April  2021, 26(4): 1843-1866. doi: 10.3934/dcdsb.2020042

Spatial pattern formation in activator-inhibitor models with nonlocal dispersal

1. 

Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong 264209, China

2. 

Department of Mathematics, William & Mary, Williamsburg, Virginia, 23187-8795, USA

3. 

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

* Corresponding author: Shanshan Chen

Received  July 2019 Revised  October 2019 Published  April 2021 Early access  February 2020

Fund Project: S. Chen is supported by National Natural Science Foundation of China (No 11771109), and J. Shi is supported by US-NSF grants DMS-1715651 and DMS-1853598, G. Zhang is supported by National Natural Science Foundation of China (No 11701472)

The stability of a constant steady state in a general reaction-diffusion activator-inhibitor model with nonlocal dispersal of the activator or inhibitor is considered. It is shown that Turing type instability and associated spatial patterns can be induced by fast nonlocal inhibitor dispersal and slow activator diffusion, and slow nonlocal activator dispersal also causes instability but may not produce stable spatial patterns. The existence of nonconstant positive steady states is shown through bifurcation theory. This suggests a new mechanism for spatial pattern formation, which has different instability parameter regime compared to Turing mechanism. The theoretical results are applied to pattern formation problems in nonlocal Klausmeier-Gray-Scott water-plant model and Holling-Tanner predator-prey model.

Citation: Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042
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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.

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L. J. S. AllenE. J. Allen and S. Ponweera, A mathematical model for weed dispersal and control, Bull. Math. Biol., 58 (1996), 815-834.  doi: 10.1007/BF02459485.

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X. L. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

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W. Chen and M. J. Ward, The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model, SIAM J. Appl. Dyn. Syst., 10 (2011), 582-666.  doi: 10.1137/09077357X.

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J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

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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. JangW.-M. Ni and M.-X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

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J.-Y. JinJ.-P. ShiJ.-J. Wei and F.-Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.

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T. KolokolnikovM. J. Ward and J.-C. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.

[22]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[23]

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.

[24]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2017-2042.  doi: 10.2307/2265698.

[25]

I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[26]

F. LiY. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[27]

S.-B. LiJ.-H. Wu and Y.-Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[28]

X. LiW.-H. Jiang and J.-P. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.

[29]

Y. LiA. Marciniak-CzochraI. Takagi and B.-Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.

[30]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[31]

A. Marciniak-CzochraS. H$\ddot{a}$rtingG. Karch and K. Suzuki, Dynamical spike solutions in a nonlocal model of pattern formation, Nonlinearity, 31 (2018), 1757-1781.  doi: 10.1088/1361-6544/aaa5dc.

[32]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures. Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.

[33]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[34]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[35]

W.-M. Ni and M.-X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.  doi: 10.1126/science.261.5118.189.

[38]

R. PengF.-Q. Yi and X.-Q. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 254 (2013), 2465-2498.  doi: 10.1016/j.jde.2012.12.009.

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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.

[40]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambers and et al., Self-organization of vegetation in arid ecosystems, Amer. Naturalist, 160 (2002), 524-530.  doi: 10.1086/342078.

[41]

M. RietkerkS. C. DekkerP. C. De Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems., Science, 305 (2004), 1926-1929.  doi: 10.1126/science.1101867.

[42]

L. A. Segel and J. L. Jackson, Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.

[43]

L. Sewalt and A. Doelman, Spatially periodic multipulse patterns in a generalized Klausmeier-Gray-Scott model, SIAM J. Appl. Dyn. Syst., 16 (2017), 1113-1163.  doi: 10.1137/16M1078756.

[44]

W.-X. Shen and X.-X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.  doi: 10.3934/dcds.2015.35.1665.

[45]

R. Sheth, L. Marcon, M. F. Bastida and M. Junco, et al., Hox genes regulate digit patterning by controlling the wavelength of a Turing-type mechanism, Science, 338 (2012), 1476-1480. doi: 10.1126/science.1226804.

[46]

S. SickS. ReinkerJ. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450.  doi: 10.1126/science.1130088.

[47]

J.-W. SunW.-T. Li and Z.-C. Wang, The periodic principal eigenvalues with applications to the nonlocal dispersal logistic equation, J. Differential Equations, 263 (2017), 934-971.  doi: 10.1016/j.jde.2017.03.001.

