
Citation: Amreeta Sarjit, Sin Mei Tan, Gary A. Dykes. Surface modification of materials to encourage beneficial biofilm formation[J]. AIMS Bioengineering, 2015, 2(4): 404-422. doi: 10.3934/bioeng.2015.4.404
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Phytoplankton-derived carbon (internal primary production) is known to be essential for the somatic growth and reproduction of zooplankton and fish. For decades, aquatic food webs were taken as systems where carbon transfer was linear from phytoplankton to zooplankton to fish [1].
With the application of isotope labeling, substantial research shows that terrestrial organic matter (TOM) plays an important role in the lake food webs. TOM can not only affect the lake ecosystem in physical and chemical way but also can be exploited by consumer as a resource [2]. And particulate organic matter (POM) of terrestrial origin can be the key factor controlling whole-lake productivity in lakes where phytoplankton productivity is low [1,3]. Besides, in lakes, the inputs of TOM often equal to or exceed internal primary production [4].
Zooplankton composed of cladocerans, cyclopoids, and calanoids represents a critical food chain link between phytoplankton and larger fish in lakes. Lot of experimental results show that zooplankton can directly consume allochthonous POM that either entered the lake in particulate form or was formed through flocculation of allochthonous dissolved organic matter (DOM) [5]. Nevertheless, it is mentioned that Daphnia only supported by allochthonous carbon can survive and give birth to offspring [6]. For the reason that phytoplankton and zooplankton are important components of lakes, plankton mechanism considering TOM is widely studied. Lots of experiments have been done by isotope labeling [5,7,8].
Mathematical modelling is an important tool to investigate the plankton mechanism [9,10,11,12,13,14]. Nutrient-phytoplankton-zooplankton models were studied by many researchers [15,16,17]. In order to enhance the knowledge of plankton mechanism, we propose phytoplankton-zooplankton models considering TOM. We focus on the ecological function of TOM in the plankton mechanism. The results of the deterministic model can fit well with some experimental results and two hypotheses supported by experimental results in [18]. The related two hypotheses are given in the following part:
(Ⅰ) Catchment deposition hypothesis: Allochthony which is the portion of a zooplankter's body carbon content that is of terrestrial origin increases as more TOM is exported from the surrounding catchment.
(Ⅱ) Algal subtraction hypothesis: Allochthony increases with the availability of TOM, where algal production becomes limited by shading more than it benefits from the nutrients associated with TOM.
The rest of the paper is organized as follows: in the next section, by taking TOM into account, formulation of two deterministic models are discussed. In Section 3, global dynamical properties of the three-dimensional ODE model are completely established using Lyapunov function method. In Section $ 4 $, by including stochastic perturbation of the white noise type, we develop a stochastic model and show that there is a stationary distribution in the stochastic model. In Section $ 5 $, the numerical simulation is given to verify the theoretical predictions. In the biological sense, the numerical results are analyzed in detail. In the last section, the conclusion is given.
The research on the interaction between phytoplankton and zooplankton is important. The phytoplankton is not only the producer of the marine ecosystem but also the base of every food web. Phytoplankton is consumed by zooplankton, which is resource for consumers of higher tropical levels. By transferring organic matter and energy into higher tropical levels, zooplankton plays a critical role in lake ecosystem. Usually, the interaction between the phytoplankton and zooplankton can be described by the Prey-Predator system,
$ dP(t)dt=P(t)(r1−a1P(t)−bZ(t)),dZ(t)dt=Z(t)(−r2+cP(t)−a2Z(t)). $
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(2.1) |
Here, $ P(t) $ and $ Z(t) $ are the population densities of phytoplankton and zooplankton, respectively. The phytoplankton grows with rate $ r_{1} $. The interaction term taken here is Holling type-Ⅰ with $ b $ and $ c $ as the respective rates of grazing and biomass conversion by the zooplankton for its growth such that $ b > c $. The natural mortality of zooplankton is $ r_{2} $. $ a_{1} $ reflects the density dependence of phytoplankton. The density dependence of predator species is wildly studied because of the environment factor [14]. In lakes where phytoplankton productivity is low, zooplankton will compete each other for the resources to survive, which means that it is more realistic to consider the density dependence of zooplankton population in system (2.1). Furthermore, the zooplankton is resource for higher tropical levels. The mortality caused by higher trophic levels is generally modeled by a nonlinear term [9]. Hence, $ a_{2} $ reflects the density dependence of zooplankton and the mortality of zooplankton caused by higher tropic. Here, it is easy to know that system (2.1) has two equilibria: $ O(0, 0) $, $ A(\frac{r_{1}}{a_{1}}, 0). $ Besides, when $ cr_{1}-a_{1}r_{2} > 0 $, there is a positive equilibrium $ E^{*}(P^{*}, Z^{*}) $, where $ P^{*} = \frac{r_{1}a_{2}+r_{2}b}{a_{1}a_{2}+bc} $, $ Z^{*} = \frac{cr_{1}-a_{1}r_{2}}{a_{1}a_{2}+bc} $, which is globally asymptotically stable. If $ cr_{1}-a_{1}r_{2}\leq 0 $, equilibrium $ A(\frac{r_{1}}{a_{1}}, 0) $ is globally asymptotically stable.
Model $ (2.1) $ ignores TOM in the lake ecosystem. Since TOM has critical influence in the lake food webs, it is more realistic to incorporate it into the phytoplankton-zooplankton model. Based on the important role of TOM in lake ecosystem, we propose the terrestrial organic matter-phytoplankton-zooplankton model. The meaning of the parameters and some assumptions are proposed in following part, firstly.
$ C(t) $ is the concentration of TOM. $ P(t) $, $ Z(t) $ are the population densities of phytoplankton and zooplankton, respectively. $ s $ is the constant input rate of TOM and $ \delta $ is its sinking rate. The interaction term between the TOM and zooplankton is Holling type-Ⅰ. $ \beta_{0} $ is TOM uptake rate by zooplankton. TOM is converted for the growth of zooplankton with rate $ \beta_{3} $. The phytoplankton grows with growth rate $ r $. $ K $ is the carrying capacity. The interaction term between the phytoplankton and zooplankton is also Holling type-Ⅰ with $ \beta_{1}, \beta_{2} $ as the respective rates of grazing and biomass conversion by the zooplankton for their growth. $ \gamma $ is the natural mortality of zooplankton. $ m $ measures the strength of competition among zooplankton and the mortality of zooplankton caused by higher tropic. Above statements can be seen clearly in Figure 1.
Based on the above statements, the following ordinary differential equations can be derived,
$ dC(t)dt=−β0C(t)Z(t)−δC(t)+s,dP(t)dt=rP(t)(1−P(t)K)−β1P(t)Z(t),dZ(t)dt=β2P(t)Z(t)+β3C(t)Z(t)−γZ(t)−mZ2(t). $
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(2.2) |
In the biological context, the population densities of phytoplankton and zooplankton are nonnegative at any time $ t $. The concentration of TOM is also nonnegative at any time $ t $. We get the following initial conditions,
$ C(0)=C0≥0,P(0)=P0≥0,Z(0)=Z0≥0. $
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(2.3) |
Proposition 2.1. The solution $ (C(t), P(t), Z(t)) $ of $ (2.2) $ with the initial condition $ (2.3) $ is nonnegative and bounded in $ R_{+}^{3} $.
Proof. From the equations of (2.2), it is easy to get that $ C(t)\geq C(0)e^{-\int_{0}^{t}\beta_{0}Z(s)+\delta ds}\geq 0 $, $ P(t) = P(0)e^{\int_{0}^{t}r\left(1-\frac{P(s)}{K}\right)-\beta_{1}Z(s)ds}\geq 0 $ and $ Z(t) = Z(0)e^{\int_{0}^{t}\beta_{2}P(s)+\beta_{3}C(s)-\gamma-mZ(s)ds}\geq 0 $. Hence, the solutions are non-negative. Clearly, $ \frac{dC}{dt}\leq -\delta C+s $ yielding $ C(t)\leq \max\left \{P_{0}, \frac{s}{\delta}\right \}: = C_{max} $. $ \frac{dP}{dt}\leq rP(1-\frac{P}{K}) $ yielding $ P(t)\leq \max\{K, P_{0}\}: = P_{max} $. Therefore, $ Z(t)\leq \beta_{2}P_{max}Z+\beta_{3}C_{max}Z-mZ^{2} $. Hence, $ Z(t)\leq \max \left \{Z_{0}, \frac{\beta_{2}P_{max}+\beta_{3}C_{max}}{m}\right \}. $ It implies the solutions $ (C(t), P(t), Z(t)) $ of $ (2.2) $ are nonnegative and bounded for all $ t\geq 0. $
The nonnegativity of the solution ensures that the model has practical significance. In the following part, we can get the equilibria of $ (2.2) $. By direct calculation, it can be known that $ (2.2) $ always has two equilibria,
$ E1=(sδ,0,0),E2=(C2,P2,0)=(sδ,K,0). $
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Proposition 2.2. If $ s > \frac{\delta \gamma}{\beta_{3}} $, there exists unique semi-trivial equilibrium $ E_{3}(C_{3}, 0, Z_{3}) $ of (2.2). If $ K\geq \frac{\gamma}{\beta_{2}} $ and $ s\in(0, s_{1}) $, or $ K < \frac{\gamma}{\beta_{2}} $ and $ s\in(s_{0}, s_{1}) $, where
$ s0=δγβ3−β2Kδβ3,s1=δγβ3+r2mβ0β3β21+r(γβ0+δm)β1β3, $
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there exists a unique positive equilibrium $ E_{4}(C_{4}, P_{4}, Z_{4}) $.
