# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022065
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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.

show all references

##### 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)
 [1] Wan-Tong Li, Yong-Hong Fan. Periodic solutions in a delayed predator-prey models with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 175-185. doi: 10.3934/dcdsb.2007.8.175 [2] Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228 [3] Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345 [4] Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75 [5] Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607 [6] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [7] Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117 [8] Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857 [9] Xinhong Zhang, Qing Yang. Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3155-3175. doi: 10.3934/dcdsb.2021177 [10] Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159 [11] Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 [12] Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203 [13] H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221 [14] Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure and Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481 [15] Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1967-1990. doi: 10.3934/dcdsb.2020041 [16] Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653 [17] Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117 [18] Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263 [19] Ziyad AlSharawi, Nikhil Pal, Joydev Chattopadhyay. The role of vigilance on a discrete-time predator-prey model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022017 [20] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082

2021 Impact Factor: 1.865