Abstract
In natural systems, the change of ecological environment can indirectly affect some key biological parameters of species and their activity patterns, among which the imprecision of biological parameters and prey refuge behavior are two more obvious embodiments. The aim of this work is to investigate the triple effects of imprecision of biological parameters, prey refuge behavior and fishing activities on the dynamics of the system. Firstly, a novel Ivlev’s type fishery model incorporating with variable prey refuge and imprecise parameters is proposed. The dynamic properties of the fishery model under continuous fishing mode are analyzed, and the effects of variable prey refuge, imprecise index and fishing effort on the existence and stability of equilibrium are given. Moreover, the bionomic equilibrium and optimal fishing strategy are obtained. Secondly, from the perspective of reasonable exploitation of fishery resources, the weighted fishing strategy is introduced into the system, and the relevant weights and thresholds can be adjusted according to the actual demand, which is conducive to the sustainable development of fishery resources. The dynamic behavior of the fishery model under weighted fishing model is discussed, including the existence and stability of predator-extinction periodic solution and co-existence order-1 periodic solution under different fishing thresholds and weights, which not only provides a possibility to fishing species according to the period, but also ensures a certain robustness of the fishing strategy. Finally, to illustrate the theoretical results, numerical verifications for two harvesting models are presented to demonstrate the impact of variable prey refugia, imprecise biological parameters and fishing efforts on the behaviors of the system. The results of this study provide a reference for the scientific and reasonable exploitation and utilization of fishery resources in an imprecise environment.
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References
Lotka, A.J.: Elements of physical biology. Am. J. Public Health 21, 341–343 (1926)
Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)
Ludwig, D., Johns, D.D., Holling, C.S.: Qualitative analysis of insect outbreak system: the spruce budworm and forest. J. Anim. Ecol. 47, 315–322 (1978)
Sivakumar, M., Sambath, M., Balachandran, K.: Stability and HOPF bifurcation analysis of a diffusive predator-prey model with Smith growth. Int. J. Biomath. 8(1), 1550013 (2015)
Ivlev, V.S.: Experimental Ecology of the Feeding of Fishes. Yale University Press, New Haven (1961)
Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. Suppl. 45, 5–60 (1965)
Kooij, R., Zegeling, A.: A predator-prey model with Ivlev’s functional response. J. Math. Anal. Appl. 198, 473–489 (1996)
Sugie, J.: Two-parameter bifurcation in a predator-prey system of Ivlev type. J. Math. Anal. Appl. 217, 349–371 (1998)
Kuang, T., Beretta. E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389-406 (1998)
Cantrell, R.S., Cosner, C.: On the dynamics of predator-prey models with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 257, 206–222 (2001)
Xiao, H.B.: Global analysis of Ivlev’s type predator-prey dynamic systems. Appl. Math. Mech. 28, 461–470 (2007)
Wang, H., Wang, W.: The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect. Chaos Solitons Fractals 38, 1168–1176 (2007)
Ling, L., Wang, W.: Dynamics of a Ivlev-type predator-prey system with constant rate harvesting. Chaos Solitons Fractals 41, 2139–2153 (2009)
Wang, H.L.: Dispersal permanence of periodic predator-prey model with Ivlev-type functional response and impulsive effects. Appl. Math. Model. 41, 2139–2153 (2010)
