• Previous Article
    Random attractor for the 2D stochastic nematic liquid crystals flows
  • CPAA Home
  • This Issue
  • Next Article
    Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent
September  2019, 18(5): 2325-2347. doi: 10.3934/cpaa.2019105

Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model

1. 

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

2. 

School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang 150080, China

3. 

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author

Received  January 2018 Revised  January 2019 Published  April 2019

By using bifurcation theory, we investigate the local asymptotical stability of non-negative steady states for a coupled dynamic system of ordinary differential equations and partial differential equations. The system models the interaction of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The asymptotic profile of positive steady states when the diffusion coefficients are sufficiently small or large are also obtained.

Citation: Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105
References:
[1]

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. Google Scholar

[2]

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. Google Scholar

[3]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton Ⅰ: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440. doi: 10.1137/07070663X. Google Scholar

[4]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton Ⅱ: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470. doi: 10.1137/070706641. Google Scholar

[5]

Y. Du and S. B. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090775105. Google Scholar

[6]

Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016. Google Scholar

[7]

Y. DuS. B. Hsu and Y. Lou, Multiple steady-states in phytoplankton population induced by photoinhibition, J. Differential Equations, 258 (2015), 2408-2434. doi: 10.1016/j.jde.2014.12.012. Google Scholar

[8]

M. Genkai-KatoY. VadeboncoeurL. Liboriussen and E. Jeppesen, Benthic-planktonic coupling, regime shifts, and whole-lake primary production in shallow lakes, Ecology, 93 (2012), 619-631. Google Scholar

[9]

S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974. doi: 10.1137/100782358. Google Scholar

[10]

S. B. HsuF. B. Wang and X. Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297. doi: 10.1016/j.jde.2013.04.006. Google Scholar

[11]

J. HuismanP. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Am. Nat., 154 (1999), 46-68. Google Scholar

[12]

J. HuismanN. N. P. ThiD. M. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum, Nature, 439 (2006), 322-325. Google Scholar

[13]

C. G. JägerS. Diehl and M. Emans, Physical determinants of phytoplankton production, algal stoichiometry, and vertical nutrient fluxes, Am. Nat., 175 (2010), 91-104. Google Scholar

[14]

C. G. Jäger and S. Diehl, Resource competition across habitat boundaries: asymmetric interactions between benthic and pelagic producers, Ecol. Monogr., 84 (2014), 287-302. Google Scholar

[15]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. Google Scholar

[16]

T. KolokolnikovC. H. Ou and Y. Yuan, Phytoplankton depth profiles and their transitions near the critical sinking velocity, J. Math. Biol., 59 (2009), 105-122. doi: 10.1007/s00285-008-0221-z. Google Scholar

[17]

L. F. Mei and X. Y. Zhang, On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients, Discrete Cont. Dyn. Syst.-B, 17 (2012), 221-243. doi: 10.3934/dcdsb.2012.17.221. Google Scholar

[18]

L. F. Mei and X. Y. Zhang, Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics, J. Differential Equations, 253 (2012), 2025-2063. doi: 10.1016/j.jde.2012.06.011. Google Scholar

[19]

L. F. MeiS. B. Hsu and F. B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Cont. Dyn. Syst.-B, 21 (2016), 607-620. doi: 10.3934/dcdsb.2016.21.607. Google Scholar

[20]

H. NieS. B. Hsu and J. H. Wu, Coexistence solutions of a competition model with two species in a water column, Discrete Cont. Dyn. Syst.-B, 20 (2015), 2691-2714. doi: 10.3934/dcdsb.2015.20.2691. Google Scholar

[21]

R. Peng and X. Q. Zhao, A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species, J. Math. Biol., 72 (2016), 755-791. doi: 10.1007/s00285-015-0904-1. Google Scholar

[22]

M. SchefferS. SzabóA. GragnaniE. H. van NesS. RinaldiN. KautskyJ. NorbergR. M. M. Roijackers and R. J. M. Franken, Floating plant dominance as a stable state, Proc. Natl. Acad. Sci. USA, 100 (2003), 4040-4045. Google Scholar

[23]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[24]

F. R. VasconcelosS. DiehlP. RodríguezP. HedströmJ. Karlsson and P. Byström, Asymmetrical competition between aquatic primary producers in a warmer and browner world, Ecology, 97 (2016), 2580-2592. Google Scholar

[25]

Y. VadeboncoeurG. PetersonM. Vander Zanden and J. Kalff, Benthic algal production across lake size gradients: interactions among morphometry, nutrients, and light, Ecology, 89 (2008), 2542-2552. Google Scholar

[26]

F. B. WangS. B. Hsu and X. Q. Zhao, A reaction-diffusion-advection model of harmful algae growth with toxin degradation, J. Differential Equations, 259 (2015), 3178-3201. doi: 10.1016/j.jde.2015.04.018. Google Scholar

[27]

F. B. Wang, A PDE system modeling the competition and inhibition of harmful algae with seasonal variations, Nonlinear Analysis RWA, 25 (2015), 258-275. doi: 10.1016/j.nonrwa.2015.02.010. Google Scholar

