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doi: 10.3934/dcdsb.2022138
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## Threshold dynamics and finite-time stability of reaction-diffusion vegetation-water systems in arid area with time-varying delay

 School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

*Corresponding author: Qimin Zhang

Received  March 2022 Revised  June 2022 Early access August 2022

Fund Project: The first author is supported by the Natural Science Foundation of China (12161068) and the Ningxia Key R & D Program Key Projects (Grant numbers 2021BEG03012)

In this paper, deterministic and stochastic reaction-diffusion vege-tation-water systems with time-varying delay are developed, respectively. For the deterministic system, we define a threshold $R_*$ and discuss the threshold dynamics of the system. When $R_*<1$, the vegetation-free equilibrium point of the system is locally asymptotically stable. For $R_*>1$, the vegetation is persistent. Besides, by Latin Hypercube Sampling (LHS) and partial rank correlation coefficients (PRCCs), global sensitivity analysis is shown. For the stochastic system driven by Markov switching and Gaussian noise, with the help of stochastic comparison principle, several sufficient conditions are obtained to ensure the finite-time stability and finite time contractive stability. Numerical simulations are carried out to support the effectiveness of theoretical results.

Citation: Zixiao Xiong, Xining Li, Qimin Zhang. Threshold dynamics and finite-time stability of reaction-diffusion vegetation-water systems in arid area with time-varying delay. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022138
##### References:
 [1] B. Bourgeois, A. Vanasse and E. González, Threshold dynamics in plant succession after tree planting in agricultural riparian zones, J. Appl. Ecol., 53 (2016), 1704-1713. [2] F. Haghnazari, H. Shahgholi and M. Feizi, Factors affecting the infiltration of agricultural soils, Int. J. Agron. Agr. Res., 6 (2015), 21-35. [3] Q. Han, T. Yang and C. Zeng, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Phys. A, 408 (2014), 96-105.  doi: 10.1016/j.physa.2014.04.015. [4] R. HilleRisLambers, M. Rietkerk and F. van den Bosch, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61. [5] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. [6] X. Li, X. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031. [7] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069.  doi: 10.1080/07362990500118637. [8] S. Mieruch, S. Nol and H. Bovensmann, Markov chain analysis of regional climates, Nonlinear Proc. Geoph., 17 (2010), 651-661. [9] S. Pan, Q. Zhang and A. Meyer-Baese, Dynamic analysis of a soil organic matter and plant system with reaction-diffusion, Chaos Soliton. Fract., 146 (2021), 110883.  doi: 10.1016/j.chaos.2021.110883. [10] M. Rietkerk, M. Boerlijst and F. van Langevelde, Self-organization of vegetation in arid ecosystems, Am. Nat., 160 (2002), 524-530. [11] K. Wang, N. Zhang and D. Niu, Periodic oscillations in a spatially explicit model with delay effect for vegetation dynamics in freshwater marshes, J. Biol. Syst., 19 (2011), 131-147.  doi: 10.1142/S0218339011003932. [12] W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942. [13] C. B. Ward, P. G. Kevrekidis and N. Whitaker, A numerical bifurcation analysis of a dryland vegetation model, Commun. Nonlinear Sci. Numer. Simul., 68 (2019), 319-335.  doi: 10.1016/j.cnsns.2018.09.003. [14] K. Wu and B. Chen, Synchronization of partial differential systems via diffusion coupling, IEEE Trans. Circuits Syst. I., 59 (2012), 2655-2668.  doi: 10.1109/TCSI.2012.2190670. [15] K. Wu, M. Na and L. Wang, Finite-time stability of impulsive reaction-diffusion systems with and without time delay, Appl. Math. Comput., 363 (2019), 124591.  doi: 10.1016/j.amc.2019.124591. [16] H. Xiang, B. Liu and Z. Li, Verification theory and approximate optimal harvesting strategy for a stochastic competitive ecosystem subject to L${\rm\acute{e}}$vy noise, J. Dyn. Control Syst., 23 (2017), 753-777.  doi: 10.1007/s10883-017-9362-y. [17] W. Xie, L. Cai and X. Yue, Information entropies and dynamics in the stochastic ecosystem of two competing species, Acta Phys. Sin-Ch. Ed., 61 (2012). [18] B. Xin and J. Zhang, Finite-time stabilizing a fractional-order chaotic financial system with market confidence, Nonlinear Dynam., 79 (2015), 1399-1409.  doi: 10.1007/s11071-014-1749-7. [19] Z. Xiong, X. Li and M. Ye, Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by L$\rm \acute{e}$vy process with time-varying delay, Math. Biosci. Eng., 18 (2021), 8462-8498.  doi: 10.3934/mbe.2021419. [20] Z. Xiong, Q. Zhang and T. Kang, Bifurcation and stability analysis of a cross-diffusion vegetation-water model with mixed delays, Math. Method. Appl. Sci., 44 (2021), 9976-9986.  doi: 10.1002/mma.7384. [21] Q. Xue, C. Liu and L. Li, Interactions of diffusion and nonlocal delay give rise to vegetation patterns in semi-arid environments, Appl. Math. Comput., 399 (2021), 126038.  doi: 10.1016/j.amc.2021.126038. [22] Q. Xue, G. Sun and C. Liu, Spatiotemporal dynamics of a vegetation model with nonlocal delay in semi-arid environment, Nonlinear Dynam., 99 (2020), 3407-3420. [23] H.-M. Yin, X. Chen and L. Wang, On a cross-diffusion system modeling vegetation spots and strips in a semi-arid or arid landscape, Nonlinear Anal., 159 (2017), 482-491.  doi: 10.1016/j.na.2017.02.022. [24] C. Zeng, Q. Han and T. Yang, Noise-and delay-induced regime shifts in an ecological system of vegetation, J. Stat. Mech-Theory. E., 2013 (2013), P10017. doi: 10.1088/1742-5468/2013/10/p10017. [25] H. Zhang, X. Liu and W. Xu, Threshold dynamics and pulse control of a stochastic ecosystem with switching parameters, J. Franklin Inst., 358 (2021), 516-532.  doi: 10.1016/j.jfranklin.2020.10.035. [26] H. Zhang, W. Xu and Y. Lei, Noise-induced vegetation transitions in the Grazing Ecosystem, Appl. Math. Model., 76 (2019), 225-237.  doi: 10.1016/j.apm.2019.06.009. [27] H. Zhang, W. Xu and Y. Lei, Early warning and basin stability in a stochastic vegetation-water dynamical system, Commun. Nonlinear Sci., 77 (2019), 258-270. [28] Forestry and grassland bureau of ningxia hui autonomous region, desertification, http://lcj.nx.-gov.cn/.

