The concept of an action threshold that depends on predator density and the rate of change is relatively novel and can engender new ideas among scholars studying predator-prey systems more effectively than earlier concepts. On this basis, a predator-prey system with an action threshold based on predator density and its change rate has been established and its dynamic behavior studied. The exact phase set and pulse set of the model were obtained conducting image analysis. The Poincaré map of the model has been constructed and the extreme value points, monotonic interval and immobility points of the Poincaré map have been studied. In addition, the nature of the periodic solution is discussed and we present simulations of the interesting dynamical behavior of the model through the use of numerical examples. An action threshold that depends on the density and rate of change of predators is more reasonable and realistic than techniques proposed in earlier studies, which is significant for the study of control strategies. It is the analytical approach adopted in this paper that allows researchers to explore other generalized predator-prey models more fully and in-depth.
Citation: Liping Wu, Zhongyi Xiang. Dynamic analysis of a predator-prey impulse model with action threshold depending on the density of the predator and its rate of change[J]. AIMS Mathematics, 2024, 9(5): 10659-10678. doi: 10.3934/math.2024520
The concept of an action threshold that depends on predator density and the rate of change is relatively novel and can engender new ideas among scholars studying predator-prey systems more effectively than earlier concepts. On this basis, a predator-prey system with an action threshold based on predator density and its change rate has been established and its dynamic behavior studied. The exact phase set and pulse set of the model were obtained conducting image analysis. The Poincaré map of the model has been constructed and the extreme value points, monotonic interval and immobility points of the Poincaré map have been studied. In addition, the nature of the periodic solution is discussed and we present simulations of the interesting dynamical behavior of the model through the use of numerical examples. An action threshold that depends on the density and rate of change of predators is more reasonable and realistic than techniques proposed in earlier studies, which is significant for the study of control strategies. It is the analytical approach adopted in this paper that allows researchers to explore other generalized predator-prey models more fully and in-depth.
| [1] |
C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European Pine Sawfly, Can. Entomol., 91 (1959), 293–320. http://dx.doi.org/10.4039/Ent91293-5 doi: 10.4039/Ent91293-5
|
| [2] |
J. M. Smith, M. Slatkin, The stability of predator-prey systems, Ecology, 54 (1973), 384–391. http://dx.doi.org/10.2307/1934346 doi: 10.2307/1934346
|
| [3] |
F. Souna, P. K. Tiwari, M. Belabbas, Anosov flows with stable and unstable differentiable distributions, Math. Method. Appl. Sci., 46 (2023), 13991–14006. http://dx.doi.org/10.1002/mma.9300 doi: 10.1002/mma.9300
|
| [4] |
M. W. Sabelis, O. Diekmann, V. A. A. Jansen, Metapopulation persistence despite local extinction: Predator-prey patch models of the Lotka-Volterra type, Biol. J. Linn. Soc., 42 (1991), 267–283. http://dx.doi.org/10.1111/j.1095-8312.1991.tb00563.x doi: 10.1111/j.1095-8312.1991.tb00563.x
|
| [5] |
M. Ruan, C. Li, X. Li, Codimension two 1:1 strong resonance bifurcation in a discrete predator-prey model with Holling Ⅳ functional response, AIMS Math., 7 (2021), 3150–3168. http://dx.doi.org/10.3934/math.2022174 doi: 10.3934/math.2022174
|
| [6] |
M. Belabbas, A. Ouahab, F. Souna, Rich dynamics in a stochastic predator-prey model with protection zone for the prey and multiplicative noise applied on both species, Nonlinear Dynam., 106 (2021), 2761–2780. http://dx.doi.org/10.1007/s11071-021-06903-4 doi: 10.1007/s11071-021-06903-4
|
| [7] |