Wang, X., Wei, J.: Diffusion-driven stability and bifurcation in a predator-prey system with Ivlev-type functional response. Appl. Anal. 92(4), 752–775 (2013)
Wang, X., Wei, J.: Dynamics in a diffusive predator-prey system with strong Allee effect and Ivlev-type functional response. J. Math. Anal. Appl. 422(2), 1447–1462 (2015)
Khan, M.S., Samreen, M., Ozair, M., Hussain, T., Elsayed, E.M., Gómez-Aguilar, J.F.: On the qualitative study of a two-trophic plant-herbivore model. J. Math. Biol. 85(4), 34 (2022). https://doi.org/10.1007/s00285-022-01809-0
Khan, M.S., Samreen, M., Gómez-Aguilar, J.F., Pérez-Careta, E.: On the qualitative study of a discrete-time phytoplankton-zooplankton model under the effects of external toxicity in phytoplankton population. Heliyon 8(12), e12415 (2022). https://doi.org/10.1016/j.heliyon.2022.e12415
Smith, J.M.: Models in Ecology. Cambridge Univ. Press, Cambridge (1974)
Mcnair, J.N.: The effects of refuges on predator-prey interactions: a reconsideration. Theor. Popul. Biol. 29, 38–63 (1986)
Ma, Z., Li, W., Zhao, Y., Wang, W., et al.: Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges. Math. Biosci. 218, 73–79 (2009)
Chang, X., Wei, J.: Stability and HOPF bifurcation in a diffusive predator-prey system incorporating a prey refuge. Math. Biosci. Eng. 10, 979 (2013)
Moustafa, M., Mohd, M.H., Ismail, A.I., et al.: Dynamical analysis of a fractional-order Rosenzweig-MacArthur model incorporating a prey refuge. Chaos Solitons Fractals 109, 1–13 (2018)
Kar, T.K.: Modelling and analysis of a harvested prey-predator system incorporating a prey refuge. Comput. Appl. Math. 185, 19–33 (2006)
Tian, Y., Guo, H., Sun, K.: Complex dynamics of two prey-predator harvesting models with prey refuge and interval-valued imprecise parameters. Math. Meth. Appl. Sci. 46(13), 14278–14298 (2023). https://doi.org/10.1002/mma.319
Mukherjee, D., Maji, C.: Bifurcation analysis of a holling type II predator-prey model with refuge. Chinese J. Phys. 65, 153–162 (2020)
Barman, D., Roy, J., Alam, S.: Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives. Eco. Inform. 67, 101483 (2022)
Liu, M., Wang, K.: Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 1114–1121 (2011)
Lv, J., Wang, K.: Asymptotic properties of a stochastic predator-prey system with Holling II functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 4037–4048 (2011)
Zhang, S., Yuan, S., Zhang, T.: Dynamic analysis of a stochastic eco-epidemiological model with disease in predators. Stud. Appl. Math. 149, 5–42 (2022)
Xu, J., Yu, Z., Zhang, T., Yuan, S.: Near-optimal control of a stochastic model for mountain pine beetles with pesticide application. Stud. Appl. Math. 149, 678–704 (2022)
Qi, H.K., Meng, X.Z.: Dynamics of a stochastic predator-prey model with fear effect and hunting cooperation. J. Appl. Math. Comput. 69, 2077–2103 (2023)
Pal, D., Mahapatra, G.S., Samanta, G.P.: Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model. Math. Biosci. 241, 181–187 (2013)
Pal, D., Mahapatra, G.S., Samanta, G.P.: Stability and bionomic analysis of fuzzy parameter based prey-predator harvesting model using UFM. Nonlinear Dyn. 79, 1939–1955 (2015)
Pal, D., Mahapatra, G.S., Samanta, G.P.: Stability and bionomic analysis of fuzzy prey-predator harvesting model in presence of toxicity: a dynamic approach. Bull. Math. Biol. 78, 1493–1519 (2016)
Pal, D., Mahapatra, G.S., Samanta, G.P.: New approach for stability and bifurcation analysis on predator-prey harvesting model for interval biological parameters with time delays. Comp. Appl. Math. 37, 3145–3171 (2018)