show all references

##### References:
 [1] B. Bourgeois, A. Vanasse and E. González, Threshold dynamics in plant succession after tree planting in agricultural riparian zones, J. Appl. Ecol., 53 (2016), 1704-1713. [2] F. Haghnazari, H. Shahgholi and M. Feizi, Factors affecting the infiltration of agricultural soils, Int. J. Agron. Agr. Res., 6 (2015), 21-35. [3] Q. Han, T. Yang and C. Zeng, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Phys. A, 408 (2014), 96-105.  doi: 10.1016/j.physa.2014.04.015. [4] R. HilleRisLambers, M. Rietkerk and F. van den Bosch, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61. [5] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. [6] X. Li, X. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031. [7] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069.  doi: 10.1080/07362990500118637. [8] S. Mieruch, S. Nol and H. Bovensmann, Markov chain analysis of regional climates, Nonlinear Proc. Geoph., 17 (2010), 651-661. [9] S. Pan, Q. Zhang and A. Meyer-Baese, Dynamic analysis of a soil organic matter and plant system with reaction-diffusion, Chaos Soliton. Fract., 146 (2021), 110883.  doi: 10.1016/j.chaos.2021.110883. [10] M. Rietkerk, M. Boerlijst and F. van Langevelde, Self-organization of vegetation in arid ecosystems, Am. Nat., 160 (2002), 524-530. [11] K. Wang, N. Zhang and D. Niu, Periodic oscillations in a spatially explicit model with delay effect for vegetation dynamics in freshwater marshes, J. Biol. Syst., 19 (2011), 131-147.  doi: 10.1142/S0218339011003932. [12] W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942. [13] C. B. Ward, P. G. Kevrekidis and N. Whitaker, A numerical bifurcation analysis of a dryland vegetation model, Commun. Nonlinear Sci. Numer. Simul., 68 (2019), 319-335.  doi: 10.1016/j.cnsns.2018.09.003. [14] K. Wu and B. Chen, Synchronization of partial differential systems via diffusion coupling, IEEE Trans. Circuits Syst. I., 59 (2012), 2655-2668.  doi: 10.1109/TCSI.2012.2190670. [15] K. Wu, M. Na and L. Wang, Finite-time stability of impulsive reaction-diffusion systems with and without time delay, Appl. Math. Comput., 363 (2019), 124591.  doi: 10.1016/j.amc.2019.124591. [16] H. Xiang, B. Liu and Z. Li, Verification theory and approximate optimal harvesting strategy for a stochastic competitive ecosystem subject to L${\rm\acute{e}}$vy noise, J. Dyn. Control Syst., 23 (2017), 753-777.  doi: 10.1007/s10883-017-9362-y. [17] W. Xie, L. Cai and X. Yue, Information entropies and dynamics in the stochastic ecosystem of two competing species, Acta Phys. Sin-Ch. Ed., 61 (2012). [18] B. Xin and J. Zhang, Finite-time stabilizing a fractional-order chaotic financial system with market confidence, Nonlinear Dynam., 79 (2015), 1399-1409.  doi: 10.1007/s11071-014-1749-7. [19] Z. Xiong, X. Li and M. Ye, Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by L$\rm \acute{e}$vy process with time-varying delay, Math. Biosci. Eng., 18 (2021), 8462-8498.  doi: 10.3934/mbe.2021419. [20] Z. Xiong, Q. Zhang and T. Kang, Bifurcation and stability analysis of a cross-diffusion vegetation-water model with mixed delays, Math. Method. Appl. Sci., 44 (2021), 9976-9986.  