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doi: 10.3934/dcdsb.2020295

Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal

Xiaoli Wang , and  Guohong Zhang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

*Corresponding author: Xiaoli Wang

Received  May 2020 Revised  August 2020 Published  October 2020

Fund Project: X.-L. Wang is partially supported by grants from National Science Foundation of China (11701472, 11871060), Fundamental Research Funds for the Central Universities(XDJK2020B050); G.-H. Zhang is partially supported by grants from National Science Foundation of China (11871403)

In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of spatial patterns. Compared with the "spatial averaging" nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

Keywords: Nonlocal dispersal, activator-inhibitor system, water-biomass model, pattern formation, bifurcation.
Mathematics Subject Classification: Primary: 35K57, 35B32; Secondary: 35B35, 35B36, 37L15, 92C15.
Citation: Xiaoli Wang, Guohong Zhang. Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020295
References:
[1]

E. J. Allen, L. J. S. Allen and X. Gilliam, Dispersal and competition models for plants, J. Math. Biol., 34 (1996), 455-481.  doi: 10.1007/BF00167944.  Google Scholar

[2]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[3]

R. S. Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38.   Google Scholar

[4]

J.-F. Cao, W.-T. Li and F.-Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar

[5]

S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. Mccalla, Crime modeling with Lévy flights, SIAM J. Appl. Math., 73 (2013), 1703-1720.  doi: 10.1137/120895408.  Google Scholar

[6]

S. Chen, J. Shi and G. Zhang, Spatial pattern formation in activator-inhibitor models with nonlocal dispersal, Discrete Contin. Dyn. Syst. Ser. B, to appear. doi: 10.3934/dcdsb.2020042.  Google Scholar

[7]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[8]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.  Google Scholar

[9]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.  Google Scholar

[10]

C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341.  Google Scholar

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[12]

L. Eigentler and J. A. Sherratt, Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal, J. Math. Biol., 77 (2018), 739-763.  doi: 10.1007/s00285-018-1233-y.  Google Scholar

[13]

P. Fife, Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions, Trends in Nonlinear Analysis, Springer, Berlin, 2003,153–191. doi: 10.1007/978-3-662-05281-5_3.  Google Scholar

[14]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.  doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[15]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[16]

K. Gowda, Y. Chen, S. Iams and M. Silber, Assessing the robustness of spatial pattern sequences in a dryland vegetation model, Proc. A, 472 (2016), 25pp. doi: 10.1098/rspa.2015.0893.  Google Scholar

[17]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Engrg. Sci., 38 (1983), 29-43.  doi: 10.1016/0009-2509(83)80132-8.  Google Scholar

[18]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $\text{A+2B}\rightarrow\text{3B}; \text{B}\rightarrow \text{C}$, Chem. Engrg. Sci., 39 (1984), 1087-1097.  doi: 10.1016/0009-2509(84)87017-7.  Google Scholar

[19]

P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behavior in isothermal reactions, J. Phys. Chem., 89 (1985), 22-32.  doi: 10.1021/j100247a009.  Google Scholar

[20]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[21]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[22]

B. J. Kealy and D. J. Wollkind, A nonlinear stability analysis of vegetative Turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat enviroment, Bull. Math. Biol., 74 (2012), 803-833.  doi: 10.1007/s11538-011-9688-7.  Google Scholar

[23]

S. Kinast, Y. R. Zelnik, G. Bel and E. Meron, Interplay between Turing mechanisms can increase pattern diversity, Phys. Rev. Lett., 112 (2014). doi: 10.1103/PhysRevLett.112.078701.  Google Scholar

[24]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.  Google Scholar

[25]

S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047.  Google Scholar

[26]

T. Kuniya and J. Wang, Global dynamics of an SIR epidemic model with nonlocal diffusion, Nonlinear Anal. Real World Appl., 43 (2018), 262-282.  doi: 10.1016/j.nonrwa.2018.03.001.  Google Scholar

[27]

