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Control effect of periodic variation on the growth of harmful algal bloom causative species
Commun Nonlinear Sci Numer Simulat 54 (2018) 185–201
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Control effect of periodic variation on the growth of harmful algal bloom causative species L. Zhang a,b,∗, S.T. Liu a, T. Liu b, C. Yu c,d, Z. Hu e,f a
College of Control Science and Engineering, Shandong University, Jinan 250061, PR China Business School, Shandong University of Political Science and Law, Jinan, 250014, PR China c Mind and Brain Theme, South Australian Health and Medical Research Institute, North Terrace, Adelaide, SA 5000, Australia d School of Medicine, Flinders University, Bedford Park, SA 5042, Australia e CAS Key Laboratory of Marine Ecology and Environmental Sciences, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, 266071, PR China f Laboratory for Marine Ecology and Environmental Science, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266071, PR China b
a r t i c l e
i n f o
Article history: Received 30 December 2016 Revised 21 April 2017 Accepted 25 May 2017 Available online 30 May 2017 Keywords: Periodic variation Causative species Control Harmful algae blooms
a b s t r a c t Blue-green algae and Dinoflagellate etc. are common types of phytoplankton as causative species which cause the harmful algal blooms (HABs). The growth process of causative species is complex according to the variation of the environmental disturbance such as the periodic factor in reality and recent studies have not revealed the secret of the growth complexity yet. Based on the empirical and theoretical results of the growth of causative species, a nonlinear controlled system with periodic factor was obtained and the different effects of the periodic factor on the control of the cell density and the growth rate of causative species were studied by three theorems using the norm theory and finite difference method. Simulations and experimental data were also used to assess the effectiveness of the controlled results. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Harmful algal blooms (HABs) are mainly produced by rapid growth of algal species named as the causative species with certain conditions and have severe impacts on public health and coastal ecosystems [1,2]. One of the most famous reproduction mode of the causative species for the unicellular algae [3,4] is the well-known fact that one cell divided into two cells by asexual reproduction. In addition, the rate of the growth is complicatedly and nonlinearly affected by different environmental factors such as the ratios of nitrogen to phosphorus [5], fluorescent and red light environments [6]. So the inherent regularity between the growth process of the causative species and the complex environmental factors still remains a mystery. Nowadays, many works focus on the control of growth process for these causative species in HABs experimentally. For instance, the higher concentration of kanamycin was found to inhibit the growth velocity of unicellular green algae Chlorella vulgaris [7]. The effect of assemblage age on the uptake rate of algal species was studied for the incorporation of cells and
∗
Corresponding author at: College of Control Science and Engineering, Shandong University, Jinan 250061, PR China. E-mail addresses: [email protected], [email protected] (L. Zhang), [email protected] (S.T. Liu), [email protected] (T. Liu), [email protected] (C. Yu), [email protected] (Z. Hu). http://dx.doi.org/10.1016/j.cnsns.2017.05.023 1007-5704/© 2017 Elsevier B.V. All rights reserved.
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eliminating differences in substrate roughness [8]. When three proteins named GsSPT1 (G. sulphuraria sugar and polyol transproter 1), GsSPT2 and GsSPT4 were studied, results were shown that the deletion of N-terminus of the protein in SPT1 affected the maximal transport velocity and released the dependency of the pH on sugar uptake, while the uptake by other two proteins was not active for the pH-dependence [9]. Moreover, according to the fact that the growth process of the causative species in reality becomes much more complex than that of growth under laboratory, the growth process can be described mathematically by dynamical systems and many interesting and important works have been obtained. For example, the dynamical behavior of algal growth with impulsive disturbance [10], such as chaos [11] and stability of attractors [12,13], was studied in colony formation, in the interaction growth and in the allelopathic of phytoplankton. In the light of the experimental results of the growth interaction such as toxin or non-toxic algae, organic pollutants, nutrients and bacteria, the coupled dynamical models were used to simulate the interactions [14–18]. The effect of allelopathic competition between two algal species on the growth rate of algae was also reported by coupled nonlinear models [19–21]. However, there exist few results about the effect of the environmental factor on the growth process of the causative species from the point view of control theory in term of the experimental results, data collection in field and the current mathematical models of causative species. In this paper, we concentrate on the control effect of the periodic environmental factor on the cell density and the growth rate of the causative species, in