September  2022, 15(9): 2747-2793. doi: 10.3934/dcdss.2022043

Dynamics of patchy vegetation patterns in the two-dimensional generalized Klausmeier model

1. 

The Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, 02903, USA

2. 

Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

* Corresponding author: Tony Wong

Received  April 2021 Revised  December 2021 Published  September 2022 Early access  March 2022

Fund Project: The second author gratefully acknowledges the financial support from the NSERC Discovery Grant Program

We study the dynamical and steady-state behavior of self-organized spatially localized patches or "spots" of vegetation for the Klausmeier reaction-diffusion (RD) system of spatial ecology that models the interaction between surface water and vegetation biomass on a 2-D spatial landscape with a spatially uniform terrain slope gradient. In this context, we develop and implement a hybrid asymptotic-numerical theory to analyze the existence, linear stability, and slow dynamics of multi-spot quasi-equilibrium spot patterns for the Klausmeier model in the singularly perturbed limit where the biomass diffusivity is much smaller than that of the water resource. From the resulting differential-algebraic (DAE) system of ODEs for the spot locations, one primary focus is to analyze how the constant slope gradient influences the steady-state spot configuration. Our second primary focus is to examine bifurcations in quasi-equilibrium multi-spot patterns that are triggered by a slowly varying time-dependent rainfall rate. Many full numerical simulations of the Klausmeier RD system are performed both to illustrate the effect of the terrain slope and rainfall rate on localized spot patterns, as well as to validate the predictions from our hybrid asymptotic-numerical theory.

Citation: Tony Wong, Michael J. Ward. Dynamics of patchy vegetation patterns in the two-dimensional generalized Klausmeier model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (9) : 2747-2793. doi: 10.3934/dcdss.2022043
References:
[1]

S. M. BaerT. Erneux and J. Rinzel, The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance, SIAM J. Appl. Math., 49 (1989), 55-71.  doi: 10.1137/0149003.

[2]

R. BastiaansenP. Carter and A. Doelman, Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems, Nonlinearity, 32 (2019), 2759-2814.  doi: 10.1088/1361-6544/ab1767.

[3]

R. Bastiaansen and A. Doelman, The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space, Physica D, 388 (2019), 45-72.  doi: 10.1016/j.physd.2018.09.003.

[4]

R. BastiaansenA. DoelmanM. B. Eppinga and M. Rietkerk, The effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation, Ecol. Lett., 23 (2020), 414-429.  doi: 10.1111/ele.13449.

[5]

R. BastiaansenO. JaïbiV. DeblauweM. B. EppingaK. SiteurE. SieroS. MermozA. BouvetA. Doelman and M. Rietkerk, Multistability of model and real dryland ecosystems through spatial self-organization, Proc. Natl. Acad. Sci. U.S.A., 115 (2018), 11256-11261.  doi: 10.1073/pnas.1804771115.

[6]

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. Sys., 10 (2011), 582-666.  doi: 10.1137/09077357X.

[7]

Y. ChenT. KolokolnikovJ. Tzou and C. Gai, Patterned vegetation, tipping points, and the rate of climate change, Europ. J. Appl. Math., 26 (2015), 945-958.  doi: 10.1017/S0956792515000261.

[8]

V. DeblauweP. CouteronO. LejeuneJ. Bogaert and N. Barbier, Environmental modulation of self-organized periodic vegetation patterns in Sudan, Ecography, 34 (2011), 990-1001.  doi: 10.1111/j.1600-0587.2010.06694.x.

[9]

M. EhudY. Hezi and G. Erez, Localized structures in dryland vegetation: Forms and functions, Chaos, 17 (2007), 037109.  doi: 10.1063/1.2767246.

[10]

T. Erneux and P. Mandel, Imperfect bifurcation with a slowly varying control parameter, SIAM J. Appl. Math., 46 (1986), 1-15.  doi: 10.1137/0146001.

[11]

P. GandhiL. WernerS. IamsK. Gowda and M. Silber, A topographic mechanism for arcing of dryland vegetation bands, J. Roy. Soc. Interface, 15 (2018), 20180508.  doi: 10.1098/rsif.2018.0508.

