolutions in Figure 1 (b) for
$m=0.190695 $ -
Figure 3.
Two-parameter bifurcation diagram of system (1.2) with respect to the conversion rate (m) and the harvesting of the prey (h). (a) The saddle-node bifurcation curve of a limit cycle
$ SNL $ $ CPL $ $ SN $ $ H_s $ $ (H_u )$ $ Hom $ $ SNL $ $ CPL $ $ GH $ $ BT $ -
Figure 4.
The corresponding phase portraits in regions Ⅰ-Ⅷ of Figure 3.
-
Figure 5.
Two-parameter bifurcation diagram of system (1.2) with respect to the conversion rate (m) and the harvesting of the predator (g). (a) The saddle-node bifurcation curve of a limit cycle
$ SNL $ $ CPL $ $ SN $ $ H_s $ $ (H_u )$ $ Hom $ $ SNL $ $ CPL $ $ GH $ $ BT $ -
Figure 6.
The corresponding phase portraits for regions Ⅰ-Ⅶ in Figure 5.
-
Figure 7.
One-parameter bifurcation diagram of system (1.2) with respect to h. (a) h vs. x. (b) Zoomed-in part of (a).
-
Figure 8.
Two-parameter bifurcation diagram of system (1.2) with respect to h and g. (a) h vs. g. (b) First zoomed-in part of (a). (c) Second zoomed-in part of (a). (d) Third zoomed-in part of (a). Here,
$ SN $ $ H_s(H_u) $ $ Hom_i(i=1, 2) $ $ SNL $ $ GH_i(i=1, 2) $ $ CPE $ $ CPL $ $ BT_i(i=1, 2) $ -
Figure 9.
The corresponding phase portraits of regions Ⅰ-Ⅵ in Figure 8.