# American Institute of Mathematical Sciences

• 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:

show all references

##### References:
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] 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 [2] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [3] 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 [4] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [5] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [6] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [7] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [8] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 [9] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [10] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 [11] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [12] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 [13] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [14] D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346 [15] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [16] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [17] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [18] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [19] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [20] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

2019 Impact Factor: 1.105