Abstract
From a biological perspective, a dynamic model describing the cultivation and flocculation of a microorganism that uses two different kinds of nutrients (carbon source and nitrogen source) is proposed. For the proposed model, there always exists a boundary equilibrium, i.e., R h o d o p s e u d o m o n a s p a l u s t r i s -free equilibrium. Furthermore, under additional conditions, the model also has five positive equilibria at most, i.e., the equilibria for which carbon source, nitrogen source, R h o d o p s e u d o m o n a s p a l u s t r i s and flocculants are coexistent. The phenomena of backward and forward bifurcations are extensively discussed by using center manifold theory. The global stability of the boundary equilibrium of the proposed model is deeply investigated. Moreover, the local stability of the positive equilibrium and the uniform persistence of the proposed model are discussed. Under additional conditions, the global stability of the positive equilibrium is studied. Some control strategies are given by the theoretical analysis. Finally, some numerical simulations are performed to confirm the correctness of the theoretical results.
Highlights
Photosynthetic bacteria, which are common microorganisms in the natural environment, have been applied in the field of environmental protection, such as in the treatment of sewage, domestic wastewater and the bioremediation of sediment mud polluted with organic matter
It should be mentioned here that, the analysis reveals that Model (1) proposed in [40] exhibits the phenomenon of backward bifurcation for the existence of positive equilibria
Motivated by the papers mentioned above, in this paper, we further consider a dynamic model describing the cultivation and flocculation of Rhodopseudomonas palustris, and the nutrients presented in [40] will be divided into carbon source and nitrogen source, which are perfectly complementary in the culture of Rhodopseudomonas palustris
Summary
Photosynthetic bacteria, which are common microorganisms in the natural environment, have been applied in the field of environmental protection, such as in the treatment of sewage, domestic wastewater and the bioremediation of sediment mud polluted with organic matter (see, for example, [1,2,3,4]). Motivated by the papers mentioned above, in this paper, we further consider a dynamic model describing the cultivation and flocculation of Rhodopseudomonas palustris, and the nutrients presented in [40] will be divided into carbon source and nitrogen source, which are perfectly complementary in the culture of Rhodopseudomonas palustris (see, for example, [5,41,42,43,44]). The term rμ (C (t))μ2 ( N (t)) is the growth rate of Rhodopseudomonas palustris, and the terms r1 μ1 (C (t))μ2 ( N (t)) and r2 μ1 (C (t))μ2 ( N (t)) represent the quantity of the decreasing of the carbon source and nitrogen source, respectively, where r1 and r2 are non-negative constants; the functions μ1 (C (t)) and μ2 ( N (t)) are nonnegative and continuous for C (t) ≥ 0, N (t) ≥ 0.
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