[12]

S. GetzinT. E. EricksonH. YizhaqM. Muñoz-RojasA. Huth and K. Wiegand, Bridging ecology and physics: Australian fairy circles regenerate following model assumptions on ecohydrological feedbacks, J. Ecol., 109 (2021), 399-416.  doi: 10.1111/1365-2745.13493.

[13]

E. GiladJ. von HardenbergA. ProvenzaleM. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation, Phys. Rev. Lett., 93 (2004), 098105.  doi: 10.1103/PhysRevLett.93.098105.

[14]

K. Gowda, Y. Chen, S. Iams and M. Silber, Assessing the robustness of spatial pattern sequences in a dryland vegetation model, Proc. Roy. Soc. A: Math., Phys. and Eng. Sci., 472 (2016), 20150893, 25 pp. doi: 10.1098/rspa.2015.0893.

[15]

K. Gowda, S. Iams and M. Silber, Signatures of human impact on self-organized vegetation in the Horn of Africa, Sci. Rep., 8 (2018), Article number: 3622. doi: 10.1038/s41598-018-22075-5.

[16]

R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), 69-106.  doi: 10.1137/0137006.

[17]

R. HilleRisLambersM. RietkerkF. van den BoschH. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61.  doi: 10.2307/2680085.

[18]

O. Jaïbi, A. Doelman, M. Chirilus-Bruckner and E. Meron, The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation, Physica D, 412 (2020), 132637, 30 pp. doi: 10.1016/j.physd.2020.132637.

[19]

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

[20]

K. Knopp, Theory and Application of Infinite Series, Dover Books on Mathematics, 2013.

[21]

T. Kolokolnikov, M. Ward, J. Tzou and J. Wei, Stabilizing a homoclinic stripe, Philos. Trans. Roy. Soc. A, 376 (2018), 20180110, 13 pp. doi: 10.1098/rsta.2018.0110.

[22]

T. KolokolnikovM. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Science, 19 (2009), 1-56.  doi: 10.1007/s00332-008-9024-z.

[23]

P. Mandel and T. Erneux, The slow passage through a steady bifurcation: Delay and memory effects, J. Stat. Phys., 48 (1987), 1059-1070.  doi: 10.1007/BF01009533.

[24]

, MATLAB version 9.4.0. (R2018a), The Mathworks, Inc., Natick, Massachusetts, 2018.

[25]

E. Meron, Modeling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187.  doi: 10.1051/mmnp/20116109.

[26] E. Meron, Nonlinear Physics of Ecosystems, CRC Press, Boca Raton, Florida, 2015.  doi: 10.1201/b18360.
[27]

M. Messaoudi, M. G. Clerc, E. Berríos-Caro, D. Pinto-Ramos, M. Khaffou, A. Makhoute and M. Tlidi, Patchy landscapes in arid environments: Nonlinear analysis of the interaction-redistribution model, Chaos, 30 (2020), 093136, 11 pp. doi: 10.1063/5.0011010.

[28]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambersJ. van de KoppelL. KumarH. H. T. Prins and A. M. de Roos, Self-organization of vegetation in arid ecosystems, Am. Nat., 160 (2002), 524-530.  doi: 10.1086/342078.

[29]

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

[30]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments Ⅰ, Nonlinearity, 23 (2010), 2657-2675.  doi: 10.1088/0951-7715/23/10/016.

[31]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments Ⅳ: Slowly moving patterns and their stability, SIAM J. Appl. Math., 73 (2013), 330-350.  doi: 10.1137/120862648.

[32]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments V: The transition from patterns to desert, SIAM J. Appl. Math., 73 (2013), 1347-1367.  doi: 10.1137/120899510.

[33]

E. Siero, A. Doelman, M. B. Eppinga, J. D. M. Rademacher, M. Rietkerk and K. Siteur, Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes, Chaos, 25 (2015), 036411, 22 pp. doi: 10.1063/1.4914450.

[34]

K. SiteurE. SieroM. B. EppingaJ. D. M. RademacherA. Doelman and M. Rietkerk, Beyond Turing: The response of patterned ecosystems to environmental change, Ecol. Complex., 20 (2014), 81-96.  doi: 10.1016/j.ecocom.2014.09.002.

