# American Institute of Mathematical Sciences

May  2017, 22(3): 717-740. doi: 10.3934/dcdsb.2017035

## Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response

 Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje 13615, C.P. 97119, Mérida, Mexico

Dedicated to Professor Stephen Cantrell on the occasion of his 60th birthday.

Received  August 2015 Revised  June 2016 Published  December 2016

In this article, the clasical Bazykin model is modifed with Bedding–ton–DeAngelis functional response, subject to self and cross-diffusion, in order to study the spatial dynamics of the model.We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We present some numerical simulations of time evolution of patterns to show the important role played by self and cross-diffusion as well as other parameters leading to complex patterns in the plane.

Citation: Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
##### References:

show all references

##### References:
Relation between $H(k^{2})$ and $\Re( \lambda )$ for $\det( \Phi_w(w^{*}) ) <0$. (a) $H(k^{2})$ (b) $\Re( \lambda )$
Relation between $H(k^{2})$ (a) and $\Re( \lambda )$ (b) for $\det( \Phi_w(w^{*}) ) >0$
Functions $s=0$ (solid line) and $g=0$ (dashed line)
Phase plane of local system for $\beta=1.1$
Prey and predator densities for $\alpha_{12}=0.0001$. a) $u$ with $t_{max}=0$, b) $v$ with $t_{max}=0$, c) $u$ with $t_{max}=50$, d) $v$ with $t_{max}=50$, e) $u$ with $t_{max}=200$, f) $v$ with $t_{max}=200$
Prey and predator densities for example 2, with $\alpha_{12}=0.0001$ and $t_{max}=400$. a) $u$ density b) $v$ density. Both figures show a pattern called holes
Prey and predator densities for example 3, with $\alpha_{12}=0.0001$ and $t_{max}=1000$. a) $u$ density b) $v$ density. Both figures show a pattern called stripes
 [1] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [2] Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020117 [3] Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042 [4] Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 [5] Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215 [6] Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027 [7] Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331 [8] Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117 [9] Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975 [10] Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 [11] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 [12] Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115 [13] Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111 [14] Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032 [15] Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure & Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022 [16] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [17] Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108 [18] 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 & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 [19] Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597 [20] Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621

2018 Impact Factor: 1.008

## Metrics

• HTML views (24)
• Cited by (0)

• on AIMS