H. Nakao and A. S. Mikhailov, Turing patterns in network-organized activator-inhibitor systems, Nature Phys., 6 (2010), 544-550.  doi: 10.1038/nphys1651.  Google Scholar

[28]

H. Ninomiya, Y. Tanaka and H. Yamamoto, Reaction, diffusion and non-local interaction, J. Math. Biol., 75 (2017), 1203-1233.  doi: 10.1007/s00285-017-1113-x.  Google Scholar

[29]

J. Pejsachowicz and P. J. Rabier, Degree theory for $\text{C}^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.  Google Scholar

[30]

J. A. Powell and N. E. Zimmermann, Multiscale analysis of active seed dispersal contributes to resolving Reid's paradox, Ecology, 85 (2004), 490-506.  doi: 10.1890/02-0535.  Google Scholar

[31]

Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems, Oikos, 117 (2008), 1522-1532.  doi: 10.1111/j.0030-1299.2008.16735.x.  Google Scholar

[32]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[33]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[34]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[35]

S. van der Stelt, A. Doelman, G. Hek and J. D. M. Rademacher, Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model, J. Nonlinear Sci., 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.  Google Scholar

[36]

X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461.  doi: 10.1006/jdeq.2001.4129.  Google Scholar

[37]

F.-Y. Yang and W.-T. Li, Dynamics of a nonlocal dispersal SIS epidemic model, Commun. Pure Appl. Anal., 16 (2017), 781-797.  doi: 10.3934/cpaa.2017037.  Google Scholar

[38]

F.-Y. Yang, W.-T. Li and S. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differential Equations, 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.  Google Scholar

show all references

References:
[1]

E. J. Allen, L. J. S. Allen and X. Gilliam, Dispersal and competition models for plants, J. Math. Biol., 34 (1996), 455-481.  doi: 10.1007/BF00167944.  Google Scholar

[2]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[3]

R. S. Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38.   Google Scholar

[4]

J.-F. Cao, W.-T. Li and F.-Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar

[5]

S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. Mccalla, Crime modeling with Lévy flights, SIAM J. Appl. Math., 73 (2013), 1703-1720.  doi: 10.1137/120895408.  Google Scholar

[6]

S. Chen, J. Shi and G. Zhang, Spatial pattern formation in activator-inhibitor models with nonlocal dispersal, Discrete Contin. Dyn. Syst. Ser. B, to appear. doi: 10.3934/dcdsb.2020042.  Google Scholar

[7]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[8]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.  Google Scholar

[9]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.  Google Scholar

[10]

C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341.  Google Scholar

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[12]

L. Eigentler and J. A. Sherratt, Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal, J. Math. Biol., 77 (2018), 739-763.  doi: 10.1007/s00285-018-1233-y.  Google Scholar

[13]

P. Fife, Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions, Trends in Nonlinear Analysis, Springer, Berlin, 2003,153–191. doi: 10.1007/978-3-662-05281-5_3.  Google Scholar

[14]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.  doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[15]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[16]

K. Gowda, Y. Chen, S. Iams and M. Silber, Assessing the robustness of spatial pattern sequences in a dryland vegetation model, Proc. A, 472 (2016), 25pp. doi: 10.1098/rspa.2015.0893.  Google Scholar

[17]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Engrg. Sci., 38 (1983), 29-43.  doi: 10.1016/0009-2509(83)80132-8.  Google Scholar

[18]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $\text{A+2B}\rightarrow\text{3B}; \text{B}\rightarrow \text{C}$, Chem. Engrg. Sci., 39 (1984), 1087-1097.  doi: 10.1016/0009-2509(84)87017-7.  Google Scholar

[19]

P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behavior in isothermal reactions, J. Phys. Chem., 89 (1985), 22-32.  doi: 10.1021/j100247a009.  Google Scholar

[20]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[21]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[22]

B. J. Kealy and D. J. Wollkind, A nonlinear stability analysis of vegetative Turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat enviroment, Bull. Math. Biol., 74 (2012), 803-833.  doi: 10.1007/s11538-011-9688-7.  Google Scholar