[35]

J. C. Tzou and L. Tzou, Spot patterns of the Schnakenberg reaction-diffusion system on a curved torus, Nonlinearity, 33 (2020), 643-674.  doi: 10.1088/1361-6544/ab5161.

[36]

J. C. Tzou and L. Tzou, Analysis of spot patterns on a coordinate-invariant model for vegetation on a curved terrain, SIAM J. Appl. Dyn. Sys., 19 (2020), 2500-2529.  doi: 10.1137/20M1326271.

[37]

J. C. TzouM. J. Ward and T. Kolokolnikov, Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems, Physica D, 290 (2015), 24-43.  doi: 10.1016/j.physd.2014.09.008.

[38]

H. UeckerD. Wetzel and J. D. M. Rademacher, pde2path-A Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.

[39]

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 Science, 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.

[40]

J. von HardenbergE. MeronM. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101.  doi: 10.1103/PhysRevLett.87.198101.

[41]

T. Wong and M. J. Ward, Spot patterns in the 2-D Schnakenberg model with localized heterogeneities, Stud. Appl. Math., 146 (2021), 779-833.  doi: 10.1111/sapm.12361.

[42]

T. Wong and M. J. Ward, Weakly nonlinear analysis of peanut-shaped deformations for localized spots of singularly perturbed reaction-diffusion systems, SIAM J. Appl. Dyn. Syst, 19 (2020), 2030-2058.  doi: 10.1137/20M1316779.

[43]

Y. R. ZelnikE. Meron and G. Bel, Gradual regime shifts in fairy circles, PNAS, 112 (2015), 12327-12331.  doi: 10.1073/pnas.1504289112.

show all references

References:
[1]

S. M. BaerT. Erneux and J. Rinzel, The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance, SIAM J. Appl. Math., 49 (1989), 55-71.  doi: 10.1137/0149003.

[2]

R. BastiaansenP. Carter and A. Doelman, Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems, Nonlinearity, 32 (2019), 2759-2814.  doi: 10.1088/1361-6544/ab1767.

[3]

R. Bastiaansen and A. Doelman, The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space, Physica D, 388 (2019), 45-72.  doi: 10.1016/j.physd.2018.09.003.

[4]

R. BastiaansenA. DoelmanM. B. Eppinga and M. Rietkerk, The effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation, Ecol. Lett., 23 (2020), 414-429.  doi: 10.1111/ele.13449.

[5]

R. BastiaansenO. JaïbiV. DeblauweM. B. EppingaK. SiteurE. SieroS. MermozA. BouvetA. Doelman and M. Rietkerk, Multistability of model and real dryland ecosystems through spatial self-organization, Proc. Natl. Acad. Sci. U.S.A., 115 (2018), 11256-11261.  doi: 10.1073/pnas.1804771115.

[6]

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. Sys., 10 (2011), 582-666.  doi: 10.1137/09077357X.

[7]

Y. ChenT. KolokolnikovJ. Tzou and C. Gai, Patterned vegetation, tipping points, and the rate of climate change, Europ. J. Appl. Math., 26 (2015), 945-958.  doi: 10.1017/S0956792515000261.

[8]

V. DeblauweP. CouteronO. LejeuneJ. Bogaert and N. Barbier, Environmental modulation of self-organized periodic vegetation patterns in Sudan, Ecography, 34 (2011), 990-1001.  doi: 10.1111/j.1600-0587.2010.06694.x.

[9]

M. EhudY. Hezi and G. Erez, Localized structures in dryland vegetation: Forms and functions, Chaos, 17 (2007), 037109.  doi: 10.1063/1.2767246.

[10]

T. Erneux and P. Mandel, Imperfect bifurcation with a slowly varying control parameter, SIAM J. Appl. Math., 46 (1986), 1-15.  doi: 10.1137/0146001.

[11]

P. GandhiL. WernerS. IamsK. Gowda and M. Silber, A topographic mechanism for arcing of dryland vegetation bands, J. Roy. Soc. Interface, 15 (2018), 20180508.  doi: 10.1098/rsif.2018.0508.