Proof. The equilibria satisfy the following system of algebraic equations,
$ −β0CZ−δC+s=0,rP(1−PK)−β1PZ=0,β2PZ+β3CZ−γZ−mZ2=0. $
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(2.4) |
If $ P = 0 $, $ Z\neq 0 $, the equilibrium is given by
$ C∗=γ+mZ∗β3,Z∗=−δC∗+sβ0C∗, $
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(2.5) |
where $ C^{*} $ and $ Z^{*} $ are constants. From the second equation of (2.5), we can get
$ a3Z2+b3Z+c3=0, $
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(2.6) |
where $ a_{3} = -m\beta_{0}, b_{3} = -(\gamma\beta_{0}+\delta m), c_{3} = \beta_{3}s-\gamma\delta $. If $ s > \frac{\delta \gamma}{\beta_{3}} $, then $ c_{3} > 0 $. Thus, $ \Delta = b_{3}^{2}-4a_{3}c_{3} > 0 $, $ Z_{31}+Z_{32} = -\frac{b_{3}}{a_{3}} < 0 $, $ Z_{31}Z_{32} = \frac{c_{3}}{a_{3}} < 0 $, which means that (2.6) exists only one positive root. It is easy to get that $ Z_{3} = \frac{-b_{3}+\sqrt{b_{3}^{2}-4a_{3}c_{3}}}{2a_{3}} $. Then $ C_{3} = \frac{\gamma+mZ_{3}}{\beta_{3}}. $
If $ P\neq0, Z\neq0, $ the equilibria of $ (2.4) $ are given by
$ H(P)=a4P2+b4P+c4, $
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(2.7) |
where
$ a4=−β0r(β2+rmβ1K),b4=(β0rK+δβ1K)(β2+rmβ1K+β0r(γ+rmβ1)),c4=K(β1β3s−(γ+rmβ1)(β0r+δβ1)). $
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If $ s < s_{1} $, then $ c_{4} < 0 $. Hence, $ H(0) = c_{4} < 0 $. If $ K\geq \frac{\gamma}{\beta_{2}} $, or $ K < \frac{\gamma}{\beta_{2}} $ and $ s > s_{0} $, $ H(K) = \beta_{3}s+\beta_{2}K\delta-\gamma\delta > 0. $ And $ \lim_{P \to +\infty} H(P) < 0. $ Therefore, function $ H(P) $ has two positive roots, one of which is larger than $ K $. However, $ Z = \frac{r}{\beta_{1}}\left(1-\frac{P}{K}\right) $. The root which is bigger than $ K $ should be discarded. Hence, there is a unique positive equilibrium.
Here, in biologic context, $ E_{1} $ means that both the phytoplankton and zooplankton go extinct eventually. $ E_{2} $ means that the zooplankton goes extinct, while the phytoplankton will exist at the density of $ K $ eventually. $ E_{3} $ means that the phytoplankton goes extinct, while the zooplankton will exist at the density of $ Z_{3} $ eventually. $ E_{4} $ means that the phytoplankton and zooplankton coexist. The existence of $ E_{3} $ and $ E_{4} $ is mainly determined by $ s $, the constant input rate of TOM.
In the next section, we will analyse the stability of those equilibria. It should be pointed out that $ (2.2) $ is totally different from the nutrient-phytoplankton-zooplankton model, for the reason that $ s $ stands for TOM, which is ingested by zooplankton.
In this section, global dynamical properties of $ (2.2) $ are established.
Firstly, at the trivial equilibrium point $ E_{1}(\frac{s}{\delta}, 0, 0) $ in (2.2), the corresponding characteristic equation is
$ (λ+δ)(λ−r)(λ+γ−β3sδ)=0. $
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(3.1) |
It is apparent that equation (3.1) has one positive eigenvalue, which implies that $ E_{1}(\frac{s}{\delta}, 0, 0) $ is always unstable.
Theorem 3.1. If $ s\geq s_{1} $, then $ E_{3}(C_{3}, 0, Z_{3}) $ is globally asymptotically stable.
Proof. Consider the following Lyapunov function
$ V1(t)=β32β0C3(C−C3)2+Z−Z3−Z3lnZZ3+β2β1P. $
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(3.2) |
Since the function $ h(z) = z-1-\ln z $ is always nonnegative, it is easy to know that $ V_{1}(t) $ is also nonnegative at any time t. Then the derivative of $ V_{1}(t) $ along the solution of $ (2.2) $ is given by
$ dV1dt=β3β0C3(C−C3)(−β0CZ−δC+s)+β2rβ1P−β2PZ−β2rβ1KP2+(Z−Z3)(β2P+β3C−mZ−γ)=β3β0C3(C−C3)(−β0CZ−δC+β0C3Z3+δC3)−m(Z3−Z)2+β3(C−C3)(Z−Z3)+β2(rβ1−Z3)P−β2rβ1KP2=−β3δβ0C3(C−C3)2−β3ZC3(C−C3)2−m(Z−Z3)2−β2rβ1KP2+β2(rβ1−Z3)P. $
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If $ s\geq s_{1} $, then $ \beta_{2}\left(\frac{r}{\beta_{1}}-Z_{3}\right)\leq 0. $ It follows that $ \frac{dV_{1}}{dt}\leq 0 $. Furthermore, $ \frac{dV_{1}}{dt} = 0 $ if and only if $ C(t) = C_{3} $, $ P(t) = 0 $ and $ Z(t) = {Z_{3}} $. Thus, the largest invariant set on which $ \frac{dV_{1}}{dt} $ is zero consists of just equilibrium $ E_{3} $. Therefore, by LaSalle's Invariance Principle, equilibrium $ E_{3} $ is globally asymptotically stable.
Theorem 3.2. If $ K\geq \frac{\gamma}{\beta_{2}} $ and $ s\in(0, s_{1}) $, or $ K < \frac{\gamma}{\beta_{2}} $ and $ s\in(s_{0}, s_{1}) $, then $ E_{4}(C_{4}, P_{4}, Z_{4}) $ is globally asymptotically stable.
Proof. Consider the following Lyapunov function
$ V2(t)=β32β0C4(C−C4)2+Z−Z4−Z4lnZZ4+β2β1(P−P4−P4lnPP4). $
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(3.3) |
Similarly, the derivative of $ V_{2}(t) $ along the solution of $ (2.2) $ is given by
$ dV2dt=β3β0C4(C−C4)(−β0CZ−δC+s)+β2β1(P−P4)(r−rPK−β1Z)+(Z−Z4)(β2P+β3C−mZ−γ)=β3β0C4(C−C4)(−β0CZ−δC+β0C4Z4+δC4)−m(Z4−Z)2+β3(C−C4)(Z−Z4)−rβ2Kβ1(P−P4)2=−β3δβ0C4(C−C4)2−β3ZC4(C−C4)2−m(Z−Z4)2−rβ2Kβ1(P−P4)2. $
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It is easy to know $ \frac{dV_{2}}{dt}\leq 0 $. $ \frac{dV_{2}}{dt} = 0 $ if and only if $ C(t) = C_{4} $, $ P(t) = P_{4} $ and $ Z(t) = Z_{4} $, and hence the largest invariant set of (2.2) in the set $ \left\{ (C(t), P(t), Z(t))|\frac{dV_{2}}{dt} = 0 \right\} $ is the singleton $ \{E_{4}\} $. Therefore, by the LaSalle's Invariance Principle, equilibrium $ E_{4} $ is globally asymptotically stable.
Theorem 3.3. If $ K < \frac{\gamma}{\beta_{2}} $ and $ s\in(0, s_{0}] $, then $ E_{2}(\frac{s}{\delta}, K, 0) $ is globally asymptotically stable.
Proof. Consider the following Lyapunov function
$ V3(t)=β32β0C2(C−C2)2+Z+β2β1(P−P2−P2lnPP2). $
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(3.4) |
Similarly, the derivative of $ V_{3}(t) $ along the solution of $ (2.2) $ is given by
$ dV3dt=β3β0C2(C−C2)(−β0CZ−δC+s)+β2β1(P−P2)(r−rPK−β1Z)+β2PZ+β3CZ−γZ−mZ2=β3β0C2(C−C2)(−β0CZ−δC+δC2)+β2β1(P−P2)(rP2K−rPK−β1Z)+β2PZ+β3CZ−γZ−mZ2=−β3δβ0C2(C−C2)2+β3β0C2(C−C2)(−β0CZ+β0C2Z−β0C2Z)−rβ2Kβ1(P−P2)2+β2P2Z+β3CZ−γZ−mZ2=−β3δβ0C2(C−C2)2−β3ZC2(C−C2)2−rβ2Kβ1(P−P4)2−mZ2+(β3C2+β2P2−γ)Z. $
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If $ s\leq\frac{\delta \gamma}{\beta_{3}}-\frac{\beta_{2}K\delta}{\beta_{3}} $, then $ \beta_{3}C_{2}+\beta_{2}P_{2}-\gamma = \frac{\beta_{3}s}{\delta}+\beta_{2}K-\gamma\leq 0 $. It is easy to know $ \frac{dV_{3}}{dt}\leq 0 $. $ \frac{dV_{3}}{dt} = 0 $ if and only if $ C(t) = \frac{s}{\delta} $, $ P(t) = K $ and $ Z(t) = 0 $. Similarly, by the LaSalle's Invariance Principle, every solution of $ (2.2) $ tends to $ M $, where $ M = \left\{ (C(t), P(t), Z(t))|\frac{dV_{3}}{dt} = 0 \right\} = \{E_{2}\} $ is the largest invariant set of (2.2). It implies that $ E_{2} $ is globally asymptotically stable.