[28]

X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. Google Scholar

[29]

X. F. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x. Google Scholar

[30]

K. YoshiyamaJ. P. MellardE. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203. Google Scholar

[31]

A. ZagarisA. DoelmanN. N. Pham Thi and B. P. Sommeijer, Blooming in a nonlocal, coupled phytoplankton-nutrient model, SIAM J. Appl. Math., 69 (2009), 1174-1204. doi: 10.1137/070693692. Google Scholar

[32]

A. Zagaris and A. Doelman, Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model, Nonlinearity, 24 (2011), 3437-3486. doi: 10.1088/0951-7715/24/12/007. Google Scholar

[33]

J. M. ZhangJ. P. Shi and X. Y. Chang, A mathematical model of algae growth in a pelagic-benthic coupled shallow aquatic ecosystem, J. Math. Biol., 76 (2018), 1159-1193. doi: 10.1007/s00285-017-1168-8. Google Scholar

show all references

References:
[1]

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. Google Scholar

[2]

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. Google Scholar

[3]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton Ⅰ: Existence, SIAM J. Math. Anal., 40 (2008), 1419-1440. doi: 10.1137/07070663X. Google Scholar

[4]

Y. Du and S. B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton Ⅱ: Limiting profile, SIAM J. Math. Anal., 40 (2008), 1441-1470. doi: 10.1137/070706641. Google Scholar

[5]

Y. Du and S. B. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090775105. Google Scholar

[6]

Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016. Google Scholar

[7]

Y. DuS. B. Hsu and Y. Lou, Multiple steady-states in phytoplankton population induced by photoinhibition, J. Differential Equations, 258 (2015), 2408-2434. doi: 10.1016/j.jde.2014.12.012. Google Scholar

[8]

M. Genkai-KatoY. VadeboncoeurL. Liboriussen and E. Jeppesen, Benthic-planktonic coupling, regime shifts, and whole-lake primary production in shallow lakes, Ecology, 93 (2012), 619-631. Google Scholar

[9]

S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974. doi: 10.1137/100782358. Google Scholar

[10]

S. B. HsuF. B. Wang and X. Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297. doi: 10.1016/j.jde.2013.04.006. Google Scholar

[11]

J. HuismanP. van Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, Am. Nat., 154 (1999), 46-68. Google Scholar

[12]

J. HuismanN. N. P. ThiD. M. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum, Nature, 439 (2006), 322-325. Google Scholar

[13]

C. G. JägerS. Diehl and M. Emans, Physical determinants of phytoplankton production, algal stoichiometry, and vertical nutrient fluxes, Am. Nat., 175 (2010), 91-104. Google Scholar

[14]

C. G. Jäger and S. Diehl, Resource competition across habitat boundaries: asymmetric interactions between benthic and pelagic producers, Ecol. Monogr., 84 (2014), 287-302. Google Scholar

[15]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. Google Scholar

[16]

T. KolokolnikovC. H. Ou and Y. Yuan, Phytoplankton depth profiles and their transitions near the critical sinking velocity, J. Math. Biol., 59 (2009), 105-122. doi: 10.1007/s00285-008-0221-z. Google Scholar

[17]

L. F. Mei and X. Y. Zhang, On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients, Discrete Cont. Dyn. Syst.-B, 17 (2012), 221-243. doi: 10.3934/dcdsb.2012.17.221. Google Scholar

[18]

L. F. Mei and X. Y. Zhang, Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics, J. Differential Equations, 253 (2012), 2025-2063. doi: 10.1016/j.jde.2012.06.011. Google Scholar

[19]

L. F. MeiS. B. Hsu and F. B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Cont. Dyn. Syst.-B, 21 (2016), 607-620. doi: 10.3934/dcdsb.2016.21.607. Google Scholar

[20]

H. NieS. B. Hsu and J. H. Wu, Coexistence solutions of a competition model with two species in a water column, Discrete Cont. Dyn. Syst.-B, 20 (2015), 2691-2714. doi: 10.3934/dcdsb.2015.20.2691. Google Scholar

[21]

R. Peng and X. Q. Zhao, A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species, J. Math. Biol., 72 (2016), 755-791. doi: 10.1007/s00285-015-0904-1. Google Scholar

[22]

M. SchefferS. SzabóA. GragnaniE. H. van NesS. RinaldiN. KautskyJ. NorbergR. M. M. Roijackers and R. J. M. Franken, Floating plant dominance as a stable state, Proc. Natl. Acad. Sci. USA, 100 (2003), 4040-4045. Google Scholar

[23]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[24]

F. R. VasconcelosS. DiehlP. RodríguezP. HedströmJ. Karlsson and P. Byström, Asymmetrical competition between aquatic primary producers in a warmer and browner world, Ecology, 97 (2016), 2580-2592. Google Scholar

[25]

Y. VadeboncoeurG. PetersonM. Vander Zanden and J. Kalff, Benthic algal production across lake size gradients: interactions among morphometry, nutrients, and light, Ecology, 89 (2008), 2542-2552. Google Scholar

[26]

