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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:

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##### 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
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