Global dynamical properties of $ (2.2) $ are shown clearly in Table 1. From biological viewpoint, Theorem 3.1 implies that if the constant input of TOM is larger than $ s_{1} $, it will result in the extinction of phytoplankton eventually which is impossible in the system (2.1). It also means that the increasing input of the TOM will inhibit the growth of phytoplankton, which fits well with the hypotheses(Ⅱ). In addition, in the extinct process of the phytoplankton, in order to survive, the zooplankton will uptake more TOM which is easy to get. That is the Allochthony of zooplankton will increase, which fits well with the hypotheses(Ⅰ). The coexistence of TOM and zooplankton is mentioned in [6]. It also means that the producers of catchment can support both the landscape and lake ecosystem, as told in [2]. Furthermore, from Theorem 3.2, it can be known that different from the system $ (2.1) $, with the enough constant input of the TOM($ s > s_{0} $), zooplankton in the deterministic setting can produce and give offsprings at small carrying capacity(K). The result fits well with the experimental result in [19]. Besides, the threshold value $ s_{1} $ is positively related to the birth rate of phytoplankton, which means that phytoplankton with high birth rate is less possible to become extinct. The result of Theorem 3.3 implies that zooplankton goes extinct if both the carrying capacity and constant input rate of TOM are small.
$ K\geq \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in(0, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ K < \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in (s_{0}, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ s \in (0, s_{0}] $ | $ E_{2} $ is G.A.S. |
In lake ecosystem, the weather, temperature and other physical factors are hard to predict. TOM, phytoplankton and zooplankton are easily affected by these factors. The deterministic model studied in the previous section does not take environmental randomness into consideration. Taking the randomness into account is more realistic. Recently, many researchers studied the stochastic biological models. Different types of stochastic perturbations are introduced into the models. Prey-predator models with fluctuations around the positive equilibrium were studied [20,21]. The stochastic perturbations which are proportional to the variables were discussed [22,23]. Fluctuations manifesting in the transmission coefficient rate were discussed [24,25].
In this section, by considering role of unpredictable environmental factors in plankton dynamics, we get following stochastic equations,
$ {dC(t)=(−β0C(t)Z(t)−δC(t)+s)dt+σ1C(t)dB1(t),dP(t)=(rP(t)(1−P(t)K)−β1P(t)Z(t))dt+σ2P(t)dB2(t),dZ(t)=(β2P(t)Z(t)−γZ(t)+β3C(t)Z(t)−mZ2(t))dt+σ3Z(t)dB3(t), $
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(4.1) |
where $ B_{i}(t), i = 1, 2, 3 $, are independent Brownian motions and $ \sigma_{i} $, $ i = 1, 2, 3 $ are the corresponding intensities of stochastic perturbations. Obviously, all the equilibria of $ (2.2) $ are no longer the equilibria of the stochastic model (4.1).
In this paper, let $ (\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq 0}, \mathrm{P}) $ be a complete probability space with filtration $ \{\mathcal{F}_{t}\}_{t\geq 0} $ satisfying the usual conditions(i.e. it is right continuous and $ \mathcal{F}_{0} $ contains all $ \mathrm{P} $-null sets). Let $ B_{i}(t) $, $ i = 1, 2, 3, $ be a standard one-dimensional Brownian motion defined on this complete probability space. Let $ R^{3}_{+} = \{x \in R^{3}: x_{i} > 0, 1\leq i \leq 3\} $.
Definition 4.1 ([26]). A non-negative random variable $ \tau(\omega) $, which is allowed to the value $ \infty $, is called stopping time(with respect to filtration $ \mathcal{F}_{t} $), if for each t, the event $ \{\omega: \tau \leq t\}\in \mathcal{F}_{t} $.
Lemma 4.2 ([26]). Let $ \{X_{t}\}_{t} $ be right-continuous on $ R^{n} $ and adapted to $ \mathcal{F}_{t} $. $ Define \inf \emptyset = \infty $. If D is an open interval on $ R^{n} $, $ \tau_{D} = \inf\{t\geq 0: X_{t}\not\in D\} $ is a $ \mathcal{F}_{t} $ stopping time.
Lemma 4.3 ([26]). Let $ x(t) = (x_{1}(t), ..., x_{n}(t)) $ be a regular adapted process $ n $-dimensional vector process. We can have any number $ n $ of process driven by a $ d $-dimensional Brownian motion
$ dx(t)=b(t)dt+σ(t)dBt, $
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where $ \sigma(t) $ is $ n\times d $ matrix valued function, $ B_{t} $ is $ d $-dimensional Brownian motion, $ x(t), \; b(t) $ are $ n $-dim vector-valued functions, the integrals with respect to Brownian motion are It$ \hat{o} $ integrals. Then $ x(t) $ is called an It$ \hat{o} $ process. The only restriction is that this dependence results in:
(i) for any $ i = 1, 2...n $, $ b_{i}(t) $ is adapted and $ \int_{0}^{T}\left|b_{i}(t)\right | < \infty $ a.s..
(ii) for any $ i = 1, 2, ...n $, $ \sigma_{ij}(t) $ is adapted and $ \int_{0}^{T}\sigma_{ij}^{2}(t) < \infty $ a.s.
Lemma 4.4 ([27]). Let $ x(t) $ be a $ l $-dimensional $ It\hat{o} $ process on $ t\geq 0 $ with stochastic differential
$ dx(t)=f(t)dt+g(t)dBt, $
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where $ f \in \mathcal{L}(R_{+}; R^{l}) $ and $ g \in \mathcal{L}^{2}(R_{+}; R^{l\times m}). $ Let $ V\in C^{2, 1}(R^{l}\times R_{+}; R) $. Then $ V(x(t), t) $ is till an $ It\hat{o} $ process with the stochastic differential given by
$ dV(x,t)=[Vt+Vx(t,x)f(t)+12Tr[gT(t)Vxxg(t)]]dt+Vx(x,t)g(t)dB(t)a.s., $
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where
$ Vt=∂V∂t,Vx=(∂V∂x1,...,∂V∂xl),Vxx=(∂2V∂xi∂xj)l×l. $
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Suppose $ V\in C^{2, 1}(R^{l}\times R_{+}; R) $, define a operator $ LV $ from $ R^l\times R_{+} $ to $ R $ by
$ LV(t,x)=Vt+Vx(t,x)f(t)+12Tr[gT(t)Vxxg(t)]. $
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Theorem 4.5. For any given initial value $ (C(0), P(0), Z(0))\in R^{3}_{+} $, (4.1) has almost surely (a.s.) a unique positive solution $ (C(t), P(t), Z(t)) $ for $ t\geqslant 0 $, and the solution remains in $ R^{3}_{+} $ with probability 1.
Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value $ (C(0), P(0), Z(0))\in R^{3}_{+} $, it can be obtained that there is a unique local solution $ (C(t), P(t), Z(t)) $ on $ t\in[0, \tau_e) $ where $ \tau_{e} $ is the explosion time. Explosion refers to the situation when the process reaches infinite values in finite time. Solution can be considered until the time of explosion[26]. If we can check that $ \tau_e = \infty $ a.s., then the solution is global. Let $ h_{0} $ be sufficient large, such that every coordinate of (C(0), P(0), Z(0)) lies within the interval $ \left[\frac{1}{h_{0}}, h_{0}\right] $. For each integer $ h\geq h_{0} $, stopping time is defined by
$ τh=inf{t∈[0,τe):P(t)∉(1h,h)orC(t)∉(1h,h)orZ(t)∉(1h,h)}, $
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where we set $ \inf \emptyset = \infty. $ Clearly, $ \tau_{h} $ is increasing as $ h\rightarrow\infty $. Let $ \tau_{\infty} = \lim_{h\rightarrow\infty}\tau_{h} $, where $ \tau_{\infty} \leq \tau_{e} $ a.s. If $ \tau_{\infty} = \infty $ a.s., then $ \tau_{e} = \infty $ and $ (C(t), P(t), Z(t))\in R^{3}_{+} $ a.s. for all $ t\geq 0 $. If not, then it exists constants $ T > 0 $ and $ \epsilon\in(0, 1) $ such that $ P\{\tau_{\infty}\leq T\} > \epsilon $. Hence, there is integer $ h_{1}\geq h_{0} $ such that
$ P\{\tau_{h}\leq T\} \gt \epsilon, \; h\geq h_{1}. $ |
Define a function $ V_{4} $: $ R^{3}_{+}\rightarrow R_{+} $ as
$ V4(C,P,Z)=β3β0(C+1−lnC)+β2β1(P+1−lnP)+Z+1−lnZ. $
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(4.2) |
We can get the non-negativity of this function from $ f(u) = u+1-\ln u \geq 0, \forall u > 0 $. Applying It$ \hat{o} $ formula, it follows that
$ dV4(C,P,Z)=(β3β0(1−1C)(−β0CZ−δC+s)+β2β1(1−1P)(rP(1−PK)−β1PZ)+(1−1Z)(β2PZ+β3CZ−γZ−mZ2)+β3σ212β0+β2σ222β1+σ232)dt+β3σ1β0(C−1)dB1(t)+β2σ2β1(P−1)dB2(t)+σ3(Z−1)dB3(t)=(β3β0(−δC+s+β0Z+δ−sC)+β2β1(rP(1−PK)−r+rPK)+γ+Z(β2−γ−mZ)−β2P−β3C+mZ+β3σ212β0+β2σ222β1+σ232)dt+β3σ1β0(C−1)dB1(t)+β2σ2β1(P−1)dB2(t)+σ3(Z−1)dB3(t). $
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Hence,
$ dV4(C,P,Z)≤(β3β0(s+δ)+γ+β3C+β2β1(r+rK+β2)P+(β2+β3+m)Z+β3σ212β0+β2σ222β1+σ232)dt+β3σ1β0(C−1)dB1(t)+β2σ2β1(P−1)dB2(t)+σ3(Z−1)dB3(t). $
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Let
$ q1=β3β0(s+δ)+γ+β3σ212β0+β2σ222β1+σ232,q2=max{β0,r+rK+β2,β2+β3+m},q3=β2+β3+m. $
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Since the function $ f $ is positive, it yields $ u_{i}\leq 2(u_{i}+1-\ln{u_{i}}) $[23]. Therefore, $ \beta_{3}C+\frac{\beta_{2}}{\beta_{1}}\left(r+\frac{r}{K}+\beta_{2}\right)P+q_{3}Z\leq 2q_{2}V_{4}(C, P, Z) $, which implies
$ dV4(C,P,Z)≤(q1+2q2V4(C,P,Z))dt+β3σ1β0(C−1)dB1(t)+β2σ2β1(P−1)dB2(t)+σ3(Z−1)dB3(t), $
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and hence,
$ dV4(C,P,Z)≤q4(1+V4(C,P,Z))dt+β3σ1β0(C−1)dB1(t)+β2σ2β1(P−1)dB2(t)+σ3(Z−1)dB3(t), $
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where $ q_{4} = \max\{q_{1}, 2q_{2}\} $. Let us define $ a\wedge b = \min\{a, b\}. $ If $ h_{1}\leq T $, then
$ ∫τh∧h10dV4(C,P,Z)≤∫τh∧h10q4(1+V4(C,P,Z))dt+∫τh∧h10β3σ1β0(C−1)dB1(t)+∫τh∧h10β2σ2β1(P−1)dB2(t)+∫τh∧h10β3σ3β0(Z−1)dB3(t). $
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Taking expectation of both sides, we can get that
$ EV4(C(τh∧h1),P(τh∧h1),Z(τh∧h1))≤V4(C(0),P(0),Z(0))+E∫τh∧h10q4(1+V4(C,P,Z))dt≤V4(C(0),P(0),Z(0))+q4h1+E∫τh∧h10V4(C,P,Z)dt≤V4(C(0),P(0),Z(0))+q4T+∫h10EV4(C(τh∧h1),P(τh∧h1),Z(τh∧h1))dt. $
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By the Gronwall inequality,
$ EV_{4}(C(\tau_{h}\wedge T), P(\tau_{h}\wedge T), Z(\tau_{h}\wedge T))\leq (V_{4}(C(0), P(0), Z(0))+q_{4}T)e^{q_{4}T}. $ |
We set $ \Omega_{h} = \{\tau_{h}\leq T\} $ for $ h\geq h_{1} $, which implies $ P(\Omega_{h})\geq\epsilon $. Note that for every $ \omega\in\Omega_{h} $, we can have that $ C(\tau_{h}, \omega) $ equals either $ h $ or $ \frac{1}{h} $ or $ P(\tau_{h}, \omega) $ equals either $ h $ or $ \frac{1}{h} $ or $ Z(\tau_{h}, \omega) $ equals either $ h $ or $ \frac{1}{h} $, and hence,
$ V4(C(τh∧T,ω),P(τh∧T,ω),Z(τh∧T,ω))≥min{β3β0(h+1−lnh),β2β1(h+1−lnh),h+1−lnh,β3β0(1h+1+lnh),β2β1(1h+1+lnh),1h+1+lnh}. $
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Then it implies that
$ (V4(C(0),P(0),Z(0))+q4T)eq4T≥E[1Ωh(ω)V4(C(τh∧T,ω),P(τh∧T,ω)Z(τh∧T,ω))]≥ϵmin{β3β0(h+1−lnh),β2β1(h+1−lnh),h+1−lnh,β3β0(1h+1+lnh),β2β1(1h+1+lnh),1h+1+lnh}, $
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where $ 1_{\Omega_{h}} $ is the indicator function of $ \Omega_{h} $. Then, we let $ h \to \infty $, which leads to the contradiction $ \infty > (V_{4}(C(0), P(0), Z(0))+q_{4}T)e^{q_{4}T} = \infty $. Therefore, $ \tau_{h} = \infty $ a.s.
Now, we give the main result of this section in the following part.
Theorem 4.6. Assume $ \sigma_{1}^{2} < \delta $, $ \sigma_{2} $, $ \sigma_{3} > 0 $ such that
$ η<min{β3β0C4(δ−σ21)C24,mZ24,rβ2Kβ1P24}. $
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Then there is a stationary distribution $ \mu(\cdot) $ for (4.1) with initial value $ (P_{0}, C_{0}, Z_{0})\in R_{+}^{3} $, which has ergodic property.
Proof. Define a Lyapunov functional
$ V5(C,P,Z)=β32β0C4(C−C4)2+Z−Z4−Z4lnZZ4+β2β1(P−P4−P4lnPP4). $
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(4.3) |
Then applying It$ \hat{o} $'s formula to (4.1), we obtain
$ LV5=−β3δβ0C4(C−C4)2−β3ZC4(C−C4)2−m(Z−Z4)2−rβ2Kβ1(P−P4)2+β3σ212β0C4C2+β2σ222β1+σ232. $
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(4.4) |
Since $ (a+b)^{2}\leq 2(a^{2}+b^{2}) $, we have that
$ LV5≤−β3β0C4(δ−σ21)(C−C4)2−β3ZC4(C−C4)2−m(Z−Z4)2−rβ2Kβ1(P−P4)2+β3σ21β0C4+β2σ222β1+σ232≤−β3β0C4(δ−σ21)(C−C4)2−m(Z−Z4)2−rβ2Kβ1(P−P4)2+η, $
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where $ \eta = \frac{\beta_{3}\sigma_{1}^{2}}{\beta_{0}}C_{4}+\frac{\beta_{2}\sigma_{2}^{2}}{2\beta_{1}}+\frac{\sigma_{3}^{2}}{2} $. When $ \eta < \min \left\{\frac{\beta_{3}}{\beta_{0}}(\delta-\sigma_{1}^{2})C_{4}, \; mZ_{4}^{2}, \; \frac{r\beta_{2}}{K\beta_{1}}P_{4}^{2}\right\} $, we can get that the ellipsoid
$ −β3β0C4(δ−σ21)(C−C4)2−m(Z−Z4)2−rβ2Kβ1(P−P4)2+η=0, $
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lies entirely in $ R_{+}^{3} $. Let $ U $ be a neighborhood of the ellipsoid with $ \bar{U}\subset E_{l} = R_{+}^{3} $, so for $ x \in U\backslash E_{l} $, $ LV_{5} < -\zeta $($ \zeta $ is a positive constant). Therefore, we have that condition (B.2) in Lemma $ 2.1 $ of [23] is satisfied($ E_{l} $ denotes euclidean $ l $-space). Besides, there is $ M > 0 $ such that
$ σ21C2ξ21+σ22P2ξ22+σ23Z2ξ23≥M|ξ2|all(C,P,Z)∈ˉU,ξ∈R3+. $
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Thus, we have that the condition (B.1) of [23] is also satisfied. Therefore, there exists a stable stationary distribution $ \mu(\cdot) $ which is ergodic in $ (4.1) $.
In this section, the numerical simulations of $ (2.2) $ and $ (4.1) $ are given to study the proposed models in details. Different simulated results can fit with different hypotheses mentioned in the first section. Unless otherwise stated, parameter values given in Table 2 are used for the simulations. In order to get biologically plausible results, many values are taken from parameter ranges found in the literature. And the numerical simulations of (2.2) are prepared by tool kit ode45 of Matlab.
Parameters | Definition | Values | Reference |
$ \beta_{0} $ | Zooplankton grazing rate on $ C $ | 0.15 | - |
$ \delta $ | Sinking rate of $ C $ | 0.2 | - |
$ r $ | Intrinsic growth rate of $ P $ | 0.7$ \rm \; d^{-1} $ | [9] |
$ \beta_{1} $ | Zooplankton grazing rate on $ P $ | 0.4$ \rm \; \mu gcl^{-1}d^{-1} $ | - |
$ \beta_{2} $ | Zooplankton grazing efficiency on $ P $ | 0.25 | - |
$ \beta_{max} $ | Maximum grazing rate on $ P $ | 6$ \rm \; \mu gcl^{-1}d^{-1} $ | [9] |
$ \beta_{3} $ | Zooplankton grazing efficiency on $ C $ | 0.05 | - |
$ \gamma $ | Zooplankton mortality | 0.3$ \rm \; d^{-1} $ | [28] |
$ s $ | Constant input rate of C | 1 | - |
$ m $ | Zooplankton competitive mortality | 0.1 | - |
$ K $ | Carrying capacity | 100$ \rm \; \mu gcl^{-1} $ | [28] |
It is easy to get $ s_{1} = 4.4, $ which is the threshold value of model $ (2.2) $. It should be noticed that since it is impossible that the absorbed energy is fully used for reproduction, the values of our parameters should satisfy $ \beta_{0} > \beta_{3} $ and $ \beta_{1} > \beta_{2} $. In the experimental results of [13], it points out that for zooplankton, the growth efficiency of assimilating TOM is lower than that of ingesting phytoplankton. In following analysis, hence, the values of our parameters should satisfy $ \beta_{1} > \beta_{0}. $
In the case where $ s = 1 < s_{1} $, Figure 2(a) shows that solutions dampen and tend to a stable steady state. The carrying capacity varies with the geographical location of the lake. Especially, the carrying capacity of poor nutrient lake is very small. If we decrease $ K $ to 1, from Theorem $ 3.1, $ we can know dynamical behavior of (2.2) is similar. However, the solution of (2.1) with the same parameter values will tend to $ A(\frac{r_{1}}{a_{1}}, 0) $, which means the extinction of zooplankton. Above statements can fit well with the experimental results of [28] that Daphnia magna(zooplankton) can use terrestrial-derived dissolved organic matter (t-DOM) to support growth and reproduction when alternative food sources are limiting.