F. B. WangS. B. Hsu and X. Q. Zhao, A reaction-diffusion-advection model of harmful algae growth with toxin degradation, J. Differential Equations, 259 (2015), 3178-3201. doi: 10.1016/j.jde.2015.04.018. Google Scholar

[27]

F. B. Wang, A PDE system modeling the competition and inhibition of harmful algae with seasonal variations, Nonlinear Analysis RWA, 25 (2015), 258-275. doi: 10.1016/j.nonrwa.2015.02.010. Google Scholar

[28]

X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. Google Scholar

[29]

X. F. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x. Google Scholar

[30]

K. YoshiyamaJ. P. MellardE. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203. Google Scholar

[31]

A. ZagarisA. DoelmanN. N. Pham Thi and B. P. Sommeijer, Blooming in a nonlocal, coupled phytoplankton-nutrient model, SIAM J. Appl. Math., 69 (2009), 1174-1204. doi: 10.1137/070693692. Google Scholar

[32]

A. Zagaris and A. Doelman, Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model, Nonlinearity, 24 (2011), 3437-3486. doi: 10.1088/0951-7715/24/12/007. Google Scholar

[33]

J. M. ZhangJ. P. Shi and X. Y. Chang, A mathematical model of algae growth in a pelagic-benthic coupled shallow aquatic ecosystem, J. Math. Biol., 76 (2018), 1159-1193. doi: 10.1007/s00285-017-1168-8. Google Scholar

Table 1.  Variables and parameters of model (1) with biological meanings
Symbol Meaning Symbol Meaning
$ t $Time$ z $Depth
$ U $Biomass density of pelagic algae$ V $Biomass density of benthic algae
$ R $Concentration of dissolved nutrients in the pelagic habitat$ W $Concentration of dissolved nutrients in the benthic habitat
$ D_u $Vertical turbulent diffusivity of pelagic algae$ D_r $Vertical turbulent diffusivity of dissolved nutrients in the pelagic habitat
$ s $Sinking or buoyant velocity of pelagic algae$ r_u,r_v $Maximum specific production rate of pelagic algae and benthic algae, respectively
$ m_u,m_v $Loss rate of pelagic and benthic algae, respectively$ \gamma_u,\gamma_v $Half saturation constant for nutrient-limited production of pelagic algae and benthic algae, respectively
$ c_u,c_v $Phosphorus to carbon quota of pelagic algae and benthic algae, respectively$ W_{sed} $Concentration of dissolved nutrients in the sediment
$ L_1 $Depth of the pelagic habitat (below water surface)$ L_2 $Vertical extent of the benthic habitat
$ a $Nutrient exchange rate between pelagic and benthic habitat$ b $Nutrient exchange rate between sediment and benthic habitat
$ \beta_u $Nutrient recycling proportion from loss of pelagic algal biomass$ \beta_v $Nutrient recycling proportion from loss of benthic algal biomass
Symbol Meaning Symbol Meaning
$ t $Time$ z $Depth
$ U $Biomass density of pelagic algae$ V $Biomass density of benthic algae
$ R $Concentration of dissolved nutrients in the pelagic habitat$ W $Concentration of dissolved nutrients in the benthic habitat
$ D_u $Vertical turbulent diffusivity of pelagic algae$ D_r $Vertical turbulent diffusivity of dissolved nutrients in the pelagic habitat
$ s $Sinking or buoyant velocity of pelagic algae$ r_u,r_v $Maximum specific production rate of pelagic algae and benthic algae, respectively
$ m_u,m_v $Loss rate of pelagic and benthic algae, respectively$ \gamma_u,\gamma_v $Half saturation constant for nutrient-limited production of pelagic algae and benthic algae, respectively
$ c_u,c_v $Phosphorus to carbon quota of pelagic algae and benthic algae, respectively$ W_{sed} $Concentration of dissolved nutrients in the sediment
$ L_1 $Depth of the pelagic habitat (below water surface)$ L_2 $Vertical extent of the benthic habitat
$ a $Nutrient exchange rate between pelagic and benthic habitat$ b $Nutrient exchange rate between sediment and benthic habitat
$ \beta_u $Nutrient recycling proportion from loss of pelagic algal biomass$ \beta_v $Nutrient recycling proportion from loss of benthic algal biomass
[1]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[2]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[3]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114

[4]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681

[5]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255

[6]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[7]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471

[8]

Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373

[9]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630

[10]

Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85

[11]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65

[12]

Zhiting Xu, Yingying Zhao. A reaction-diffusion model of dengue transmission. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2993-3018. doi: 10.3934/dcdsb.2014.19.2993

[13]

Feng-Bin Wang. A periodic reaction-diffusion model with a quiescent stage. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 283-295. doi: 10.3934/dcdsb.2012.17.283

[14]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[15]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[16]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[17]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415

[18]

Feng-Bin Wang, Sze-Bi Hsu, Wendi Wang. Dynamics of harmful algae with seasonal temperature variations in the cove-main lake. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 313-335. doi: 10.3934/dcdsb.2016.21.313

[19]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[20]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (51)
  • HTML views (203)
  • Cited by (0)

Other articles
by authors

[Back to Top]