[12]

S. GetzinT. E. EricksonH. YizhaqM. Muñoz-RojasA. Huth and K. Wiegand, Bridging ecology and physics: Australian fairy circles regenerate following model assumptions on ecohydrological feedbacks, J. Ecol., 109 (2021), 399-416.  doi: 10.1111/1365-2745.13493.

[13]

E. GiladJ. von HardenbergA. ProvenzaleM. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation, Phys. Rev. Lett., 93 (2004), 098105.  doi: 10.1103/PhysRevLett.93.098105.

[14]

K. Gowda, Y. Chen, S. Iams and M. Silber, Assessing the robustness of spatial pattern sequences in a dryland vegetation model, Proc. Roy. Soc. A: Math., Phys. and Eng. Sci., 472 (2016), 20150893, 25 pp. doi: 10.1098/rspa.2015.0893.

[15]

K. Gowda, S. Iams and M. Silber, Signatures of human impact on self-organized vegetation in the Horn of Africa, Sci. Rep., 8 (2018), Article number: 3622. doi: 10.1038/s41598-018-22075-5.

[16]

R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), 69-106.  doi: 10.1137/0137006.

[17]

R. HilleRisLambersM. RietkerkF. van den BoschH. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61.  doi: 10.2307/2680085.

[18]

O. Jaïbi, A. Doelman, M. Chirilus-Bruckner and E. Meron, The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation, Physica D, 412 (2020), 132637, 30 pp. doi: 10.1016/j.physd.2020.132637.

[19]

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

[20]

K. Knopp, Theory and Application of Infinite Series, Dover Books on Mathematics, 2013.

[21]

T. Kolokolnikov, M. Ward, J. Tzou and J. Wei, Stabilizing a homoclinic stripe, Philos. Trans. Roy. Soc. A, 376 (2018), 20180110, 13 pp. doi: 10.1098/rsta.2018.0110.

[22]

T. KolokolnikovM. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Science, 19 (2009), 1-56.  doi: 10.1007/s00332-008-9024-z.

[23]

P. Mandel and T. Erneux, The slow passage through a steady bifurcation: Delay and memory effects, J. Stat. Phys., 48 (1987), 1059-1070.  doi: 10.1007/BF01009533.

[24]

, MATLAB version 9.4.0. (R2018a), The Mathworks, Inc., Natick, Massachusetts, 2018.

[25]

E. Meron, Modeling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187.  doi: 10.1051/mmnp/20116109.

[26] E. Meron, Nonlinear Physics of Ecosystems, CRC Press, Boca Raton, Florida, 2015.  doi: 10.1201/b18360.
[27]

M. Messaoudi, M. G. Clerc, E. Berríos-Caro, D. Pinto-Ramos, M. Khaffou, A. Makhoute and M. Tlidi, Patchy landscapes in arid environments: Nonlinear analysis of the interaction-redistribution model, Chaos, 30 (2020), 093136, 11 pp. doi: 10.1063/5.0011010.

[28]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambersJ. van de KoppelL. KumarH. H. T. Prins and A. M. de Roos, Self-organization of vegetation in arid ecosystems, Am. Nat., 160 (2002), 524-530.  doi: 10.1086/342078.

[29]

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

[30]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments Ⅰ, Nonlinearity, 23 (2010), 2657-2675.  doi: 10.1088/0951-7715/23/10/016.

[31]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments Ⅳ: Slowly moving patterns and their stability, SIAM J. Appl. Math., 73 (2013), 330-350.  doi: 10.1137/120862648.

[32]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments V: The transition from patterns to desert, SIAM J. Appl. Math., 73 (2013), 1347-1367.  doi: 10.1137/120899510.

[33]

E. Siero, A. Doelman, M. B. Eppinga, J. D. M. Rademacher, M. Rietkerk and K. Siteur, Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes, Chaos, 25 (2015), 036411, 22 pp. doi: 10.1063/1.4914450.

[34]

K. SiteurE. SieroM. B. EppingaJ. D. M. RademacherA. Doelman and M. Rietkerk, Beyond Turing: The response of patterned ecosystems to environmental change, Ecol. Complex., 20 (2014), 81-96.  doi: 10.1016/j.ecocom.2014.09.002.