The catchment areas around different lakes are different. The constant input of TOM is mainly determined by the catchment areas. It is reasonable to consider the variation of parameter value $ s. $ The values of other parameters are kept invariant.
In the case where $ s = 5 > s_{1} $, we see from Figure 3(a) that the phytoplankton goes extinct eventually. In particular, because of the extinction of phytoplankton, the growth of zooplankton is determined by TOM. That is, the zooplankton shows a preference for TOM, which fits well with the hypothesis (Ⅰ).
In addition, it also means the increasing input rate of TOM will hinder the growth of phytoplankton, which fits well with the hypothesis (Ⅱ).
In this part, we decrease $ K = 100 $ to $ 1 $ and $ s = 1 $ to $ 0.1 $. The other parameter values are the same as before. The result of numerical simulation is shown in Figure 4(a). We can notice that the density of zooplankton decreases to zero, which means if both environmental capacity and constant input of TOM are small, the zooplankton goes extinct in the end.
Finally, we consider the stochastic model. the values of the parameters are same as those for Figure 2. And $ \sigma_{1} = \sigma_{2} = \sigma_{3} = 0.1 $. The following discretization equations in Milsteins type[29] are used to iteratively calculate the approximate solutions of stochastic system (4.1) in Matlab programs:
$ Ci+1=Ci+(−δCi−β0CiZi+s)Δt+σ1Ciξ1,i√Δt+σ212Ci(ξ21,iΔt−Δt),Pi+1=Pi+(r(Pi−PiK)−β1PiZi)Δt+σ2Piξ2,i√Δt+σ222Pi(ξ22,iΔt−Δt),Zi+1=Zi+(β2PiZi+β3CiZi−γZi−mZ2i)Δt+σ3Ziξ3,i√Δt+σ232Zi(ξ23,iΔt−Δt), $
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(5.1) |
where $ \xi_{1, i}, \xi_{2, i}, $ and $ \xi_{3, i} $ are $ N(0, 1) $-distributed independent Gaussian random variable, $ \sigma_{1}, \sigma_{2} $ and $ \sigma_{3} $ are intensities of white noise and time increment $ \Delta t > 0 $. The simulation results are shown in Figure 5.
In this paper, firstly, phytoplankton-zooplankton model including TOM is proposed to investigate effects of TOM upon planktonic dynamics. By constructing Lyapunov functions and using LaSalle's Invariance Principle, global stability of the equilibria which is determined by threshold values $ s_{0} $ and $ s_{1} $ is established. The analytical results of $ (2.2) $ indicate that TOM has significant effects on the lake ecosystem stability. Different constant input rates of TOM in the model can result in different results, which represents different biological meanings.
Plankton populations which reside in lake ecosystem are persistently influenced by the environmental fluctuations. In order to make more precise biological findings, we develop a stochastic model. We get that when the perturbation is small, the plankton populations reach a stationary distribution, which is ergodic. Then we simulate model (2.2) with many parameter values taken from the literature by varying the constant input rate of TOM along with the carrying capacity value. The simulations indicate that the magnitude of constant input rate of TOM and carrying capacity affect the persistence of phytoplankton populations greatly. We also simulate the stochastic model with the same parameter values. We can notice that mean value trajectory of (4.1) which is the average of ten trajectories oscillates randomly surrounding the solution of system (2.2), rather than reach the stable steady state (Figure 5). And the simulations also show that the trajectory of (4.1) is totally different from that of (2.2).
In summary, we set up exploratory models to consider the potential role of TOM that mediates interactions between trophic levels in a simple plankton food-chain. We have shown that, in principle, TOM plays an important role in influencing interactions between phytoplankton and zooplankton. The theoretical results of our deterministic model fit well with some experimental results. Further, when the perturbation is small, the stochastic model exists a stationary distribution which is ergodic. We emphasize that the qualitative behaviour of our models that we are interested in and this study are only meant as an initial exploratory attempt to consider the influence of TOM on the lake ecosystem.
This research is supported by the National Natural Sciences Foundation of China (Grant No. 11571326).
The authors declare there is no conflict of interest.
[1] |
Garrett TR, Bhakoo M, Zhang Z (2008) Bacterial adhesions and biofilms on surfaces. Prog Nat Sci 18: 1049-1056. doi: 10.1016/j.pnsc.2008.04.001
![]() |
[2] |
Heydorn A, Ersboll BK, Hentzer M (2000) Experimental reproducibility in flow chamber biofilms. Microbiology 146: 2409-2415. doi: 10.1099/00221287-146-10-2409
![]() |
[3] |
Singh R, Paul D, Jain RK (2006) Biofilms: implications in bioremediation. Trends Microbiol 14: 389-397. doi: 10.1016/j.tim.2006.07.001
![]() |
[4] |
Spector MP, Kenyon WJ (2012) Resistance and survival strategies of Salmonella enterica to environmental stresses. Food Res Int 45: 455-481. doi: 10.1016/j.foodres.2011.06.056
![]() |
[5] |
Carpentier B, Cerf O (1993) Biofilms and their consequences, with particular reference to hygiene in the food industry. J Appl Bacteriol 75: 499-511. doi: 10.1111/j.1365-2672.1993.tb01587.x
![]() |
[6] |
Berlowska J, Kregiel D, Ambroziak W (2013) Enhancing adhesion of yeast brewery strains to chamotte carriers through aminosilane surface modification. World J Microbiol Biotechnol 29: 1307-1316. doi: 10.1007/s11274-013-1294-4
![]() |
[7] |
Kang CS, Eaktasang N, Kwon DY, et al. (2014) Enhanced current production by Desulfovibrio desulfuricans biofilm in a mediator-less microbial fuel cell. Bioresour Technol 165: 27-30. doi: 10.1016/j.biortech.2014.03.148
![]() |
[8] |
Lackner S, Holmberg M, Terada A, et al. (2009) Enhancing the formation and shear resistance of nitrifying biofilms on membranes by surface modification. Water Res 43: 3469-3478. doi: 10.1016/j.watres.2009.05.011
![]() |
[9] | Wang Y, Lee SM, Dykes GA (2014) The physicochemical process of bacterial attachment to abiotic surfaces: Challenges for mechanistic studies, predictability and the development of control strategies. Crit Rev Microbiol. http://dx.doi.org/10.3109/1040841X.2013.866072. |
[10] |
Dunne WM (2002) Bacterial adhesion: Seen any good biofilms lately? Clin Microbiol Rev 15: 155-166. doi: 10.1128/CMR.15.2.155-166.2002
![]() |
[11] |
Costerton JW, Cheng KJ, Geesey GG, et al. (1987) Bacterial biofilms in nature and disease. Annu Rev Microbiol 41: 435-464. doi: 10.1146/annurev.mi.41.100187.002251
![]() |
[12] |
Costerton JW, Lewandowski Z, Caldwell DE, et al. (1995) Microbial biofilms. Annu Rev Microbiol 49: 711-745. doi: 10.1146/annurev.mi.49.100195.003431
![]() |
[13] |
Elder MJ, Stapleton F, Evans E, et al. (1995) Biofilm-related infections in ophthalmology. Eye 9: 102-109. doi: 10.1038/eye.1995.16
![]() |
[14] | Ofek I, Doyle RJ (2000) Bacterial adhesion to cells and tissues. London/New York: Chapman & Hall. |
[15] | van der Aa BC, Dufrêne YF (2002) In situ characterization of bacterial extracellular polymeric substances by AFM. Colloid Surfaces B 23:173-182. |
[16] |
Donlan RM (2002) Biofilms: microbial life on surfaces. Emerg Infect Dis 8: 881-890. doi: 10.3201/eid0809.020063
![]() |
[17] | Kanematsu H, Barry DM (2015) Conditioning films, In: Kanematsu H., Barry D.M., Eds, Biofilm and Materials Science, New York: Springer, 9-16. |
[18] |
Lorite GS, Rodrigues CM, de Souza AA, et al. (2011) The role of conditioning film formation and surface chemical changes on Xylella fastidiosa adhesion and biofilm evolution. J Colloid Interf Sci 359: 289-295. doi: 10.1016/j.jcis.2011.03.066
![]() |
[19] |
Bos R, van der Mei HC, Busscher HJ (1999) Physico-chemistry of initial microbial adhesive interactions - its mechanisms and methods for study. FEMS Microbiol Rev 23: 179-230. doi: 10.1111/j.1574-6976.1999.tb00396.x
![]() |
[20] | An YH, Dickinson RB, Doyle RJ (2000) Mechanisms of bacterial adhesion and pathogenesis of implant and tissue infections, In: An, Y.H., Friedman, R.J., Eds, Handbook of bacterial adhesion: principles, methods, and applications, New Jersey: Humana Press, 1-27. |
