Abstract
Effects of predator-taxis on the dynamics of a predator–prey model with Michaelis–Menten type nonlinear harvesting are considered in this paper. Through theoretical analysis of quasilinear parabolic equations, the local existence, global existence and boundedness of solutions to the system are first established. Then the formation mechanisms of spatiotemporal patterns in such model are explored. It is found that only the attractive predator-taxis will lead to the Turing bifurcation and the corresponding spatiotemporal solutions; meanwhile, no such phenomenon occurs with the repulsive taxis or in the absence of predator-taxis. Moreover, the diffusion ratio can affect the Turing bifurcation thresholds, and nonlinear harvesting can affect the spatial distribution of prey and predator species. Further, the stability of the nonconstant steady state bifurcated from the Turing bifurcation is analyzed through the amplitude equation, so that the direction of the Turing bifurcation is determined. Effectiveness of the analysis is illustrated in the numerical simulations.
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References
Bentout, S., Djilali, S., Atangana, A.: Bifurcation analysis of an age-structured prey-predator model with infection developed in prey. Math. Methods Appl. Sci. 45, 1189–1208 (2022)
Chen, M.X., Wu, R.C., Wang, X.H.: Non-constant steady states and Hopf bifurcation of a species interaction model. Commun. Nonlinear Sci. Numer. Simul. 116, 106846 (2023)
Adak, D., Bairagi, N., Hakl, R.: Chaos in delay-induced Leslie-Gower prey-predator-parasite model and its control through prey harvesting. Nonlinear Anal. RWA 51, 102998 (2020)
Alidousti, J.: Stability and bifurcation analysis for a fractional prey-predator scavenger model. Appl. Math. Model. 81, 342–355 (2020)
Zhang, T.R., Wang, W.D., Wang, K.F.: Minimal wave speed for a class of non-cooperative diffusion-reaction system. J. Differ. Equ. 260(3), 2763–2791 (2016)
Dong, F.D., Li, W.T., Zhang, G.B.: Invasion traveling wave solutions of a predator-prey model with nonlocal dispersal. Commun. Nonlinear Sci. Numer. Simul. 79, 104926 (2019)
Chowdhury, P.R., Petrovskii, S., Banerjee, M.: Oscillations and pattern formation in a slow-fast prey-predator system. Bull. Math. Biol. 83, 110 (2021)
Jana, D., Batabyal, S., Lakshmanan, M.: Self-diffusion-driven pattern formation in prey-predator system with complex habitat under fear effect. Eur. Phys. J. Plus 135, 884 (2020)
Bai, D.Y., Kang, Y., Ruan, S.G., et al.: Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey. Nonlinear Anal. RWA 58, 103206 (2021)
Hung, K.C.: Bifurcation curves of a Dirichlet problem with geometrically concave nonlinearity and an application to the generalized logistic growth model. Proc. Am. Math. Soc. 149, 1117–1126 (2021)
Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. 97, 5–60 (1965)
Yi, F.Q., Wei, J.J., Shi, J.P.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differ. Equ. 246, 1944–1977 (2009)
Ko, W., Ryu, K.: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J. Differ. Equ. 231, 534–550 (2006)
Medvinsky, S.V., Petrovskii, I.A., Tikhonova, H., et al.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44, 311–370 (2002)
Dai, G., Tang, M.: Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58, 193–210 (1998)
Xiao, D., Jennings, L.: Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J. Appl. Math. 65, 737–753 (2005)
Zhang, X.B., Zhao, H.Y.: Dynamics analysis of a delayed reaction-diffusion predator-prey system with non-continuous threshold harvesting. Math. Biosci. 289, 130–141 (2017)
Lv, Y.F., Pei, Y.Z., Wang, Y.: Bifurcations and simulations of two predator-prey models with nonlinear harvesting. Chaos Solitons Fractals 120, 158–170 (2019)
Tiwari, V., Tripathi, J.P., Abbas, S., et al.: Qualitative analysis of a diffusive Crowley-Martin predator-prey model: the role of nonlinear predator harvesting. Nonlinear Dyn. 98, 1169–1189 (2019)
Chen, M.X., Wu, R.C., Liu, B., et al.: Pattern selection in a predator-prey model with Michaelis-Menten type nonlinear predator harvesting. Ecol. Complex. 36, 239–249 (2018)