F. Souna, A. Lakmeche, S. Djilali, Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting, Chaos Soliton. Fract., 140 (2020), 110180. http://dx.doi.org/10.1016/j.chaos.2020.110180 doi: 10.1016/j.chaos.2020.110180
|
| [8] |
X. Meng, F. Meng, Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting, AIMS Math., 6 (2021), 5695–5719. http://dx.doi.org/10.3934/math.2021336 doi: 10.3934/math.2021336
|
| [9] |
Y. Tian, Y. Gao, K. Sun, A fishery predator-prey model with anti-predator behavior and complex dynamics induced by weighted fishing strategies, Math. Biosci. Eng., 20 (2023), 1558–1579. http://dx.doi.org/10.3934/mbe.2023071 doi: 10.3934/mbe.2023071
|
| [10] |
H. Li, Y. Tian, Dynamic behavior analysis of a feedback control predator-prey model with exponential fear effect and Hassell-Varley functional response, J. Franklin I., 360 (2023), 3479–3498. http://dx.doi.org/10.1016/j.jfranklin.2022.11.030 doi: 10.1016/j.jfranklin.2022.11.030
|
| [11] |
F. Souna, A. Lakmeche, Spatiotemporal patterns in a diffusive predator-prey system with Leslie-Gower term and social behavior for the prey, Math. Method. Appl. Sci., 44 (2021), 13920–13944. http://dx.doi.org/10.1002/mma.7666 doi: 10.1002/mma.7666
|
| [12] |
E. Accinelli, A, García, L. Policardo, C. Sánchez, A predator-prey economic system of tax evasion and corrupt behavior, J. Dyn. Games, 10 (2023), 181–207. http://dx.doi.org/10.3934/jdg.2022025 doi: 10.3934/jdg.2022025
|
| [13] |
Z. Xiang, S. Tang, C. Xiang, J. Wu, On impulsive pest control using integrated intervention strategies, Appl. Math. Comput., 269 (2015), 930–946. http://dx.doi.org/10.1016/J.AMC.2015.07.076 doi: 10.1016/J.AMC.2015.07.076
|
| [14] |
S. Tang, R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257–292. http://dx.doi.org/10.1007/S00285-004-0290-6 doi: 10.1007/S00285-004-0290-6
|
| [15] |
Y. Wu, F. Chen, F. Ma, D. Qian, Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects, AIMS Math., 8 (2023), 20080–20096. http://dx.doi.org/10.3934/math.20231023 doi: 10.3934/math.20231023
|
| [16] |
Z. Zhao, L. Pang, X. Song, D. Wang, Q. Li, Impact of the impulsive releases and Allee effect on the dispersal behavior of the wild mosquitoes, J. Appl. Math. Comput., 68 (2022), 1527–1544. http://dx.doi.org/10.1007/s12190-021-01569-y doi: 10.1007/s12190-021-01569-y
|
| [17] |
Z. Xiang, D. Long, X. Song, A delayed Lotka-Volterra model with birth pulse and impulsive effect at different moment on the prey, Appl. Math. Comput., 219 (2013), 10263–10270. http://dx.doi.org/10.1016/j.amc.2013.03.129 doi: 10.1016/j.amc.2013.03.129
|
| [18] |
Z. Xiang, X. Song, The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response, J. Am. Math. Soc., 39 (2009), 2282–2293. http://dx.doi.org/10.1016/j.chaos.2007.06.124 doi: 10.1016/j.chaos.2007.06.124
|
| [19] |
Q. Zhang, S. Tang, Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincare map defined in phase set, Commun. Nonlinear Sci., 108 (2022), 1007–5704. http://dx.doi.org/10.1016/j.cnsns.2021.106212 doi: 10.1016/j.cnsns.2021.106212
|
| [20] |
I. U. Khan, S. Tang, The impulsive model with pest density and its change rate dependent feedback control, Discrete Dyn. Nat. Soc., 2020 (2020), 1–20. http://dx.doi.org/10.1155/2020/4561241 doi: 10.1155/2020/4561241
|
| [21] |
Y. Tian, S. Tang, Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control, Math. Biosci. Eng., 1 (20821), 7318–7343. http://dx.doi.org/10.3934/mbe.2021362 doi: 10.3934/mbe.2021362
|
| [22] |
I. U. Khan, S. Ullah, E. Bonyah, A. A. Basem, M. A. Ahmed, A state-dependent impulsive nonlinear system with ratio-dependent action threshold for investigating the pest-natural Enemy model, Complexity, 2022 (2022), 1–18. http://dx.doi.org/10.1155/2022/7903450 doi: 10.1155/2022/7903450
|
| [23] |
H. Cheng, H. Xu, J. Fu, Dynamic analysis of a phytoplankton-fish model with the impulsive feedback control depending on the fish density and its changing rate, Math. Biosci. Eng., 205 (2023), 8103–8123. http://dx.doi.org/10.3934/mbe.2023352 doi: 10.3934/mbe.2023352
|
| [24] |
