American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022065
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Qualitative structure of a discrete predator-prey model with nonmonotonic functional response

 1 Minnan Science and Technology University, Quanzhou, Fujian 362332, China 2 School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

* Corresponding author: Shengfu Deng

This paper is dedicated to Professor Jibin Li for his 80th birthday

Received  December 2021 Revised  February 2022 Early access March 2022

In this paper, we study the qualitative structure of a discrete predator-prey model with nonmonotonic functional response near a degenerate fixed point whose eigenvalues are $\pm1$. Firstly, the model is transformed into an ordinary differential system by using the normal form theory and the Takens's theorem. Then, the qualitative properties of this ordinary differential system near the degenerate equilibrium are analyzed with the blowing-up method. Finally, according to the conjugacy between the discrete model and the time-one mapping of the vector field, the qualitative structure of this discrete model is obtained. Numerical simulations are also given.

Citation: Yanlin Zhang, Qi Cheng, Shengfu Deng. Qualitative structure of a discrete predator-prey model with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022065
References:
 [1] T. Agrawal and M. Saleem, Complex dynamics in a ratio-dependent two-predator one-prey model, Comput. Appl. Math., 34 (2015), 265-274.  doi: 10.1007/s40314-014-0115-1. [2] M. H. Al-Towaiq, Qualitative study of an adaptive three species predator-prey model, Far East J. Appl. Math., 70 (2012), 123-138. [3] M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416. [4] A. A. Berryman, The origins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535. [5] Q. Chen and Z. Teng, Codimension-two bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, J. Difference Equ. Appl., 23 (2017), 2093-2115.  doi: 10.1080/10236198.2017.1395418. [6] Q. Cheng, Y. Zhang and S. Deng, Qualitative analysis of a degenerate fixed point of a discretepredator-prey model with cooperative hunting, Math. Meth. Appl. Sci., 44 (2021), 11059-11075.  doi: 10.1002/mma.7468. [7] Y.-H. Chou, Y. Chow, X. Hu and S. R.-J. Jang, A Ricker-type predator-prey system with hunting cooperation in discrete time, Math. Comput. Simulation, 190 (2021), 570-586.  doi: 10.1016/j.matcom.2021.06.003. [8] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, New York, 1994. doi: 10.1017/CBO9780511665639. [9] J. Dhar and K. S. Jatav, Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories, Ecol. Complex., 16 (2013), 59-67. [10] F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006. [11] F. Dumortier, P. R. Rodrigues and R. Roussarie, Germs of Diffeomorphisms in the Plane, Lecture Notes in Mathematics, 902. Springer-Verlag, Berlin-New York, 1981. [12] L. Fei, X. Chen and B. Han, Bifurcation analysis and hybrid control of a discrete-time predator-prey model, J. Difference Equ. Appl., 27 (2021), 102-117.  doi: 10.1080/10236198.2021.1876038. [13] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980. [14] B.-S. Goh, Management and Analysis of Biological Populations, Elsevier, Amsterdam, 1980. [15] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl., 12 (2011), 2356-2377.  doi: 10.1016/j.nonrwa.2011.02.009. [16] G. Izzo and A. Vecchio, A discerete time version for models of population dynamics in the presence of an infection, J. Comput. Appl. Math., 210 (2007), 210-221.  doi: 10.1016/j.cam.2006.10.065. [17] Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27 (2006), 259-277.  doi: 10.1016/j.chaos.2005.03.040. [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2$^{nd}$ edition, Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [19] A. J. Lotka, Elements of physical biology, Nature, 116 (1925), 341-343. [20] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001. [21] J. D. Murray, Mathematical Biology, 2$^{nd}$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [22] S. Pal, N. Pal and J. Chattopadhyay, Hunting cooperation in a discrete-time predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850083, 22 pp. doi: 10.1142/S0218127418500839. [23] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896. [24] A. Singh and P. Deolia, Bifurcation and chaos in a discrete predator-prey model with Holling type-Ⅲ functional response and harvesting effect, J. Biol. Systems, 29 (2021), 451-478.  doi: 10.1142/S021833902140009X. [25] Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity, 17 (2004), 1407-1426.  doi: 10.1088/0951-7715/17/4/015. [26] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0. [27] J. Zhang and J. Zhong, Qualitative structures of a degenerate fixed point of a Ricker competition model, J. Difference Equ. Appl., 25 (2019), 430-454.  doi: 10.1080/10236198.2019.1581181. [28] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.