[48]

J.-W. SunW.-T. Li and F.-Y. Yang, Blow-up profiles for positive solutions of nonlocal dispersal equation, Appl. Math. Lett., 42 (2015), 59-63.  doi: 10.1016/j.aml.2014.11.009.

[49]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.

[50]

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.

[51]

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.

[52]

J.-F. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, J. Dynam. Differential Equations, 29 (2017), 1383-1409.  doi: 10.1007/s10884-016-9517-7.

[53]

J.-F. WangJ.-P. Shi and J.-J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[54]

F.-Q. YiJ.-J. Wei and J.-P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

show all references

References:
[1]

M. AlfaroH. Izuhara and M. Mimura, On a nonlocal system for vegetation in drylands, J. Math. Biol., 77 (2018), 1761-1793.  doi: 10.1007/s00285-018-1215-0.

[2]

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.

[3]

L. J. S. AllenE. J. Allen and S. Ponweera, A mathematical model for weed dispersal and control, Bull. Math. Biol., 58 (1996), 815-834.  doi: 10.1007/BF02459485.

[4]

X. L. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

[5]

X.-L. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35pp. doi: 10.1007/s00526-018-1419-6.

[6]

K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal., 24 (1995), 1713-1725.  doi: 10.1016/0362-546X(94)00218-7.

[7]

W. Chen and M. J. Ward, The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model, SIAM J. Appl. Dyn. Syst., 10 (2011), 582-666.  doi: 10.1137/09077357X.

[8]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.

[9]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

[10]

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.

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.

[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]

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.

[14]

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

[15]

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.

[16]

J. JangW.-M. Ni and M.-X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[17]

J.-Y. JinJ.-P. ShiJ.-J. Wei and F.-Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.

[18]

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 environment, Bull. Math. Biol., 74 (2012), 803-833.  doi: 10.1007/s11538-011-9688-7.

[19]

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

[20]

T. KolokolnikovM. J. Ward and J.-C. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.

[21]

T. KolokolnikovM. J. Ward and J.-C. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.

[22]

S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[23]

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.

[24]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2017-2042.  doi: 10.2307/2265698.

[25]

I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[26]

F. LiY. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[27]

S.-B. LiJ.-H. Wu and Y.-Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[28]

X. LiW.-H. Jiang and J.-P. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.

[29]

Y. LiA. Marciniak-CzochraI. Takagi and B.-Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.

[30]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[31]

A. Marciniak-CzochraS. H$\ddot{a}$rtingG. Karch and K. Suzuki, Dynamical spike solutions in a nonlocal model of pattern formation, Nonlinearity, 31 (2018), 1757-1781.  doi: 10.1088/1361-6544/aaa5dc.

[32]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures. Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.

[33]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[34]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[35]

W.-M. Ni and M.-X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.  doi: 10.1126/science.261.5118.189.

[38]

R. PengF.-Q. Yi and X.-Q. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 254 (2013), 2465-2498.  doi: 10.1016/j.jde.2012.12.009.

[39]

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.

[40]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambers and et al., Self-organization of vegetation in arid ecosystems, Amer. Naturalist, 160 (2002), 524-530.  doi: 10.1086/342078.

[41]

M. RietkerkS. C. DekkerP. C. De Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems., Science, 305 (2004), 1926-1929.  doi: 10.1126/science.1101867.

[42]

L. A. Segel and J. L. Jackson, Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.

[43]

L. Sewalt and A. Doelman, Spatially periodic multipulse patterns in a generalized Klausmeier-Gray-Scott model, SIAM J. Appl. Dyn. Syst., 16 (2017), 1113-1163.  doi: 10.1137/16M1078756.

[44]

W.-X. Shen and X.-X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.  doi: 10.3934/dcds.2015.35.1665.

[45]

R. Sheth, L. Marcon, M. F. Bastida and M. Junco, et al., Hox genes regulate digit patterning by controlling the wavelength of a Turing-type mechanism, Science, 338 (2012), 1476-1480. doi: 10.1126/science.1226804.