[35]

J. C. Tzou and L. Tzou, Spot patterns of the Schnakenberg reaction-diffusion system on a curved torus, Nonlinearity, 33 (2020), 643-674.  doi: 10.1088/1361-6544/ab5161.

[36]

J. C. Tzou and L. Tzou, Analysis of spot patterns on a coordinate-invariant model for vegetation on a curved terrain, SIAM J. Appl. Dyn. Sys., 19 (2020), 2500-2529.  doi: 10.1137/20M1326271.

[37]

J. C. TzouM. J. Ward and T. Kolokolnikov, Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems, Physica D, 290 (2015), 24-43.  doi: 10.1016/j.physd.2014.09.008.

[38]

H. UeckerD. Wetzel and J. D. M. Rademacher, pde2path-A Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.

[39]

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 Science, 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.

[40]

J. von HardenbergE. MeronM. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101.  doi: 10.1103/PhysRevLett.87.198101.

[41]

T. Wong and M. J. Ward, Spot patterns in the 2-D Schnakenberg model with localized heterogeneities, Stud. Appl. Math., 146 (2021), 779-833.  doi: 10.1111/sapm.12361.

[42]

T. Wong and M. J. Ward, Weakly nonlinear analysis of peanut-shaped deformations for localized spots of singularly perturbed reaction-diffusion systems, SIAM J. Appl. Dyn. Syst, 19 (2020), 2030-2058.  doi: 10.1137/20M1316779.

[43]

Y. R. ZelnikE. Meron and G. Bel, Gradual regime shifts in fairy circles, PNAS, 112 (2015), 12327-12331.  doi: 10.1073/pnas.1504289112.