[21] | Boland T, Latour RA, Sutzenberger FJ (2000) Molecular basis of bacterial adhesion, In: An, Y.H., Friedman, R.J., Eds, Handbook of bacterial adhesion: principles, methods, and applications, New Jersey: Humana Press, 29-41. |
[22] | Brading MG, Jass J, Lappin-Scott HM (1995) Dynamics of bacterial biofilm formation, In: Lappin-Scott, H.M., Costerton, J.W., Eds, Microbial biofilms, New York: Cambridge University Press, 46-63. |
[23] |
Costerton JW, Stewart PS, Greenberg EP (1999) Bacterial biofilms: A common cause of persistent infections. Science 284: 1318-1322. doi: 10.1126/science.284.5418.1318
![]() |
[24] | Mortensen KP, Conley SN (1994) Film fill fouling in counterflow cooling towers: Mechanisms and design. CTI J 15: 10-25. |
[25] |
McDonogh R, Schaule G, Flemming HC (1994) The permeability of biofouling layers on membranes. J Membr Sci 87: 199-217. doi: 10.1016/0376-7388(93)E0149-E
![]() |
[26] |
Goulter RM, Gentle IR, Dykes GA (2009) Issues in determining factors influencing bacterial attachment: A review using the attachment of Escherichia coli to abiotic surfaces as an example. Lett Appl Microbiol 49: 1-7. doi: 10.1111/j.1472-765X.2009.02591.x
![]() |
[27] |
Bazaka K, Jacob MV, Truong VK, et al. (2011) The effect of polyterpenol thin film surfaces on bacterial viability and adhesion. Polymers 3: 388-404. doi: 10.3390/polym3010388
![]() |
[28] |
Bos R, Van der Mei HC, Gold J, et al. (2000) Retention of bacteria on a substratum surface with micro-patterned hydrophobicity. FEMS Microbiol Lett 189: 311-315. doi: 10.1111/j.1574-6968.2000.tb09249.x
![]() |
[29] | Shi X, Zhu X (2009) Biofilm formation and food safety in food industries. Trends Food Sci Tech 20: 407-413. |
[30] | Christensen BE, Characklis WG (1990) Physical and chemical properties of biofilms. In: Characklis W.G., Marshall K.C., Eds, Biofilms, New York,Wiley, 93-130. |
[31] |
Sheng G-P, Yu H-Q, Li X-Y (2010) Extracellular polymer substances (EPS) of microbial aggregates in biological wastewater treatment systems: A review. Biotechnology Adv 28: 882-894. doi: 10.1016/j.biotechadv.2010.08.001
![]() |
[32] | Flemming H-C, Wingender J (2010) The biofilm matrix. Nature Rev Microbiol 8:623-633. |
[33] |
Whitchurch CB, Tolker-Nielsen T, Ragas PC, et al. (2002) Extracellular DNA required for bacterial biofilm formation. Science 295:1487. doi: 10.1126/science.295.5559.1487
![]() |
[34] | Vilain S, Pretorius JM, Theron J, et al. (2009) DNA as an adhesion: Bacillus cereus requires extracellular DNA to form biofilms. Appl Environ Microbiol 75: 2861-2868. |
[35] | Das T, Sharma PK, Busscher HJ, et al. (2010) Role of extracellular DNA in initial bacterial adhesion and surface aggregation. Appl Environ Microbiol 76: 3405-3408. |
[36] | Das T, Sharma PK, Krom BP, et al. (2011) Role of eDNA on the adhesion forces between Streptococcus mutans and substratum surfaces: influence of ionic strength and substratum hydrophobicity. Langmuir 27: 10113-10118. |
[37] |
Harmsen M, Lappann M, Knochel S, et al (2010) Role of extracellular DNA during biofilm formation by Listeria monocytogenes. Appl Environ Microbiol 76: 2271-2279. doi: 10.1128/AEM.02361-09
![]() |
[38] |
Thomas VC, Thurlow LR, Boyle D, et al (2008) Regulation of autolysis-dependent extracellular DNA release by Enterococcus faecalis extracellular proteases influences biofilm development. J Bacteriol 190: 5690-5698. doi: 10.1128/JB.00314-08
![]() |
[39] | Polson EJ, Buckman JO, Bowen D, et al (2010) An environmental-scanning electron microscope investigation into the effect of biofilm on the wettability of quartz. Soc Pet J 15: 223-227. |
[40] | Neu TR (1996) Significance of bacterial surface-active compounds in interaction of bacteria with interfaces. Microbiol Rev 60: 151-166. |
[41] |
Olofsson A-C, Hermansson M, Elwing H (2003) N-acetyl-L-cysteine affects growth, extracellular polysaccharide production, and bacterial biofilm formation on solid surfaces. Appl Environ Microbiol 69: 4814-4822. doi: 10.1128/AEM.69.8.4814-4822.2003
![]() |
[42] |
Epifanio M, Inguva S, Kitching M, et al. (2015) Effects of atmospheric air plasma treatment of graphite and carbon felt electrodes on the anodic current from Shewanella attached cells. Bioelectrochemistry 106: 186-193. doi: 10.1016/j.bioelechem.2015.03.011
![]() |
[43] | Bhattacharjee S, Ko CH, Elimelech M (1998) DLVO interaction between rough surfaces. Langmuir 14: 3365-3375. |
[44] | Czarnecki J, Warszyński P (1987) The evaluation of tangential forces due to surface in homogeneties in the particle deposition process. Colloid Surface 22: 197-205. |
[45] |
Scheuerman TR, Camper AK, Hamilton MA (1998) Effects of substratum topography on bacterial adhesion. J Colloid Interface Sci 208: 23-33. doi: 10.1006/jcis.1998.5717
![]() |
[46] |
Chia TWR, Goulter RM, McMeekin T, et al. (2009) Attachment of different Salmonella serovars to materials commonly used in a poultry processing plant. Food Microbiol 26: 853-859. doi: 10.1016/j.fm.2009.05.012
![]() |
[47] |
Howell D, Behrends B (2006) A review of surface roughness in antifouling coatings illustrating the importance of cutoff length. Biofouling 22: 401-410. doi: 10.1080/08927010601035738
![]() |
[48] |
Scardino A J, Harvey E, De Nys R (2006) Testing attachment point theory: diatom attachment on microtextured polyimide biomimics. Biofouling 22: 55-60. doi: 10.1080/08927010500506094
![]() |
[49] | Guðbjörnsdóttir B, Einarsson H, Thorkelsson G (2005) Microbial adhesion to processing lines for fish fillets and cooked shrimp: influence of stainless steel surface finish and presence of gram-negative bacteria on the attachment of Listeria monocytogenes. Food Technol Biotech 43: 55-61. |
[50] |
Li B, Logan BE (2004). Bacterial adhesion to glass and metal-oxide surfaces. Colloids Surface B 36: 81-90. doi: 10.1016/j.colsurfb.2004.05.006
![]() |
[51] |
Kristich CJ, Li YH, Cvitkovitch, DG, et al. (2004) Esp-independent biofilm formation by Enterococcus faecalis. J Bacteriol 186: 154-163. doi: 10.1128/JB.186.1.154-163.2004
![]() |
[52] |
Stoodley P, Cargo R, Rupp CJ. (2002) Biofilm material properties as related to shear-induced deformation and detachment phenomena. J Ind Microbiol Biotechnol 29: 361-367. doi: 10.1038/sj.jim.7000282
![]() |
[53] | Liu YJ, Tay JH (2002) Metabolic response of biofilm to shear stress in fixed-film culture. J Appl Microbiol 90: 337-342. |
[54] | Ohashi A, Harada H (2004) Adhesion strength of biofilm developed in an attached-growth reactor. Water Sci Technol 29: 281-288. |
[55] |
Chen MJ, Zhang Z, Bott TR (1998) Direct measurement of the adhesive strength of biofilms in pipes by micromanipulation. Biotechnol Tech 12: 875-880. doi: 10.1023/A:1008805326385
![]() |
[56] |
Morikawa M (2006) Beneficial biofilm formation by industrial bacteria Bacillus subtilis and related species. J Biosci Bioeng 101: 1-8. doi: 10.1263/jbb.101.1
![]() |
[57] | Singh P, Cameotra SS (2004) Enhancement of metal bioremediation by use of microbial surfactants. Biochem Biophys Res Commun 317: 291-297. |
[58] | Quereshi FM. (2005) Genetic manipulation of genes for environmental bioremediation and construction of strains with multiple environmental bioremediation properties. Karachi, Pakistan: University of Karachi. |
[59] |
Salah KA, Sheleh G, Levanon D, et al. (1996) Microbial degradation of aromatic and polyaromatic toxic compounds adsorbed on powdered activated carbon. J Biotechnol 51: 265-272. doi: 10.1016/S0168-1656(96)01605-7
![]() |
[60] |
Nishijima W, Speital G (2004) Fate of biodegradable dissolved organic carbon produced by ozonation on biological activated carbon. Chemosphere 56: 113-119. doi: 10.1016/j.chemosphere.2004.03.009
![]() |
[61] |
Scholz M, Martin R (1997) Ecological equilibrium on biological active carbon. Water Res 31: 2959-2968. doi: 10.1016/S0043-1354(97)00155-3
![]() |
[62] | Takeuchi Y, Mochidzuki K, Matsunobu N, et al. (1997) Removal of organic substances from water by ozone treatment followed by biological active carbon treatment. Water Sci Technol 35: 171-178. |