Yu, X.W., Yuan, S.L., Zhang, T.H.: About the optimal harvesting of a fuzzy predator-prey system: a bioeconomic model incorporating prey refuge and predator mutual interference. Nonlinear Dyn. 94, 2143–2160 (2018)
Das, S., Mahato, P., Mahato, S.K.: A prey predator model in case of disease transmission via pest in uncertain environment. Differ. Equ. Dyn. Syst. (2020). https://doi.org/10.1007/s12591-020-00551-7
Xiao, Q.Z., Dai, B.X., Wang, L.: Analysis of a competition fishery model with interval-valued parameters: extinction, coexistence, bionomic equilibria and optimal harvesting policy. Nonlinear Dyn. 80, 1631–1642 (2015)
Tian, Y., Li, C.X., Liu, J.: Complex dynamics and optimal harvesting strategy of competitive harvesting models with interval-valued imprecise parameters. Chaos Solitons Fractals 167, 113084 (2023)
Worm, B., Barbier, E.B., Beaumont, N., et al.: Impacts of biodiversity loss on ocean ecosystem services. Science 314, 787–790 (2006)
Lv, Y.F., Yuan, R., Pei, Y.Z.: A prey-predator model with harvesting for fishery resource with reserve area. Appl. Math. Model. 37(5), 3048–3062 (2013)
Hu, D.P., Cao, H.J.: Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting. Nonlinear Anal. Real. 33, 58–82 (2017)
Ang, T.K., Safuan, H.M.: Dynamical behaviors and optimal harvesting of an intraguild prey-predator fishery model with Michaelis-Menten type predator harvesting. Biosystems 202, 104357 (2021)
Meng, X.Y., Li, J.: Dynamical behavior of a delayed prey-predator-scavenger system with fear effect and linear harvesting. Int. J. Biomath. 14, 2150024 (2021)
Debnath, S., Majumdar, P., Sarkar, S., et al.: Global dynamics of a prey-predator model with holling type III functional response in the presence of harvesting. J. Biol. Syst. 30, 225–260 (2022)
Nie, L.F., Teng, Z.D., Lin, H., Peng, J.G.: The dynamics of a Lotka-Volterra predator-prey model with state dependent impulsive harvest for predator. Biosystems 98, 67–72 (2009)
Guo, H.J., Chen, L.S., Song, X.Y.: Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property. Appl. Math. Comput. 271, 905–922 (2015)
Tian, Y., Li, H.M.: The study of a predator-prey model with fear effect based on state-dependent harvesting strategy. Complexity 2022, 9496599 (2022). https://doi.org/10.1155/2022/9496599
Tian, Y., Gao, Y., Sun, K.B.: Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy. Chaos Soliton Fract 164, 112597 (2022)
Tian, Y., Gao, Y., Sun, K.B.: A fishery predator-prey model with anti-predator behavior and complex dynamics induced by weighted fishing strategies. Math. Biosci. Eng. 20(2), 1558–1579 (2023)
Tian, Y., Gao, Y., Sun, K.B.: Qualitative analysis of exponential power rate fishery model and complex dynamics guided by a discontinuous weighted fishing strategy. Commun. Nonlinear Sci. Numer. Simul. 118, 107011 (2023)
Li, H.M., Tian, Y.: Dynamic behavior analysis of a feedback control predator-prey model with exponential fear effect and Hassell-Varley functional response. J. Franklin Inst. 360, 3479–3498 (2023)
Liu, X.N., Chen, L.S.: Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator. Chaos Soliton Fract 16(2), 311–320 (2004)
Liu, B., Zhang, Y.J., Chen, L.S.: Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control. Chaos Soliton Fract 22(1), 123–134 (2004)
Zhao, Z., Yang, L., Chen, L.: Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes. J. Appl. Math. Comput. 35, 119–134 (2011). https://doi.org/10.1007/s12190-009-0346-2
Jiang, G.R., Lu, Q.S., Qian, L.N.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Soliton Fract 31(2), 448–461 (2007)