Figure 1.  Numerical results for $ \chi(S) $, $ \gamma(S) $, and $ \mu(S) $, as defined in (2.4b) and (2.29), as computed numerically from the core problem (2.4a) and the homogeneous adjoint problem (2.9)
Figure 3.  Left panel: For $ \varepsilon = 0.02 $ and $ a = 36 $ in (4.1a), we plot $ S_1 $ versus $ x_1 $ for various values of $ H $ along the midline of the unit square. Right panel: We fix $ \varepsilon = 0.02 $, $ a = 32 $ so that $ S_1<\Sigma_2\approx 4.3 $ and no spot-splitting occurs. The curves from bottom to top are the $ x $-coordinates of the spot trajectory from simulations (full PDE (1.2), solid curves; DAE (4.1), solid markers) for $ H = 0 $, $ H = 0.5 $ and $ H = 1 $, respectively. We do not show the $ y $-coordinates of the spot. Both PDE and DAE simulations show that the $ y $-coordinates are very close to $ y = 1/2 $
Figure 4.  We fix $ \varepsilon = 0.02 $, $ H = 0.5 $ and $ a = 36 $ for a one-spot pattern. The spot is initially centered at $ x_1 = 0.2 $ on the midline $ y = 1/2 $, for which the initial source strength $ S_1 $ exceeds the threshold $ \Sigma_2 $. This results in the initial peanut-shaped deformation of the spot at $ t = 40 $. However, as the spot slowly drifts to the right along the midline, $ S_1 $ eventually decreases below $ \Sigma_2 $ (see Fig. 3a), and no spot self-replication event occurs
Figure 2.  Full PDE results computed from (1.2) for a two-spot quasi-equilibrium pattern in the unit square for $ \varepsilon = 0.02 $, $ m = 1 $ and $ H = 0.5 $. The spots are centered at $ (0.7,0.25)^T $ and $ (0.7,0.75)^T $. Left panel: For the rainfall rate $ a = 20 $, the amplitudes of the two spots, as represented by the solid and dashed lines, show a nonlinear spot-annihilation event. Right panel: For $ a = 30 $, a solid line and red dots are used to represent the two, nearly indistinguishable, spot amplitudes
Figure 5.  A steady-state one-spot solution with spot location at $ \mathbf{{x}}_e = (x_e,1/2)^T $ and source strength $ S_1 $ when $ \varepsilon = 0.02 $. Left panel: The $ x $-coordinates of the equilibrium position versus the rainfall rate $ a $ for various $ H $. Middle panel: The source strength $ S_1 $ at the steady-state location. Right panel: The blue-shaded linear stability region is where $ S_1<\Sigma_2 $ in the $ a $ versus $ H $ parameter space
Figure 6.  We fix $ \varepsilon = 0.02 $, $ H = 0.5 $ and $ a = 39 $, for which $ S_1\approx 4.485 >\Sigma_2 $. The single spot is initially at its steady-state location $ x_{1e} \approx 0.6077 $ on the midline $ y = 1/2 $. The steady-state spot is predicted to be unstable to a local peanut-shape deformation. PDE simulations of (1.2) shows that the linear instability triggers spot-splitting
Figure 7.  $ H = 0.4 $, $ a = 40 $, $ \varepsilon = 0.02 $, with the initial spot location $ \mathbf{{x}}_1 = \left(0.6,0.5\right)^T $. After spot-splitting, there is a very favorable comparison between the trajectories of the two spots from the DAE (2.30) (red solid curve) and from the PDE simulation of (1.2) (black markers)
Figure 8.  Same caption as in Fig. 7 except that $ H $ is increased to $ H = 0.7 $. The two steady-state spots are now vertically aligned
Figure 9.  For $ \varepsilon = 0.02 $ and $ a = 40 $, the steady-state locations for a two-spot pattern, labeled by $ \left(x_{1e} \,, y_{1e}\right) $ and $ \left( x_{2e} \,, y_{2e} \right) $, in the unit square computed from the steady-state DAE system (4.3) as $ H $ is varied. Left panel: The bifurcation diagram of $ x_{1e}-x_{2e} $ versus $ H $. The trivial branch, for which $ x_{1e} = x_{2e} $ and $ y_{1e} = 0.25, y_{2e} = 0.75 $, becomes linearly stable as a solution of the DAE dynamics when $ H > H_c \approx 0.6076 $. The lower (non-trivial) branch is due to the symmetry across the midline $ y = 1/2 $. Other panels: Visualization of some two-spot equilibria (either open or filled circles) corresponding to the left-panel
Figure 10.  Linear stability (blue) region for a vertically aligned two-spot steady-state in the $ a $ versus $ H $ parameter space, as computed from (4.3). In the blue shaded region the two-spot equilibrium is linearly stable as a steady-state of the DAE dynamics (2.30). The boundary is the bifurcation threshold $ H_c = H_c(a) $ that predicts the minimum value of $ H $ where vertical alignment of two-spot steady-states occurs