[63] |
Zhang S, Huck P (1996) Parameter estimation for biofilm processes in biological water treatment. Water Res 30: 456-464. doi: 10.1016/0043-1354(95)00162-X
![]() |
[64] |
Rabaey K, Clauwaert P, Aelterman P, et al. (2005) Tubular microbial fuel cells for efficient electricity generation. Environ Sci Technol 39: 8077-8082. doi: 10.1021/es050986i
![]() |
[65] |
Upadhyayula VKK, Gadhamshetty V (2010) Appreciating the role of carbon nanotubes composites in preventing biofouling and promoting biofilms on material surfaces in environmental engineering: A review. Biotechnol Adv 28: 802-816. doi: 10.1016/j.biotechadv.2010.06.006
![]() |
[66] |
Kriegel D (2014) Advances in biofilm control for food and beverage industry using organo-silane technology: A review. Food Control 40: 32-40. doi: 10.1016/j.foodcont.2013.11.014
![]() |
[67] | Mittal KL (2009) Silanes coupling agents in Silanes and other coupling agents. Netherland: Koninklijke Brill NV, 1-176. |
[68] | Van Ooij WJ, Child T (1998) Protecting metals with silane coupling agents. Chemtech 28: 26-35. |
[69] |
Subramanian PR, van Ooij WJ (1999) Silane based metal pretreatments as alternatives to chromating. Surface Eng 15: 168-172. doi: 10.1179/026708499101516407
![]() |
[70] |
Van Schaftinghen T, LePen C, Terryn H, et al. (2004) Investigation of the barrier properties of silanes on cold rolled steel. Electrochimica Acta 49: 2997-3004. doi: 10.1016/j.electacta.2004.01.059
![]() |
[71] | Materne T, de Buyl F, Witucki GL (2006) Organosilane technology in coating applications: Review and perspectives. USA: Dow Corning Corporation. |
[72] | Carré A, Birch W, Lacarriére V (2007) Glass substrate modified with organosilanes for DNA immobilization, In: Mittal, K.L., Ed., Silanes and other coupling agents, Netherlands: VSP Utrecht, 1-14. |
[73] |
Li N, Ho C (2008) Photolithographic patterning of organosilane monolayer for generating large area two-dimensional B lymphocyte arrays. Lab on a Chip 8: 2105-2112. doi: 10.1039/b810329a
![]() |
[74] |
Saal K, Tätte T, Tulp I, et al. (2006) Solegel films for DNA microarray applications. Mater Lett 60: 1833-1838. doi: 10.1016/j.matlet.2005.12.035
![]() |
[75] |
Seo JH, Shin D, Mukundan P, et al. (2012) Attachment of hydrogel microstructures and proteins to glass via thiol-terminated silanes. Colloids Surfaces B 98: 1-6. doi: 10.1016/j.colsurfb.2012.03.025
![]() |
[76] | Shriver-Lake LC, Charles PT, Taitt CR. (2008) Immobilization of biomolecules onto silica and silica-based surfaces for use in planar array biosensors. Methods Mol Biol 504: 419-440. |
[77] |
Yamaguchi M, Ikeda K, Suzuki M, et al. (2011) Cell patterning using a template of microstructured organosilane layer fabricated by vacuum ultraviolet light lithography. Langmuir 27: 12521-12532. doi: 10.1021/la202904g
![]() |
[78] |
Khramov AN, Balbyshev VN, Voevodin NN, et al. (2003) Nanostructured sol-gel derived conversion coatings based on epoxy- and amino-silanes. Prog Org Coat 47: 207-213. doi: 10.1016/S0300-9440(03)00140-1
![]() |
[79] |
White JS, Walker GM. (2011) Influence of cell surface characteristics on adhesion of Saccharomyces cerevisiae to the biomaterial hydroxylapatite. Antonie van Leeuwenhoek Int J Gen Mol Microbiol 99: 201-209. doi: 10.1007/s10482-010-9477-6
![]() |
[80] |
Bekers M, Ventina E, Karsakevich A, et al. (1999) Attachment of yeast to modified stainless steel wire spheres growth of cells and ethanol production. Process Biochem 35: 523-530. doi: 10.1016/S0032-9592(99)00100-4
![]() |
[81] |
Karsakevich A, Ventina E, Vina I, et al. (1998) The effect of chemical treatment of stainless steel wire surface on Zymomonas mobilis cell attachment and product synthesis. Acta Biotechnol 18: 255-265. doi: 10.1002/abio.370180310
![]() |
[82] | Friedrich J (2012) Plasma, In: Friedrich, J. Plasma chemistry of polymer surfaces advanced techniques for surface design, Ed., Weinheim, Germany: Wiley-VCH Verlag GmBH & Co. KGaA, 35-53. |
[83] |
Goddard JM, Hotchkiss JH (2007) Polymer surface modification for the attachment of bioactive compounds. Prog Polymer Sci 32: 698-725. doi: 10.1016/j.progpolymsci.2007.04.002
![]() |
[84] |
Cha S, Park YS (2014) Plasma in dentistry. Clin Plasma Med 2: 4-10. doi: 10.1016/j.cpme.2014.04.002
![]() |
[85] |
Xiong Z, Cao Y, Lu X, et al. (2011) Plasmas in tooth root canal. IEEE Trans Plasma Sci 39: 2968-2969. doi: 10.1109/TPS.2011.2157533
![]() |
[86] |
Okajima K, Ohta K, Sudoh M (2005) Capacitance behaviour of activated carbon fibers with oxygen-plasma treatment. Electrochim Acta 50: 2227-2231. doi: 10.1016/j.electacta.2004.10.005
![]() |
[87] |
Díaz-Benito B, Velasco F (2013) Atmospheric plasma torch treatment of aluminium: Improving wettability with silanes. Appl Surface Sci 287: 263-269. doi: 10.1016/j.apsusc.2013.09.138
![]() |
[88] |
Kamgang JO, Naitali M, Herry JM, et al. (2009) Increase in the hydrophilicity and Lewis acid-base properties of solid surfaces achieved by electric gliding discharge in humid air: Effects on bacterial adherence. Plasma Sci Technol 11: 187-193. doi: 10.1088/1009-0630/11/2/11
![]() |
[89] |
Flexer V, Marque M, Donose BC, et al. (2013) Plasma treatment of electrodes significantly enhances the development of anodic electrochemically active biofilms. Electrochim Acta 108: 566-574. doi: 10.1016/j.electacta.2013.06.145
![]() |
[90] | He YR, Xiao X, Li WW, et al. (2012) Enhanced electricity production from microbial fuel cells with plasma-modified carbon paper anode. Phys Chem Chem Phys 14: 9966-9971. |
[91] |
Ploux L, Beckendorff S, Nardin M, et al. (2007) Quantitative and morphological analysis of biofilm formation on self-assembled monolayers. Colloid Surface B 57:174-181. doi: 10.1016/j.colsurfb.2007.01.018
![]() |
[92] |
Lijima S (1991) Helical microtubules of graphitic carbon. Nature 354: 56-58. doi: 10.1038/354056a0
![]() |
[93] |
Baughman RH, Zakhidov AA, de Heer WA. (2002) Carbon nanotubes—the route toward applications. Science 297: 787-792. doi: 10.1126/science.1060928
![]() |
[94] | Li YH, Di ZC, Luan ZK, et al. (2004) Removal of heavy metals from aqueous solution by carbon nanotubes: adsorption equilibrium and kinetics. J Environ Sci (China) 16: 208-211. |
[95] |
Yan XM, Shi BY, Lu JJ, et al. (2008) Adsorption and desorption of atrazine on carbon nanotubes. J Colloid Interface Sci 321: 30-38. doi: 10.1016/j.jcis.2008.01.047
![]() |
[96] |
Gotovac S, Yang CM, Hattori Y, et al. (2007) Adsorption of poly aromatic hydrocarbons on single walled carbon nanotubes of different functionalities and diameters. J Colloid Interface Sci 314: 18-24. doi: 10.1016/j.jcis.2007.04.080
![]() |
[97] |
Lu C, Chung YL, Chang KF (2005) Adsorption of trihalomethanes from water with carbon nanotubes. Water Res 39: 1183-1189. doi: 10.1016/j.watres.2004.12.033
![]() |
[98] | Pan B, Lin D, Mashayekhi H, et al. (2008) Adsorption and hysteresis of bisphenol A and 17-α-ethinyl estradiol on carbon nanomaterials. Environ Sci Technol 2008: 15. |
[99] |
Upadhyayula VKK, Deng S, Mitchell MC, et al. (2009) Application of carbon nanotube technology for removal of contaminants in drinking water: A review. Sci Total Environ 408: 1-13. doi: 10.1016/j.scitotenv.2009.09.027
![]() |
[100] |
Hyung H, Kim JH (2008) Natural organic matter (NOM) adsorption to multi walled carbon nanotubes: effect on NOM characteristics and water quality parameters. Environ Sci Technol 42: 4416-4421. doi: 10.1021/es702916h
![]() |
[101] |
Yan H, Gong A, He H, et al. (2006) Adsorption of microcystins by carbon nanotubes. Chemosphere 62: 142-148. doi: 10.1016/j.chemosphere.2005.03.075
![]() |
[102] |
Corry B (2008) Designing carbon nanotube membrane for efficient water desalination. J Phys Chem B 112: 1427-1434. doi: 10.1021/jp709845u
![]() |
[103] |
Raval HD, Gohil JM (2009) Carbon nanotube membrane for water desalination. Int J Nucl Desal 3: 360-368. doi: 10.1504/IJND.2009.028863
![]() |
[104] |