doi: 10.1002/mma.7384. [21] Q. Xue, C. Liu and L. Li, Interactions of diffusion and nonlocal delay give rise to vegetation patterns in semi-arid environments, Appl. Math. Comput., 399 (2021), 126038.  doi: 10.1016/j.amc.2021.126038. [22] Q. Xue, G. Sun and C. Liu, Spatiotemporal dynamics of a vegetation model with nonlocal delay in semi-arid environment, Nonlinear Dynam., 99 (2020), 3407-3420. [23] H.-M. Yin, X. Chen and L. Wang, On a cross-diffusion system modeling vegetation spots and strips in a semi-arid or arid landscape, Nonlinear Anal., 159 (2017), 482-491.  doi: 10.1016/j.na.2017.02.022. [24] C. Zeng, Q. Han and T. Yang, Noise-and delay-induced regime shifts in an ecological system of vegetation, J. Stat. Mech-Theory. E., 2013 (2013), P10017. doi: 10.1088/1742-5468/2013/10/p10017. [25] H. Zhang, X. Liu and W. Xu, Threshold dynamics and pulse control of a stochastic ecosystem with switching parameters, J. Franklin Inst., 358 (2021), 516-532.  doi: 10.1016/j.jfranklin.2020.10.035. [26] H. Zhang, W. Xu and Y. Lei, Noise-induced vegetation transitions in the Grazing Ecosystem, Appl. Math. Model., 76 (2019), 225-237.  doi: 10.1016/j.apm.2019.06.009. [27] H. Zhang, W. Xu and Y. Lei, Early warning and basin stability in a stochastic vegetation-water dynamical system, Commun. Nonlinear Sci., 77 (2019), 258-270. [28] Forestry and grassland bureau of ningxia hui autonomous region, desertification, http://lcj.nx.-gov.cn/.
Schematic diagram of the model with delay
State trajectories of vegetation-water system with time-varying delay when $R_*<1$. $(u(0), v(0), w(0)) = (0.5, 0.12,$ $0.68)$
State trajectories of vegetation-water system with time-varying delay when $R_*>1$. $(u(0), v(0), w(0)) = (0.5, 0.12,$ $0.68)$
Sensitivity analysis on $R_*$
The sample paths of Markov chain $\varsigma(t)$
State trajectories of Markov switching stochastic vegetation-water system with delay. $(u_0, v_0, w_0) = (5, 10, 12)$
State trajectories of Markov switching stochastic vegetation-water system with delay. $(u_0, v_0, w_0) = (5, 10, 12)$. $\pi_1(1) = 0.15, \ \pi_2(1) = 0.1, \ \pi_3(1) = 0.3, \ \pi_1(2) = 0.1, \ \pi_2(2) = 0.1, \ \pi_3(2) = 0.2$
State trajectories of Markov switching stochastic vegetation-water system with delay. $(u_0, v_0, w_0) = (5, 10, 12)$. $\bar{\tau} = \tau(t) = 6, \ t\in[0, T]$
State trajectories of Markov switching stochastic vegetation-water system with delay. $(u_0, v_0, w_0) = (5, 10, 12)$. $\bar{\tau} = \tau(t) = 7, \ t\in[0, T]$
State trajectories of Markov switching stochastic vegetation-water system with delay. $(u_0, v_0, w_0) = (5, 10, 12)$. $\sigma_1(1) = 0.7, \ \sigma_2(1) = 0.7, \ \sigma_3(1) = 0.7, \ \sigma_1(2) = 0.5, \ \sigma_2(2) = 0.5, \ \sigma_3(2) = 0.5$
The meaning of each item