Figure 11.  Full PDE results of (1.2) in the unit square for $ a = 46 $ and $ \varepsilon = 0.02 $ for two-spot patterns with the initial spot locations $ (0.3,0.3)^T $ and $ (0.7,0.7)^T $. For $ H = 0.64 $, there is a very favorable comparison between the full PDE results (black markers) and the DAE results (solid red curves), as computed from (2.30), for the $ x $ and $ y $ coordinates of the two-spot pattern as shown in (a) and (b), respectively. A similar comparison for $ H = 0.62 $ is shown in (d) and (e). At $ t = 10^5 $, the two-spot quasi-equilibria become very close to their steady-states, as shown in (c) and (f) for $ H = 0.64 $ and $ H = 0.62 $, respectively. Vertical alignment occurs only for $ H = 0.64 $. From Fig. 10, the predicted threshold for vertical alignment from the asymptotic theory is $ H_c\approx 0.631 $ when $ a = 46 $
Figure 12.  For $ \varepsilon = 0.02 $ and $ a = 60 $, steady-state three-spot patterns of the form $ \left\{ (x_{1e} \,, 1/2)^T \,, (x_{2e} \,, y_{2e})^T \,, (x_{2e} \,, 1-y_{2e})^T\right\} $ in the unit square are computed from the steady-state of the DAE dynamics (2.30) and (2.26) as $ H $ is varied. In the top and bottom panels, we show the dependence of $ x_{1e}, x_{2e} $ and $ y_{2e} $ with respect to $ H $ for the cases $ x_{1e}<x_{2e} $ and $ x_{1e}>x_{2e} $, respectively. Branches of equilibria for $ x_{1e}<x_{2e} $ (top panels) are all linearly stable. For $ x_{1e}>x_{2e} $ (bottom panels), the solid branches are unstable as steady-states of the DAE dynamics. The dashed branches are unstable to spot amplitude perturbations from the GCEP (3.15b)
Figure 13.  Full PDE simulation of (1.2) for $ H = 1 $, $ a = 60 $, and $ \varepsilon = 0.02 $ for a three-spot steady-state pattern with spots centered at $ (x_{1e},{1/2}) $, $ (x_{2e},y_{2e}) $, and $ (x_{2e},1-y_{2e}) $, with $ x_{1e} \approx 0.9046 $, $ x_{2e} \approx 0.5630 $, and $ y_{2e} \approx 0.7635 $, corresponding to a point on the dashed curves in Figs. 12c and 12d where the steady-state is unstable due to a positive real eigenvalue of the GCEP (3.15b). The snapshots of the $ v $-component in (1.2) show that the weaker spot on the uphill side of the terrain gradient is rapidly annihilated
Figure 14.  Snapshots of full PDE simulations of (1.2) with $ H = 0.5 $ and $ \varepsilon = 0.02 $ for an initial three-spot pattern that is vertically aligned with spots centered at $ \mathbf{{x}}_1 = (1/2,1/2)^T $, $ \mathbf{{x}}_2 = (1/2, 5/6)^T $ and $ \mathbf{{x}}_3 = (1/2, 1/6)^T $. The vertical alignment breaks down as time increase, with the pattern eventually tending to the linearly stable steady-state in Figs. 12a and 12b after first approaching the unstable steady-state in Figs. 12c and 12d
Figure 15.  Steady-states of the DAE dynamics (2.30) and (2.26) for a three-spot equilibrium of the form $ \left\{ (x_{1e}, 1/2)^T, (x_{1e}, 5/6)^T, (x_{1e}, 1/6)^T \right\} $. Left panel: $ x_{1e} $ versus $ H $ for $ a = 58.231 $. These steady-states are unstable as equilibria of the DAE dynamics (2.30) and (2.26). Right panel: $ x_{1e} $ versus $ a $ with $ H = 0.5077 $. Along the solid portion the equilibria are unstable for the DAE dynamics, while along the dashed portion ($ 30 < a < 35.9284 $) the spot amplitudes are unstable owing to a positive real eigenvalue of the GCEP (3.15b)
Figure 16.  Full PDE simulation of (1.2) with $ a = 34.8720 $ and $ H = 0.5077 $ for an initial three-spot steady-state with spots centered at $ \left\{ (x_{1e}, 1/2)^T, (x_{1e}, 5/6)^T, (x_{1e}, 1/6)^T \right\} $, with $ x_{1e} = 0.5697 $, corresponding to a point on the dashed branch Fig. 15b where a competition instability in the spot amplitudes is predicted. The snapshots of $ v $ confirm this linear instability, and that it triggers the annihilation of two spots on an $ {\mathcal O}(1) $ time-scale
Figure 17.  Snapshots of full PDE results for (1.2) with $ \varepsilon = 0.02 $ and $ H = 0.5 $. The dynamic rainfall rate is $ a = \max(40- \delta t, 26) $ with $ \delta = 0.01 $. The initial condition is a spot on the right and two vertically aligned spots on the left. The spot on the midline disappears around $ t = 550 $, after which the remaining two vertically aligned spots drift slowly up the terrain slope to their steady-state locations
