Abstract
This paper concerns a diffusive predation model with nonlinear growth, cross-diffusion and protection zone terms. The main purpose is to investigate the effects of nonlinear growth and cross-diffusion on the coexistent solution when protection zone is present. Firstly, a priori estimate and the existence of positive solutions are discussed, including local and global existence. Then, some asymptotic properties of coexistent solutions induced by the mortality rate, nonlinear growth of predator and cross-diffusion are analyzed. It is revealed that there exist critical values related to certain principal eigenvalues such that the nonlinear growth, cross-diffusion and protection zone all have significant effects on the coexistent solutions; as far as the nonlinear growth concerned, we find that it has important influences on the coexistence region of two species undoubtedly. Biologically, this implies that these critical values greatly affect the survival of species.
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References
Du, Y., Shi, J.: Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans. Am. Math. Soc. 359, 4557–4593 (2007)
Dancer, E.N., Du, Y.: Effects of certain degeneracies in the predator-prey model. SIAM J. Math. Anal. 34, 292–314 (2002)
Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006)
Du, Y., Peng, R., Wang, M.: Effect of a protection zone in the diffusive Leslie predator-prey model. J. Differ. Equ. 246, 3932–3956 (2009)
He, X., Zheng, S.: Protection zone in a diffusive predator-prey model with Beddington–DeAngelis functional response. J. Math. Biol. 75, 239–257 (2017)
Li, S., Wu, J.: The effects of diffusion on the dynamics of a Lotka–Volterra predator–prey model with a protection zone. Calc. Var. Partial Differ. Equ. 61, 1–27 (2022)
Wang, Y.: Effects of protection zone and nonlinear growth on a predator-prey model. Acta Appl. Math. 176, 15 (2021)
Oeda, K.: Effect of cross-diffusion on the stationary problem of a prey–predator model with a protection zone. J. Differ. Equ. 250, 3988–4009 (2011)
Carrillo, J.A., Huang, Y., Schmidtchen, M.: Zoology of a non-local cross-diffusion model for two species. SIAM J. Appl. Math. 78, 1078–1104 (2018)
Jia, Y., Xue, P.: Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system. Nonlinear Anal. Real World Appl. 32, 229–241 (2016)
Oeda, K.: Coexistence states of a prey–predator model with cross-diffusion and a protection zone. Adv. Math. Sci. Appl. 22, 501–520 (2012)
Li, S., Wu, J., Liu, S.: Effect of cross-diffusion on the stationary problem of a Leslie prey–predator model with a protection zone. Calc. Var. Partial Differ. Equ. 56, 82 (2017)
Li, S., Yamada, Y.: Effect of cross-diffusion in the diffusion prey–predator model with a protection zone II. J. Math. Anal. Appl. 461, 971–992 (2018)
Dong, Y., Li, S., Li, Y.: Effects of cross-dffusion for a prey–predator system in a heterogeneous environment. Electron. J. Differ. Equ. 44, 1–14 (2019)
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theoret. Biol. 79, 83–99 (1979)
Chen, S., Yu, J.: Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator. J. Differ. Equ. 260, 7923–7939 (2016)
Wei, Z., Xia, Y., Zhang, T.: Stability and bifurcation analysis of a commensal model with additive Allee effect and nonlinear growth rate. Int. J. Bifur. Chaos 31, 2150204 (2021)
Yang, W.: Existence asymptotic behavior of solutions for a predator–prey system with a nonlinear growth rate. Acta Appl. Math. 152, 57–72 (2017)
Feng, Z., Chen, G.: Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete Contin. Dyn. Syst. 24, 763–780 (2009)
Beverton, R.J.H., Holt, S.J.: On the dynamics of exploited fish populations. Rev. Fish Biol. Fisher. 4, 259–260 (1994)
De la Sen, M.: The generalized Beverton–Holt equation and the control of populations. Appl. Math. Model. 32, 2312–2328 (2008)
De la Sen, M., Alonso-Quesada, S.: Control issues for the Beverton–Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases. Appl. Math. Comput. 215, 2616–2633 (2009)
Tang, S., Cheke, R.A., Xiao, Y.: Optimal implusive harvesting on non-autonomous Beverton–Holt difference equations. Nonlinear Anal. 65, 2311–2341 (2006)
Lou, Y., Ni, W.-M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)
Lin, C., Ni, W.-M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72, 1–27 (1988)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
López-Gómez, J.: Spectral Theory and Nonlinear Functional Analysis. Chapman and Hall, Boca Raton (2001)
Shi, J., Wang, X.: On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246, 2788–2812 (2009)
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The work is supported in part by the National Natural Science Foundations of China (12171296), the Natural Science Basic Research Program of Shaanxi (2024JC-YBQN-0006) and the Youth Innovation Team of Shaanxi Universities.
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Daoxin Qiu contributed to conceptualization, writing—original draft, investigation, software, and formal analysis. Yunfeng Jia contributed to conceptualization, methodology, investigation, writing—review, and formal analysis. Jingjing Wang contributed to conceptualization, investigation, and formal analysis. All authors reviewed the manuscript
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Qiu, D., Jia, Y. & Wang, J. Effects of nonlinear growth, cross-diffusion and protection zone on a diffusive predation model. Z. Angew. Math. Phys. 75, 109 (2024). https://doi.org/10.1007/s00033-024-02254-3
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DOI: https://doi.org/10.1007/s00033-024-02254-3