[46]

S. SickS. ReinkerJ. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450.  doi: 10.1126/science.1130088.

[47]

J.-W. SunW.-T. Li and Z.-C. Wang, The periodic principal eigenvalues with applications to the nonlocal dispersal logistic equation, J. Differential Equations, 263 (2017), 934-971.  doi: 10.1016/j.jde.2017.03.001.

[48]

J.-W. SunW.-T. Li and F.-Y. Yang, Blow-up profiles for positive solutions of nonlocal dispersal equation, Appl. Math. Lett., 42 (2015), 59-63.  doi: 10.1016/j.aml.2014.11.009.

[49]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.

[50]

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.

[51]

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.

[52]

J.-F. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, J. Dynam. Differential Equations, 29 (2017), 1383-1409.  doi: 10.1007/s10884-016-9517-7.

[53]

J.-F. WangJ.-P. Shi and J.-J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[54]

F.-Q. YiJ.-J. Wei and J.-P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

Figure 1.  Diagram for parameters regions $ R_1 $ and $ R_2 $. Here only nonlocal model (3) could exhibit complex patterns in region (Ⅰ), only reaction-diffusion model (4) could exhibit complex patterns in region (Ⅱ), and both model (3) and (4) could exhibit complex patterns in region (Ⅲ)
Figure 2.  The solution of model (50) converges to the constant positive equilibrium $ (u_1,v_1) $ for $ c>b $. Here $ d = 6 $, $ A = 4 $, $ b = 1.8 $, $ L = 1 $, $ c = 10 $, and the initial values $ u(x,0) = 1.5+0.001x(1-x) $, and $ v(x,0) = 1.1+0.001\cos x $. (Left) $ u(x,t) $; (Right) $ v(x,t) $
Figure 3.  The solution of model (50) forms a one-spike spatial pattern for $ c<b $, and the upper panels show the profile of $ u $ and $ v $ at time $ t = 12 $ and $ t = 30 $, respectively. Here $ d = 6 $, $ A = 4 $, $ b = 1.8 $, $ L = 1 $, $ c = 1 $, and the initial values $ u(x,0) = 1.5+0.001x(1-x) $, and $ v(x,0) = 1.1+0.001\cos x $. (Left) $ u(x,t) $; (Right) $ v(x,t) $
Figure 4.  The solution of model (50) forms to a two-spike spatial pattern for $ c<b $, and the upper panels show the profile of $ u $ and $ v $ at time $ t = 13 $ and $ t = 30 $, respectively. Here $ d = 6 $, $ A = 4 $, $ b = 1.8 $, $ L = 2 $, $ c = 1 $, and the initial values $ u(x,0) = 1.5+0.001x(2-x) $, and $ v(x,0) = 1.1+0.001\cos x $. (Left) $ u(x,t) $; (Right) $ v(x,t) $
Figure 5.  The solution of model (54) converges to the constant steady state (respectively, a nonconstant stationary pattern) for $ c<c_0 $ (respectively, $ c>c_0 $), and the lower panel show the profile of the nonconstant stationary pattern with $ c = 4 $. Here $ \beta = 0.2 $, $ m = 2 $, $ L = \pi $, $ s = 1 $, $ d = 0.03 $, and the initial values $ u(x,0) = 0.5+0.05\cos x $, and $ v(x,0) = 0.3+0.02\cos x $. (Upper) $ c = 2 $; (Middle) $ c = 4 $; (Left) $ u(x,t) $; (Right) $ v(x,t) $
Figure 6.  The solution converges to a nonconstant stationary pattern for the nonlocal model (54), whereas the solution converges to the constant steady state for the reaction-diffusion model (59). Here the initial values $ u(x,0) = 0.3+0.05\cos x/2 $, and $ v(x,0) = 0.5+0.03\cos x/2 $, $ \beta = 0.2 $, $ m = 2 $, $ L = 2\pi $, $ s = 1 $, $ d = 0.15 $, $ c = 4 $, and $ (c,d)\in R_2\backslash R_1 $, where $ R_1 $ and $ R_2 $ are defined as in Eqs. (44) and (39), respectively. (Left) $ u(x,t) $; (Right) $ v(x,t) $
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