Figure 18.  Comparison of spot trajectories between the DAE simulations of (2.30) and (2.26) (solid lines), with the zero-eigenvalue detection criterion of the GCEP, and the full PDE results from (1.2) (red markers). Parameters as in Fig. 17
Figure 19.  Snapshots of full PDE results for (1.2) with $ \varepsilon = 0.02 $, $ H = 0.5 $, and dynamic rainfall rate $ a = \max(40- \delta t, 26) $ with $ \delta = 0.01 $. The initial condition has one spot on the left and two vertically aligned spots on the right. One spot remains on the midline $ y = 1/2 $ until one of the two vertically aligned spots disappears at around $ t = 590 $. The two surviving spots undergo another spot-annihilation event at around $ t = 1445 $. The sole remaining spot then approaches the midline where it slowly drifts up the terrain slope to its steady-state location
Figure 20.  Comparison of spot trajectories between the DAE simulations of (2.30) and (2.26) (solid lines), with the zero-eigenvalue detection criterion of the GCEP, and the full PDE results from (1.2) (red markers). Parameters as in Fig. 19. We observe very good agreement both before and after the two spot-annihilation events
Figure 21.  Snapshots of full PDE results for (1.2) with $ \varepsilon = 0.02 $, $ H = 1.0 $, and the dynamic rainfall rate $ a = \max(70- \delta t, 55) $ with $ \delta = 0.01 $. The five-spot initial pattern has one spot on the left and two pairs of vertically aligned spots. Two spot-annihilation events occur at later times and the final steady-state has a spot on the midline and a pair of vertically aligned spots on the uphill side of the terrain slope
Figure 22.  Same caption as in Fig. 20 except that the parameters now correspond to the initial five-spot pattern shown in Fig. 21. The spot trajectories from the DAE simulations and the PDE (1.2) compare very favorably both before and after the two spot-annihilation events
Figure 23.  The bifurcation diagram for a quasi steady-state spot centered at the midpoint of the unit square for (5.2). The black solid curve is the spot amplitude $ V(0) $ versus the rainfall rate $ a $ from the NAS (4.1a). The PDE simulation results with $ \varepsilon = 0.02 $ and $ a(t) = 16 - \varepsilon \, t $, are represented by the red dots. The asymptotic prediction $ a_0 \approx 13.0757 $ for the jump value of $ a $ (dashed vertical line) is seen to compare favorably with the PDE results
Figure 24.  Left panel: The source strength $ S_1 $ versus the rainfall rate $ a $, computed from the NAS (5.37) for a one spot solution centered at the midpoint $ ({1/2},{1/2}) $ of the unit square with $ H = 0 $, $ \varepsilon = 0.02 $, and $ m = 1 $. The static onset and delayed onset for the peanut-splitting instability are shown by the open and filled circles, respectively. Right panel: The deviation (5.38) from local radial symmetry of the PDE numerical solution computed from (1.2) with the dynamic rainfall rate $ a(t) = 34+0.02\, t $. The deviation begins to increase very rapidly near our asymptotic prediction $ t_{\star} \approx 236.81 $ for the delayed onset time
Figure 25.  Snapshots of the full PDE solution of (1.2) with $ H = 0 $, $ m = 1 $, $ \varepsilon = 0.02 $, and $ a(t) = 34+0.02\, t $ showing a delayed spot-splitting event for a spot centered at the midpoint of the unit square. The results correspond to the deviation metric shown in the right panel of Fig. 24
Figure 26.  Global bifurcation diagrams for the $ L_2 $ norm of the $ v $-component versus the rainfall rate $ a $ for a one-spot steady-state solution of (1.2) in the unit square with $ \varepsilon = 0.03 $ and $ m = 1 $. A non-zero terrain slope gradient perturbs the pitchfork bifurcation that exists when $ H = 0 $ into a saddle-node structure
Figure 27.  Contour plot (zoomed) of the $ v $-component at the indicated points in the global bifurcation diagram for $ H = 1 $ in Fig. 26f. (a, b): elongated spot on the linearly stable branch. (c): elongated spot on the unstable branch. (d, e): elongated spots on the isolated unstable branch
Table 1.  The bifurcation values in $ a $, as indicated by the open circles in Fig. 26. The corresponding spot source strength $ S_1 $ is given
$ H $ $ a $ $ S_1 $
0 36.01 4.5122
0.1 36.01 4.5078
0.3 36.18 4.4988
0.5 36.57 4.4887
0.7 37.16 4.4780
1 38.40 4.4647
$ H $ $ a $ $ S_1 $
0 36.01 4.5122
0.1 36.01 4.5078
0.3 36.18 4.4988
0.5 36.57 4.4887
0.7 37.16 4.4780
1 38.40 4.4647
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