
-
Previous Article
Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system
- DCDS-B Home
- This Issue
-
Next Article
Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps
Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
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.
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. |
[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. |
[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.
|
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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 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. |
[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. 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. |
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. |
[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. |
[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.
|
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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 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. |
[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. 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. |




[1] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[2] |
Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 |
[3] |
Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 |
[4] |
Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156 |
[5] |
Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116 |
[6] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[7] |
Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157 |
[8] |
Yong-Jung Kim, Hyowon Seo, Changwook Yoon. Asymmetric dispersal and evolutional selection in two-patch system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3571-3593. doi: 10.3934/dcds.2020043 |
[9] |
Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu, Li Zhang. Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1031-1060. doi: 10.3934/dcdsb.2020152 |
[10] |
Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003 |
[11] |
H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020433 |
[12] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[13] |
Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020316 |
[14] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174 |
[15] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
[16] |
Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020357 |
[17] |
Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021003 |
[18] |
Robert Stephen Cantrell, King-Yeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020361 |
[19] |
Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146 |
[20] |
Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048 |
2019 Impact Factor: 1.27
Tools
Article outline
Figures and Tables
[Back to Top]