Huang S, Maynor B, Cai X, et al. (2003) Ultralong well-aligned single-walled carbon nanotube architecture on surfaces. Adv Mater 15: 1651-1655. doi: 10.1002/adma.200305203
![]() |
[105] |
Agnihotri S, Mota JPB, Rostam-Abadi M, et al. (2005) Structural characterization of single walled carbon nanotube bundles by experiment and molecular simulation. Langmuir 21: 896-904. doi: 10.1021/la047662c
![]() |
[106] |
Benny TH, Bandosz TJ, Wong SS (2008) Effect of ozonolysis on the pore structure, surface chemistry, and bundling of single walled carbon nanotubes. J Colloid Interface Sci 317: 375-382. doi: 10.1016/j.jcis.2007.09.064
![]() |
[107] |
Chen Y, Liu C, Li F, et al. (2006) Pore structures of multi walled carbon nanotubes activated by air, CO2 and KOH. J Porous Mater 13: 141-146. doi: 10.1007/s10934-006-7017-6
![]() |
[108] |
Liao Q, Sun J, Gao L (2008) Adsorption of chlorophenols by multi walled carbon nanotubes treated with HNO3 and NH3. Carbon 46: 544-561. doi: 10.1016/j.carbon.2007.12.005
![]() |
[109] |
Mauter SM, Elimelech M (2008) Environmental applications of carbon based nanomaterials. Environ Sci Technol 42: 5843-5859. doi: 10.1021/es8006904
![]() |
[110] |
Niu JJ, Wang JN, Jiang Y, et al. (2007) An approach to carbon nanotubes with high surface area and large pore volume. Microporous Mesoporous Mater 100: 1-5. doi: 10.1016/j.micromeso.2006.10.009
![]() |
[111] |
Deng S, Upadhyayula VKK, Smith GB, et al. (2008) Adsorption equilibrium and kinetics of microorganisms on single walled carbon nanotubes. IEEE Sens 8: 954-962. doi: 10.1109/JSEN.2008.923929
![]() |
[112] |
Malhotra BD, Chaubey A, Singh SP (2006) Prospects of conducting polymers in biosensors. Anal Chim Acta 578: 59-74. doi: 10.1016/j.aca.2006.04.055
![]() |
[113] |
Zou Y, Xiang C, Yang L, et al. (2008) A mediator less microbial fuel cell using polypyrrole coated carbon nanotubes composite as anode material. Int J Hydrogen Energ 33: 4856-4862. doi: 10.1016/j.ijhydene.2008.06.061
![]() |
[114] |
Sharma T, Mohana Reddy AL, Chandra TS, et al. (2008) Development of carbon nanotubes and nanofluids based microbial fuel cell. Int J Hydrogen Energ 33: 6749-6754. doi: 10.1016/j.ijhydene.2008.05.112
![]() |
[115] |
Odaci D, Timur S, Telefoncu A (2009) A microbial biosensor based on bacterial cells immobilized on chitosan matrix. Bioelectrochemistry 75: 77-82. doi: 10.1016/j.bioelechem.2009.01.002
![]() |
[116] |
Qiao Y, Li CM, Bao SJ, et al. (2007) Carbon nanotube/polyaniline composite as anode material for microbial fuel cells. J Power Sources 170: 79-84. doi: 10.1016/j.jpowsour.2007.03.048
![]() |
[117] | Nambiar S, Togo CA, Limson JL (2009) Application of multi-walled carbon nanotubes to enhance anodic performance of an Enterobacter cloacae based fuel cell. Afr J Biotechnol 8: 6927-6932. |
[118] | Chen J, Yu Z, Sun J, et al. (2008) Preparation of biofilm electrode with Xanthomonas sp. and carbon nanotubes and the application to rapid biochemical oxygen demand analysis in high salt condition. Water Environ Res 80: 699-702. |
[119] |
Timur S, Anik U, Odaci D, et al. (2007) Development of microbial biosensor based on carbon nanotube (CNT) modified electrodes. Electrochem Commun 9: 1810-1815. doi: 10.1016/j.elecom.2007.04.012
![]() |
[120] | Kanepalli S, Donna FE (2006) Enhancing the remediation of trichloroethane (TCE) using double-walled carbon nanotubes (DWNT): United States Geological Survey. |
[121] | Pumera M (2007) Carbon nanotubes contain residual metal catalyst nanoparticles even after washing with nitric acid at elevated temperatures because these metal nanoparticles are sheathed by several graphene sheets. Langmuir 23: 6453-6458. |
[122] |
Logan BE, Hamelers B, Rozendal R, et al. (2006) Microbial fuel cells: methodology and technology. Environ Sci Technol 40: 5181-5192. doi: 10.1021/es0605016
![]() |
[123] |
Tang X, Guo K, Li H, et al. (2011) Electrochemical treatment of graphite to enhance electron transfer from bacteria to electrodes. Bioresour Technol 102: 3558-3560. doi: 10.1016/j.biortech.2010.09.022
![]() |
[124] |
Cercado-Quezada B, Delia ML, Bergel A (2011) Electrochemical microstructuring of graphite felt electrodes for accelerated formation of electroactive biofilms on microbial anodes. Electrochem Commun 13: 440-443. doi: 10.1016/j.elecom.2011.02.015
![]() |
[125] |
Cheng S, Logan BE (2007) Ammonia treatment of carbon cloth anodes to enhance power generation of microbial fuel cells. Electrochem Commun 9: 492-496. doi: 10.1016/j.elecom.2006.10.023
![]() |
[126] |
Park DH, Zeikus JG (2003) Improved fuel cell and electrode designs for producing electricity from microbial degradation. Biotechnol Bioeng 81: 348-355. doi: 10.1002/bit.10501
![]() |
[127] |
Zhou M, Chi M, Wang H, et al. (2012) A new practical method to improve the performance of microbial fuel cells. Biochem Eng J 60: 151-155. doi: 10.1016/j.bej.2011.10.014
![]() |
[128] |
Jin T, Luo J, Yang J, et al. (2012) Coupling of anodic and cathodic modification for increased power generation in microbial fuel cells. J Power Sources 219: 358-363. doi: 10.1016/j.jpowsour.2012.07.066
![]() |
[129] |
Popov AL, Kim JR, Dinsdale RM, et al. (2012) The effect of physic-chemically immobilized methylene blue and neutral red on the anode of microbial fuel cell. Biotechnol Bioprocess Eng 17: 361-370. doi: 10.1007/s12257-011-0493-9
![]() |
[130] |
Guo K, Chen X, Freguia BC, et al. (2013) Spontaneous modification of carbon surface with neutral red from its diazonium salts for bioelectrochemical systems. Biosens Bioelectron 47: 184-189. doi: 10.1016/j.bios.2013.02.051
![]() |
1. | Tiancai Liao, Hengguo Yu, Chuanjun Dai, Min Zhao, Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics, 2019, 2019, 1076-2787, 1, 10.1155/2019/8205696 | |
2. | Gauri Agrawal, Alok Kumar Agrawal, Arvind Kumar Misra, Effects of human population and forestry trees on the hydrologic cycle: A modeling-based study, 2025, 22, 1551-0018, 2072, 10.3934/mbe.2025076 |
$ K\geq \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in(0, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ K < \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in (s_{0}, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ s \in (0, s_{0}] $ | $ E_{2} $ is G.A.S. |
Parameters | Definition | Values | Reference |
$ \beta_{0} $ | Zooplankton grazing rate on $ C $ | 0.15 | - |
$ \delta $ | Sinking rate of $ C $ | 0.2 | - |
$ r $ | Intrinsic growth rate of $ P $ | 0.7$ \rm \; d^{-1} $ | [9] |
$ \beta_{1} $ | Zooplankton grazing rate on $ P $ | 0.4$ \rm \; \mu gcl^{-1}d^{-1} $ | - |
$ \beta_{2} $ | Zooplankton grazing efficiency on $ P $ | 0.25 | - |
$ \beta_{max} $ | Maximum grazing rate on $ P $ | 6$ \rm \; \mu gcl^{-1}d^{-1} $ | [9] |
$ \beta_{3} $ | Zooplankton grazing efficiency on $ C $ | 0.05 | - |
$ \gamma $ | Zooplankton mortality | 0.3$ \rm \; d^{-1} $ | [28] |
$ s $ | Constant input rate of C | 1 | - |
$ m $ | Zooplankton competitive mortality | 0.1 | - |
$ K $ | Carrying capacity | 100$ \rm \; \mu gcl^{-1} $ | [28] |
$ K\geq \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in(0, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ K < \frac{d}{\beta_{2}} $ | $ s\in [s_{1}, +\infty) $ | $ E_{3} $ is G.A.S. |
$ s\in (s_{0}, s_{1}) $ | $ E_{4} $ is G.A.S. | |
$ s \in (0, s_{0}] $ | $ E_{2} $ is G.A.S. |
Parameters | Definition | Values | Reference |
$ \beta_{0} $ | Zooplankton grazing rate on $ C $ | 0.15 | - |
$ \delta $ | Sinking rate of $ C $ | 0.2 | - |
$ r $ | Intrinsic growth rate of $ P $ | 0.7$ \rm \; d^{-1} $ | [9] |
$ \beta_{1} $ | Zooplankton grazing rate on $ P $ | 0.4$ \rm \; \mu gcl^{-1}d^{-1} $ | - |
$ \beta_{2} $ | Zooplankton grazing efficiency on $ P $ | 0.25 | - |
$ \beta_{max} $ | Maximum grazing rate on $ P $ | 6$ \rm \; \mu gcl^{-1}d^{-1} $ | [9] |
$ \beta_{3} $ | Zooplankton grazing efficiency on $ C $ | 0.05 | - |
$ \gamma $ | Zooplankton mortality | 0.3$ \rm \; d^{-1} $ | [28] |
$ s $ | Constant input rate of C | 1 | - |
$ m $ | Zooplankton competitive mortality | 0.1 | - |
$ K $ | Carrying capacity | 100$ \rm \; \mu gcl^{-1} $ | [28] |