August  2021, 26(8): 4459-4477. doi: 10.3934/dcdsb.2020295

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

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

*Corresponding author: Xiaoli Wang

Received  May 2020 Revised  August 2020 Published  August 2021 Early access  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.

Citation: Xiaoli Wang, Guohong Zhang. Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4459-4477. doi: 10.3934/dcdsb.2020295
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P. W. BatesP. C. FifeX. 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.

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J.-F. CaoW.-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.

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

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[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.

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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.

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

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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.

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[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.

[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.

[20]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[21]

C.-Y. KaoY. 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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[28]

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

[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.

[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.

[31]

Y. PueyoS. KéfiC. 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.

[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.

[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.

[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.

[35]

S. van der SteltA. DoelmanG. 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.

[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.

[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.

[38]

F.-Y. YangW.-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.

show all references

References:
[1]

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

[2]

P. W. BatesP. C. FifeX. 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.

[3]

R. S. CantrellC. CosnerY. Lou and D. Ryan, Evolutionary stability of ideal dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38. 

[4]

J.-F. CaoW.-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.

[5]

S. ChaturapruekJ. BreslauD. YazdiT. Kolokolnikov and S. G. Mccalla, Crime modeling with Lévy flights, SIAM J. Appl. Math., 73 (2013), 1703-1720.  doi: 10.1137/120895408.

[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.

[7]

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

[8]

C. CortázarM. ElguetaJ. 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.

[9]

C. CortázarM. ElguetaJ. 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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[20]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[21]

C.-Y. KaoY. 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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[28]

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

[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.

[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.

[31]

Y. PueyoS. KéfiC. 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.

[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.

[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.

[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.

[35]

S. van der SteltA. DoelmanG. 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.

[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.

[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.

[38]

F.-Y. YangW.-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.

Figure 1.  The kernel (1.5) with $ l = 20 $, and $ d_w = 1 $ (blue), $ d_w = 3 $ (red)
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) $
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) $
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 $
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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