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References:
 [1] T. Agrawal and M. Saleem, Complex dynamics in a ratio-dependent two-predator one-prey model, Comput. Appl. Math., 34 (2015), 265-274.  doi: 10.1007/s40314-014-0115-1. [2] M. H. Al-Towaiq, Qualitative study of an adaptive three species predator-prey model, Far East J. Appl. Math., 70 (2012), 123-138. [3] M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416. [4] A. A. Berryman, The origins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535. [5] Q. Chen and Z. Teng, Codimension-two bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, J. Difference Equ. Appl., 23 (2017), 2093-2115.  doi: 10.1080/10236198.2017.1395418. [6] Q. Cheng, Y. Zhang and S. Deng, Qualitative analysis of a degenerate fixed point of a discretepredator-prey model with cooperative hunting, Math. Meth. Appl. Sci., 44 (2021), 11059-11075.  doi: 10.1002/mma.7468. [7] Y.-H. Chou, Y. Chow, X. Hu and S. R.-J. Jang, A Ricker-type predator-prey system with hunting cooperation in discrete time, Math. Comput. Simulation, 190 (2021), 570-586.  doi: 10.1016/j.matcom.2021.06.003. [8] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, New York, 1994. doi: 10.1017/CBO9780511665639. [9] J. Dhar and K. S. Jatav, Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories, Ecol. Complex., 16 (2013), 59-67. [10] F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006. [11] F. Dumortier, P. R. Rodrigues and R. Roussarie, Germs of Diffeomorphisms in the Plane, Lecture Notes in Mathematics, 902. Springer-Verlag, Berlin-New York, 1981. [12] L. Fei, X. Chen and B. Han, Bifurcation analysis and hybrid control of a discrete-time predator-prey model, J. Difference Equ. Appl., 27 (2021), 102-117.  doi: 10.1080/10236198.2021.1876038. [13] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980. [14] B.-S. Goh, Management and Analysis of Biological Populations, Elsevier, Amsterdam, 1980. [15] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl., 12 (2011), 2356-2377.  doi: 10.1016/j.nonrwa.2011.02.009. [16] G. Izzo and A. Vecchio, A discerete time version for models of population dynamics in the presence of an infection, J. Comput. Appl. Math., 210 (2007), 210-221.  doi: 10.1016/j.cam.2006.10.065. [17] Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27 (2006), 259-277.  doi: 10.1016/j.chaos.2005.03.040. [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2$^{nd}$ edition, Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [19] A. J. Lotka, Elements of physical biology, Nature, 116 (1925), 341-343. [20] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001. [21] J. D. Murray, Mathematical Biology, 2$^{nd}$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [22] S. Pal, N. Pal and J. Chattopadhyay, Hunting cooperation in a discrete-time predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850083, 22 pp. doi: 10.1142/S0218127418500839. [23] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896. [24] A. Singh and P. Deolia, Bifurcation and chaos in a discrete predator-prey model with Holling type-Ⅲ functional response and harvesting effect, J. Biol. Systems, 29 (2021), 451-478.  doi: 10.1142/S021833902140009X. [25] Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity, 17 (2004), 1407-1426.  doi: 10.1088/0951-7715/17/4/015. [26] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0. [27] J. Zhang and J. Zhong, Qualitative structures of a degenerate fixed point of a Ricker competition model, J. Difference Equ. Appl., 25 (2019), 430-454.  doi: 10.1080/10236198.2019.1581181. [28] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.
Phase portraits of system (20) in the rigion $U_{+}$
The phase portraits for $0<D<\frac{8}{3}$
The phase portraits for $D = \frac{8}{3}$
The phase portraits for $D>\frac{8}{3}$
Phase portraits of system (1)
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