 Term Biological significance $d_u\Delta u$ Plant dispersal $\frac{v}{v+1}u$ Vegetation biomass growth (due to the presence of soil water $v$ $-su$ Vegetation biomass average decay rate $-\sigma_1u$ Vegetation biomass decay rate (due to environmental factors) $d_v\Delta v$ Soil water dispersal $\alpha\frac{u+f}{u+1}w$ The rate of infiltration of surface water into soil water (soil water increase) $-\gamma\frac{v}{v+1}u$ The rate at which vegetation absorbs soil water $-zv$ Soil water average evaporation rate $-\sigma_2 v$ Soil water evaporation rate (due to environmental factors) $d_w\Delta w$ Surface water dispersal $R$ Precipitation rate $-\alpha\frac{u+f}{u+1}w$ The rate of infiltration of surface water into soil water (surface water decrease) $-pw$ Surface water average loss rate $-\sigma_3 w$ Surface water loss rate (due to environmental factors)
 Term Biological significance $d_u\Delta u$ Plant dispersal $\frac{v}{v+1}u$ Vegetation biomass growth (due to the presence of soil water $v$ $-su$ Vegetation biomass average decay rate $-\sigma_1u$ Vegetation biomass decay rate (due to environmental factors) $d_v\Delta v$ Soil water dispersal $\alpha\frac{u+f}{u+1}w$ The rate of infiltration of surface water into soil water (soil water increase) $-\gamma\frac{v}{v+1}u$ The rate at which vegetation absorbs soil water $-zv$ Soil water average evaporation rate $-\sigma_2 v$ Soil water evaporation rate (due to environmental factors) $d_w\Delta w$ Surface water dispersal $R$ Precipitation rate $-\alpha\frac{u+f}{u+1}w$ The rate of infiltration of surface water into soil water (surface water decrease) $-pw$ Surface water average loss rate $-\sigma_3 w$ Surface water loss rate (due to environmental factors)
PRCC value of $R_*$, which are ranked from the most sensitive to the least
 Parameters Mean Standard deviation Source PRCC p-value $R$ 4/5 0.3 [10] 0.9181 0 $s$ 0.25 0.05 [10] -0.8059 0 $f$ 0.2 0.04 [10] 0.7311 0 $\alpha$ 0.2 0.03 [10] 0.6148 0 $z$ 0.4 0.05 [10] -0.5897 0 $p$ 0.4 0.05 [10] -0.5258 0
 Parameters Mean Standard deviation Source PRCC p-value $R$ 4/5 0.3 [10] 0.9181 0 $s$ 0.25 0.05 [10] -0.8059 0 $f$ 0.2 0.04 [10] 0.7311 0 $\alpha$ 0.2 0.03 [10] 0.6148 0 $z$ 0.4 0.05 [10] -0.5897 0 $p$ 0.4 0.05 [10] -0.5258 0
The corresponding system parameter value when $\varsigma(t) = 1\ \text{or}\ 2$
 Parameters Values Parameters Values Parameters Values Parameters Values $d_u(1)$ 0.0066 $\alpha(1)$ 0.1 $d_u(2)$ 0.001 $\alpha(2)$ 0.1 $d_v(1)$ 0.01 $f(1)$ 0.1 $d_v(2)$ 0.01 $f(2)$ 0.09 $d_w(1)$ 1 $\gamma(1)$ 0.1 $d_w(2)$ 1 $\gamma(2)$ 0.1 $R(1)$ 2 $z(1)$ 0.4 $R(2)$ 1 $z(2)$ 0.4 $s(1)$ 0.4 $p(1)$ 0.6 $s(2)$ 0.4 $p(2)$ 0.5 $\sigma_i(1)\ (i=1, 2, 3)$ 0.5 $S(1)$ 2 $\sigma_i(2)\ (i=1, 2, 3)$ 0.3 $S(2)$ 2 $\pi_i(1)\ (i=1, 2, 3)$ 0 $\pi_i(2)\ (i=1, 2, 3)$ 0
 Parameters Values Parameters Values Parameters Values Parameters Values $d_u(1)$ 0.0066 $\alpha(1)$ 0.1 $d_u(2)$ 0.001 $\alpha(2)$ 0.1 $d_v(1)$ 0.01 $f(1)$ 0.1 $d_v(2)$ 0.01 $f(2)$ 0.09 $d_w(1)$ 1 $\gamma(1)$ 0.1 $d_w(2)$ 1 $\gamma(2)$ 0.1 $R(1)$ 2 $z(1)$ 0.4 $R(2)$ 1 $z(2)$ 0.4 $s(1)$ 0.4 $p(1)$ 0.6 $s(2)$ 0.4 $p(2)$ 0.5 $\sigma_i(1)\ (i=1, 2, 3)$ 0.5 $S(1)$ 2 $\sigma_i(2)\ (i=1, 2, 3)$ 0.3 $S(2)$ 2 $\pi_i(1)\ (i=1, 2, 3)$ 0 $\pi_i(2)\ (i=1, 2, 3)$ 0
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