# 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 and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
##### References:
 [1] D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28-34. [2] J. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, 44 (1975), 331-340.  doi: 10.2307/3866. [3] Q. Bie, Q. Wang and Z.-a. Yao, Cross-diffusion induced instability and pattern formation for a Holling type-Ⅱ predator–prey model, Applied Mathematics and Computation, 247 (2014), 1-12.  doi: 10.1016/j.amc.2014.08.088. [4] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations John Wiley & Sons, 2003. [5] D. L. DeAngelis, R. Goldstein and R. O'neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Vol. 42, Springer-Verlag, New York, 1990. [7] G.-P. Hu and X.-L. Li, Turing patterns of a predator–prey model with a modified Leslie–Gower term and cross-diffusions, International Journal of Biomathematics, 5 (2012), 1250060, 17 pp.  doi: 10.1142/S179352451250060X. [8] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory vol. 112, Springer Science & Business Media, 2004. [9] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation Chapman & Hall/CRC Press London, 2008. [10] E. A. McGehee, N. Schutt, D. A. Vasquez and E. Peacock-Lopez, Bifurcations, and temporal and spatial patterns of a modified lotka–volterra model, International Journal of Bifurcation and Chaos, 18 (2008), 2223-2248.  doi: 10.1142/S0218127408021671. [11] J. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications Intercisciplinary Applied Mathematics: Mathematical Biology, Springer, 2003. [12] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives Springer, 2001. [13] S. Sarwardi, M. Haque and P. K. Mandal, Persistence and global stability of bazykin predator–prey model with beddington–deangelis response function, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 189-209.  doi: 10.1016/j.cnsns.2013.05.029. [14] J. Shen and Y. M. Jung, Geometric and stochastic analysis of reaction-diffusion patterns, Int J Pure Appl Math, 19 (2005), 195-248. [15] H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator–prey models with mutual interference, IMA Journal of Applied Mathematics, 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006. [16] E. Tulumello, M. C. Lombardo and M. Sammartino, Cross-diffusion driven instability in a predator-prey system with cross-diffusion, Acta Applicandae Mathematicae, 132 (2014), 621-633.  doi: 10.1007/s10440-014-9935-7. [17] A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [18] R. K. Upadhyay, P. Roy and J. Datta, Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and turing instability, Nonlinear Dynamics, 79 (2015), 2251-2270.  doi: 10.1007/s11071-014-1808-0. [19] X. Wang and Y. Cai, Cross-diffusion-driven instability in a reaction-diffusion harrison predator-prey model, in Abstract and Applied Analysis Hindawi Publishing Corporation, 2013 (2013), Art. ID 306467, 9 pp. [20] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [21] E. Zemskov, Nonlinear analysis of a reaction-diffusion system: Amplitude equations, Journal of Experimental and Theoretical Physics, 115 (2012), 729-732.  doi: 10.1134/S1063776112090178. [22] E. Zemskov, K. Kassner, M. Hauser and W. Horsthemke, Turing space in reaction-diffusion systems with density-dependent cross diffusion, Physical Review E, 87 (2013), 032906.  doi: 10.1103/PhysRevE.87.032906. [23] J.-F. Zhang, W.-T. Li and Y.-X. Wang, Turing patterns of a strongly coupled predator–prey system with diffusion effects, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 847-858.  doi: 10.1016/j.na.2010.09.035. [24] J. Zhou, Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type Ⅲ functional response, Nonlinear Dynamics, 81 (2015), 1535-1552.  doi: 10.1007/s11071-015-2088-z.

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##### References:
 [1] D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28-34. [2] J. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, 44 (1975), 331-340.  doi: 10.2307/3866. [3] Q. Bie, Q. Wang and Z.-a. Yao, Cross-diffusion induced instability and pattern formation for a Holling type-Ⅱ predator–prey model, Applied Mathematics and Computation, 247 (2014), 1-12.  doi: 10.1016/j.amc.2014.08.088. [4] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations John Wiley & Sons, 2003. [5] D. L. DeAngelis, R. Goldstein and R. O'neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Vol. 42, Springer-Verlag, New York, 1990. [7] G.-P. Hu and X.-L. Li, Turing patterns of a predator–prey model with a modified Leslie–Gower term and cross-diffusions, International Journal of Biomathematics, 5 (2012), 1250060, 17 pp.  doi: 10.1142/S179352451250060X. [8] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory vol. 112, Springer Science & Business Media, 2004. [9] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation Chapman & Hall/CRC Press London, 2008. [10] E. A. McGehee, N. Schutt, D. A. Vasquez and E. Peacock-Lopez, Bifurcations, and temporal and spatial patterns of a modified lotka–volterra model, International Journal of Bifurcation and Chaos, 18 (2008), 2223-2248.  doi: 10.1142/S0218127408021671. [11] J. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications Intercisciplinary Applied Mathematics: Mathematical Biology, Springer, 2003. [12] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives Springer, 2001. [13] S. Sarwardi, M. Haque and P. K. Mandal, Persistence and global stability of bazykin predator–prey model with beddington–deangelis response function, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 189-209.  doi: 10.1016/j.cnsns.2013.05.029. [14] J. Shen and Y. M. Jung, Geometric and stochastic analysis of reaction-diffusion patterns, Int J Pure Appl Math, 19 (2005), 195-248. [15] H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator–prey models with mutual interference, IMA Journal of Applied Mathematics, 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006. [16] E. Tulumello, M. C. Lombardo and M. Sammartino, Cross-diffusion driven instability in a predator-prey system with cross-diffusion, Acta Applicandae Mathematicae, 132 (2014), 621-633.  doi: 10.1007/s10440-014-9935-7. [17] A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [18] R. K. Upadhyay, P. Roy and J. Datta, Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and turing instability, Nonlinear Dynamics, 79 (2015), 2251-2270.  doi: 10.1007/s11071-014-1808-0. [19] X. Wang and Y. Cai, Cross-diffusion-driven instability in a reaction-diffusion harrison predator-prey model, in Abstract and Applied Analysis Hindawi Publishing Corporation, 2013 (2013), Art. ID 306467, 9 pp. [20] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [21] E. Zemskov, Nonlinear analysis of a reaction-diffusion system: Amplitude equations, Journal of Experimental and Theoretical Physics, 115 (2012), 729-732.  doi: 10.1134/S1063776112090178. [22] E. Zemskov, K. Kassner, M. Hauser and W. Horsthemke, Turing space in reaction-diffusion systems with density-dependent cross diffusion, Physical Review E, 87 (2013), 032906.  doi: 10.1103/PhysRevE.87.032906. [23] J.-F. Zhang, W.-T. Li and Y.-X. Wang, Turing patterns of a strongly coupled predator–prey system with diffusion effects, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 847-858.  doi: 10.1016/j.na.2010.09.035. [24] J. Zhou, Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type Ⅲ functional response, Nonlinear Dynamics, 81 (2015), 1535-1552.  doi: 10.1007/s11071-015-2088-z.
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
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