Li, Y., Wang, M.X.: Hopf bifurcation and global stability of a delayed predator-prey model with prey harvesting. Comput. Math. Appl. 69, 398–410 (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)
Kaur, M., Rani, R., Bhatia, R.: Dynamical study of quadrating harvesting of a predator-prey model with Monod-Haldane functional response. J. Appl. Math. Comput. 66, 397–422 (2020)
Clark, C.W.: Aggregation and fishery dynamics: a theoretical study of schooling and the purse seine tuna fisheries. Fish. Bull. 77(2), 317–337 (1979)
Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting. J. Math. Anal. Appl. 398(1), 278–295 (2013)
Kareiva, P., Odell, G.: Swarms of predators exhibit prey taxis if individual predators use area-restricted search. Am. Nat. 130, 233–270 (1987)
Chen, M.X., Zheng, Q.Q.: Predator-taxis creates spatial pattern of a predator-prey model. Chaos Solitons Fractals 161(3), 112332 (2022)
Fuest, M.: Global solution near homogeneous steady state in a multidimensional population model with both predator-and prey-taxis. SIAM J. Math. Anal. 52, 5865–5891 (2020)
Li, D.: Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete Contin. Dyn. Syst. 41, 1681–1705 (2021)
Mishra, P., Wrzosek, D.: Repulsive chemotaxis and predator evasion in predator-prey models with diffusion and prey-taxis. Math. Model. Methods Appl. Sci. 32, 1–42 (2022)
Gao, J.P., Guo, S.J.: Effect of prey-taxis and diffusion on positive steady states for a predator-prey system. Math. Methods Appl. Sci. 41, 3570–3587 (2018)
Ahn, I., Yoon, C.: Global solvability of prey-predator models with indirect predator-taxis. Z. Angew. Math. Phys. 72, 29 (2021)
Wu, S.N., Wang, J.F., Shi, J.P.: Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis. Math. Model. Methods Appl. Sci. 28, 2275–2312 (2018)
Dong, Y.X., Wu, D.Y., Shen, C.S., et al.: Influence of fear effect and predator-taxis sensitivity on dynamical behavior of a predator-prey model. Z. Angew. Math. Phys. 73, 25 (2021)
Amann, H.: Dynamic theory of quasilinear parabolic equations II. Differ. Integral Equ. 3(1), 13–75 (1990)
Alikakos, N.D.: L\(^p\) bounds of solutions of reaction-diffusion equations. Commun. Part. Differ. Equ. 48(8), 827–868 (1979)
Amann, H.: Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems (1993). https://doi.org/10.1007/978-3-663-11336-2_1
Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola. Norm. Sup. Pisa. 13, 115–162 (1959)
Tao, Y., Winkler, M.: Blow-up prevention by quadratic degradation in a two-dimension Leller-Segel-Navier-Stokes system. Z. Angew. Math. Phys. 67(6), 138 (2016)
Bai, X., Winkler, M.: Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65, 553–583 (2016)
Chen, M.X., Wu, R.C., Liu, B., et al.: Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing-Hopf bifurcation point. Commun. Nonlinear Sci. Numer. Simul. 77, 141–167 (2019)
Geng, D.X., Jiang, W.H., Lou, Y., et al.: Spatiotemporal patterns in a diffusive predator-prey system with nonlocal intraspecific prey competition. Stud. Appl. Math. 148, 396–432 (2021)
Xu, X.F., Wei, J.J.: Turing-Hopf bifurcation of a classs of modified Leslie-Gower model with diffusion. Discrete Contin. Dyn. Syst. B 23, 765–783 (2018)
Song, Y.L., Jiang, H.P., Yuan, Y.: Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model. J. Appl. Anal. Comput. 9, 1132–1164 (2019)
Acknowledgements
The authors are very grateful to the referees for their detailed comments and valuable suggestions, which greatly improved the present manuscript. This work was supported by the National Natural Science Foundation of China (No. 11971032) and China Postdoctoral Science Foundation (No. 2021M701118).
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Communicated by Shangjiang Guo.
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Chen, M., Wu, R. Dynamics of a Harvested Predator–Prey Model with Predator-Taxis. Bull. Malays. Math. Sci. Soc. 46, 76 (2023). https://doi.org/10.1007/s40840-023-01470-w
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DOI: https://doi.org/10.1007/s40840-023-01470-w