Tian, Y., Sun, K.B., Chen, L.S.: Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system. Int. J. Biomath. 7, 1450018 (2014)
Tang, S.Y., Pang, W.H., Cheke, R.A., et al.: Global dynamics of a state-dependent feedback control system. Adv. Differ. Equ. 2015, 322 (2015). https://doi.org/10.1186/s13662-015-0661-x
Tang, S.Y., Tang, B., Wang, A.L., Xiao, Y.N.: Holling II predator-prey impulsive semi-dynamic model with complex Poincaré map. Nonlinear Dyn. 81, 1575–1596 (2015)
Zhang, T.Q., Ma, W.B., Meng, X.Z., Zhang, T.H.: Periodic solution of a prey-predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95–107 (2015)
Yang, J., Tang, S.Y.: Holling type II predator-prey model with nonlinear pulse as state-dependent feedback control. J. Comput. Appl. Math. 291, 225–241 (2016)
Chen, L.S., Liang, X.Y., Pei, Y.Z.: The periodic solutions of the impulsive state feedback dynalical system. Commun. Math. Biol. Neurosci. 2018, 14 (2018)
Tang, S., Li, C., Tang, B., Wang, X.: Global dynamics of a nonlinear state-dependent feedback control ecological model with a multiple-hump discrete map. Commun. Nonlinear Sci. Numer. Simul. 79, 104900 (2019)
Zhang, Q., Tang, B., Cheng, T., Tang, S.: Bifurcation analysis of a generalized impulsive Kolmogorov model with applications to pest and disease control. SIAM J. Appl. Math. 80, 1796–1819 (2020)
Zhang, Q., Tang, S., Zou, X.: Rich dynamics of a predator-prey system with state-dependent impulsive controls switching between two means. J. Differ. Equ. 364, 336–377 (2023)
Li, W., Ji, J., Huang, L.: Global dynamic behavior of a predator-prey model under ratio-dependent state impulsive control. Appl. Math. Model. 77, 1842–1859 (2020)
Xu, J., Huang, M., Song, X.: Dynamical analysis of a two-species competitive system with state feedback impulsive control. Int. J. Biomath. 13(05), 2050007 (2020)
Xu, J., Huang, M., Song, X.: Dynamics of a guanaco-sheep competitive system with unilateral and bilateral control. Nonlinear Dyn. 107(3), 3111–3126 (2022)
Zhang, Q., Tang, S.: Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set Commun. Nonlinear Sci. Numer. Simul. 108, 106212 (2022)
Pontryagin, L.S.: The Mathematical Theory of Optimal Processes. CRC press (1987)
Jan, M.N., Zaman, G., Ali, N., Ahmad, I., Shah, Z.: Optimal control application to the epidemiology of HBV and HCV co-infection. Int. J. Biomath. 15(03), 2150101 (2022)
Zhang, Z.Z., Rahman, G., Gómez-Aguilar, J.F., Torres-Jiménez, J.: Dynamical aspects of a delayed epidemic model with subdivision of susceptible population and control strategies. Chaos Soliton Fract 160, 112194 (2022)
Liu, R., Liu, G.: Complex dynamics and optimal harvesting for a stochastic food-web model with intraguild predation and time delays. Int. J. Biomath. 15(07), 2250050 (2022)
Chien, F.S., Nik, H.S., Shirazian, M., Gómez-Aguilar, J.F.: The global stability and optimal control of the Covid-19 epidemic model. Int. J. Biomath. 17(1), 2350002 (2024). https://doi.org/10.1142/S179352452350002X
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The work was supported by the National Natural Science Foundation of China (Nos. 12071407, 12191193).
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Hua, G., Yuan, T., Kaibiao, S. et al. Study on dynamic behavior of two fishery harvesting models: effects of variable prey refuge and imprecise biological parameters. J. Appl. Math. Comput. 69, 4243–4268 (2023). https://doi.org/10.1007/s12190-023-01925-0
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DOI: https://doi.org/10.1007/s12190-023-01925-0
Keywords
- Bionomic equilibrium
- Imprecise parameters
- Optimal fishing strategy
- Variable prey refuge
- Weighted fishing strategy