[23]

S. Kinast, Y. R. Zelnik, G. Bel and E. Meron, Interplay between Turing mechanisms can increase pattern diversity, Phys. Rev. Lett., 112 (2014). doi: 10.1103/PhysRevLett.112.078701.  Google Scholar

[24]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.  Google Scholar

[25]

S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047.  Google Scholar

[26]

T. Kuniya and J. Wang, Global dynamics of an SIR epidemic model with nonlocal diffusion, Nonlinear Anal. Real World Appl., 43 (2018), 262-282.  doi: 10.1016/j.nonrwa.2018.03.001.  Google Scholar

[27]

H. Nakao and A. S. Mikhailov, Turing patterns in network-organized activator-inhibitor systems, Nature Phys., 6 (2010), 544-550.  doi: 10.1038/nphys1651.  Google Scholar

[28]

H. Ninomiya, Y. Tanaka and H. Yamamoto, Reaction, diffusion and non-local interaction, J. Math. Biol., 75 (2017), 1203-1233.  doi: 10.1007/s00285-017-1113-x.  Google Scholar

[29]

J. Pejsachowicz and P. J. Rabier, Degree theory for $\text{C}^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.  Google Scholar

[30]

J. A. Powell and N. E. Zimmermann, Multiscale analysis of active seed dispersal contributes to resolving Reid's paradox, Ecology, 85 (2004), 490-506.  doi: 10.1890/02-0535.  Google Scholar

[31]

Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems, Oikos, 117 (2008), 1522-1532.  doi: 10.1111/j.0030-1299.2008.16735.x.  Google Scholar

[32]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[33]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[34]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[35]

S. van der Stelt, A. Doelman, G. Hek and J. D. M. Rademacher, Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model, J. Nonlinear Sci., 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.  Google Scholar

[36]

X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461.  doi: 10.1006/jdeq.2001.4129.  Google Scholar

[37]

F.-Y. Yang and W.-T. Li, Dynamics of a nonlocal dispersal SIS epidemic model, Commun. Pure Appl. Anal., 16 (2017), 781-797.  doi: 10.3934/cpaa.2017037.  Google Scholar

[38]

F.-Y. Yang, W.-T. Li and S. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differential Equations, 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.  Google Scholar

Figure 1.  The kernel (1.5) with $ l = 20 $, and $ d_w = 1 $ (blue), $ d_w = 3 $ (red)
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Figure 2.  (a): Parameter space for Turing instability when the activator has a nonlocal dispersal. (b): The effect of nonlocal dispersal on the parameter region of Turing instability. Here, $ A = 1, B = 0.45 $, $ d_w = 1(cyan) $, and $ d_w = 3(green) $
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Figure 3.  Graph of $ D(d_v,p) $ when $ d_u<f_u $, $ d_u = f_u $ and $ d_u>f_u $. Here, $ A = 1, B = 0.45, l = 20 $, $ d_w = 1(cyan), 3(green) $, and $ d_u = 0.2 $ in $ (a) $, $ d_u = 0.45 $ in $ (b) $, $ d_u = 1 $ in $ (c) $
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Figure 4.  $ (a), (c), (e): $ Spiky or bump pattern formation of model (4.2). $ (b), (d), (f): $ Antiphase or inphase distributions of plant $ u $ and water $ v $ of model (4.2). Here $ A = 1, B = 0.45, d_v = 30, d_w = 1 $ and $ d_u = 0.2, 0.45, 1 $
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Figure 5.  $ (a) $ Turing instability region of the classical reaction-diffusion model. $ (b) $ Turing instability region of the "spatial averaging" nonlocal dispersal model when the inhibitor has a nonlocal dispersal. $ (c) $ Turing instability region of the nonlocal model when the inhibitor has a nonlocal dispersal with the kernel function (1.5). Here, the constant equilibrium solution is stable in region "S" and the spatial scale could induce instability in region "U"
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