I. U. Khan, S. Tang, B. Tang, The state-dependent impulsive model with action threshold depending on the pest density and its changing rate, Complexity, 2019 (2019). http://dx.doi.org/10.1155/2019/6509867 doi: 10.1155/2019/6509867
|
| [25] |
Z. Shi, H. Cheng, Y. Liu, Y. Wang, Optimization of an integrated feedback control for a pest management predator-prey model, Math. Biosci. Eng., 16 (2019), 7963–7981. http://dx.doi.org/10.3934/mbe.2019401 doi: 10.3934/mbe.2019401
|
| [26] |
T. Li, W. Zhao, Periodic solution of a neutral delay leslie predator-prey model and the effect of random perturbation on the smith growth model, Complexity, 2020 (2020), 1–15. http://dx.doi.org/10.1155/2020/8428269 doi: 10.1155/2020/8428269
|
| [27] |
M. Huang, J. Li, X. Song, H. Guo, Modeling impulsive injections of insulin: Towards artificial pancreas, J. Am. Math. Soc., 72 (2012), 1524–1548. http://dx.doi.org/10.1137/110860306 doi: 10.1137/110860306
|
| [28] |
G. Wang, M. Yi, S. Tang, Dynamics of an antitumour model with pulsed radioimmunotherapy, Comput. Math. Method. M., 2022 (2022). http://dx.doi.org/10.1155/2022/4692772 doi: 10.1155/2022/4692772
|
| [29] |
J. Lou, Y. Lou, J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623–652. http://dx.doi.org/10.1007/s00285-011-0474-9 doi: 10.1007/s00285-011-0474-9
|
| [30] |
W. Wang, X. Lai, Global stability analysis of a viral infection model in a critical case, Math. Biosci. Eng., 17 (2020), 1442–1449. http://dx.doi.org/10.3934/mbe.2020074 doi: 10.3934/mbe.2020074
|
| [31] |
E. M. Bonotto, M. Federson, Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Differ. Equations, 244 (2008), 2334–2349. http://dx.doi.org/10.1016/J.JDE.2008.02.007 doi: 10.1016/J.JDE.2008.02.007
|
| [32] |
E. M. Bonotto, Lyapunov stability of closed sets in impulsive semidynamical systems, Electron. J. Differ. Eq., 2010 (2010), 1–18. http://dx.doi.org/10.1007/s10589-009-9245-6 doi: 10.1007/s10589-009-9245-6
|
| [33] |
Y. Choh, M. Ignacio, M. W. Sabelis, A. Janssen, Predator-prey role reversals, juvenile experience and adult antipredator behaviour, Sci. Rep.-UK, 2 (2012), 1–6. http://dx.doi.org/10.1038/srep00728 doi: 10.1038/srep00728
|
| [34] |
J. K. B. Ford, R. R. Reeves, Fight or flight: Antipredator strategies of baleen whales, Mammal Rev., 38 (2008), 50–86. http://dx.doi.org/10.1111/J.1365-2907.2008.00118.X doi: 10.1111/J.1365-2907.2008.00118.X
|
| [35] |
S. Magalhaes, A. Janssen, M. Montserrat, W. S. Maurice, Prey attack and predators defend: Counterattacking prey trigger parental care in predators, P. Roy. Soc. B-Biol. Sci., 272 (2005), 1929–1933. http://dx.doi.org/10.1098/rspb.2005.3127 doi: 10.1098/rspb.2005.3127
|
| [36] |
F. S. Garduño, P. Miramontes, T. T. M. Lago, Role reversal in a predator-prey interaction, Roy. Soc. Open Sci., 1 (2014), 140186. http://dx.doi.org/10.1098/rsos.140186 doi: 10.1098/rsos.140186
|
| [37] |
B. Tang, Y. Xiao, Bifurcation analysis of a predator-prey model with anti-predator behaviour, Chaos Soliton. Fract., 70 (2015), 58–68. http://dx.doi.org/10.1016/J.CHAOS.2014.11.008 doi: 10.1016/J.CHAOS.2014.11.008
|
| [38] |
A. Kent, C. P. Doncaster, T. Sluckin, Consequences for predators of rescue and Allee effects on prey, Ecol. Model., 162 (2003), 233–245. http://dx.doi.org/10.1016/S0304-3800(02)00343-5 doi: 10.1016/S0304-3800(02)00343-5
|
| [39] |
G. A. K. van Voorn, L. Hemerik, M. P. Boer, B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451–469. http://dx.doi.org/10.1016/J.MBS.2007.02.006 doi: 10.1016/J.MBS.2007.02.006
|
| [40] |
C. Wei, L. Chen, Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting, Comput. Math. Method. M., 76 (2014), 1109–1117. http://dx.doi.org/10.1007/S11071-013-1194-Z doi: 10.1007/S11071-013-1194-Z
|
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Liping Wu, Zhongyi Xiang. Dynamic analysis of a predator-prey impulse model with action threshold depending on the density of the predator and its rate of change[J]. AIMS Mathematics, 2024, 9(5): 10659-10678. doi: 10.3934/math.2024520
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