which the periodic environmental factor include variation of seasons, light intensity, temperature and some inner cycle clock, etc. [22,23]. Our work can play vital roles for revealing the secret of the HABs. The paper is organized as follows. In Section 2, the nonlinear controlled system for the causative species growth is established and the stable condition of the system is presented. Three theorems about the relationships between the periodic factor and the cell density and the growth rate of the causative species are studied in Section 3, respectively. Simulations and experimental data in Section 4 and in Section 5 also illustrate the control effect of the periodic factor on the cell density and the growth rate of causative species theoretically and experimentally. Conclusions and discussions are given in Section 6. 2. Problem assumption This section revolves around the conditions like exist and unique solution and stable theory for the nonlinear reactiondiffusion system with initial and boundary conditions based on the experimental results of growth of the causative species. In general, the growth process of the causative species dynamically satisfies the following system:
∂ P (x, y, z, t ) = f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )), ∂t
(2.1)
where P(x, y, z, t) is the cell density of the causative species,
P ∈ Hilbert space H ([ξ , η], Rn ),
(x, y, z, t ) ∈ R3 × (0, +∞ ), f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )) is the factor including the birth rate α (x, y, z, t), the death rate β (x, y, z, t) and the controller as u(x, y, z, t), like gene modulating factors and the strength of sunlight, etc. Based on the models in Ref.[24,25] and the fact that the main reproductive mode of single cell species in the causative species is binary fission and the growth velocity varies with different spatial location and the different environmental factors, the dynamical system (2.1) for the growth process of the causative species can be represented as follows:
∂ P (x, y, z, t ) = f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )) ∂t P 2 (x, y, z, t ) = (α + u(t ) − β (t ))P (x, y, z, t ) − (α + u(t )) + DP (x, y, z, t ), PA
(2.2)
where
(α + u(t ) − β (t ))P (x, y, z, t ) − (α + u(t ))
P 2 (x, y, z, t ) PA
means the binary fission term,
∂ 2 P (x, y, z, t ) ∂ 2 P (x, y, z, t ) ∂ 2 P (x, y, z, t ) DP (x, y, z, t ) = D + + ∂ x2 ∂ y2 ∂ z2
stands for the spatial diffusion term. Since the velocity of the cell division of the causative species is affected by different periodic factors like strength of sunlight, temperature and inner cycle clock, etc., we introduce the controller as
L. Zhang et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 185–201
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u = Asin(ωt + θ ), where A is the strength of the periodic factors. And ω and θ represent the amplitude and phase in periodic environmental factors like strength of sunlight, temperature and inner cycle clock, etc., respectively. D indicates the diffusion coefficient. In order to find the control effect of the controller u on the growth process of the causative species easily, we assume the following initial and boundary conditions for system (2.2):
P (x, y, z, 0 ) = (φ (x, y, z ), 0, 0, · · · , 0 ), P (0, y, z, t ) = P (x, 0, z, t ) = P (x, y, 0, t ) = (0, 0, · · · , 0, a0 ), P (l, y, z, t ) = P (x, l, z, t ) = P (x, y, l, t ) = (1, 1, · · · , 1, a1 ),
where φ (x, y, z) is any continuous function, a0 and a1 are constant parameters and a1 > a0 ≥ 0. The stable condition will be discussed for the system (2.2) due to the exponential stability in this paragraph. On the basis of the condition of (2.2) and the reason is clear that the function
f1 (P, α , β , u ) = (α + u − β )P − (α + u ) PPA
2
satisfies
f1 (P0 , α , β , u ) = (0, 0, · · · , 0 )T , where P0 = (0, 0, · · · , 0 )T is the initial growth density. Then we assume f1 (P, α , β , u) is locally Lipschitz continuous, i.e. there exist positive real numbers K, K0 for any P1 , P2 ∈ H satisfying the following condition:
max{ P1 , P2
} ≤ K.
It means that
f1 (P1 , α , β , u ) − f1 (P2 , α , β , u ) ≤ K0 P1 − P2 , where P1 , P2 ∈ Hilbert space H ([ξ , η], Rn ). In addition, the parameters of (2.2) satisfy the following condition:
(α + u ) + P ) K0 ≥ (α + u − β ) − . φ
(2.3)
Because the norm theory will be used in the proof of the following theorems in next section, the following lemma is presented: Lemma 1. For different norm definitions:
P 1 =
n n
|Pi, j |,
i=1 j=1
and
P 2 =
n n
|Pi, j |2 ,
i=1 j=1
the two norms have the following inequality relationship:
P 1 ≤ nP 2 . 3. Main theorem Without loss of generality and in order to calculate easily, we omit the panel effect and consider only the water depth so that the system (2.2) becomes the following form
∂ P (z, t ) = f (P (z, t ), α (z, t ), β (z, t ), u(z, t )) ∂t = (α (z, t ) + u(z, t )) − β (z, t ))P (z, t ) − (α (z, t ) + u(z, t ))
P 2 (z, t ) ∂ 2 P (z, t ) +D , PA ∂ z2
(3.1)
with the initial and boundary conditions shown below as:
P (z, 0 ) = (0, 0, · · · , 0, φ (z )), P ( 0, t ) = ( 0, 0, · · · , 0, a0 ), P ( 1, t ) = ( 1, 1, · · · , 0, a1 ).
As the complexity of the form of f in system (3.1), it is difficult to solve the solution analytically. So we next use the finite difference method for the system of (3.1), which is the famous difference method for PDEs [26,27]. Then system (3.1) is discretized as follows:
Pk, j − Pk−1, j = rPk−1, j+1 − 2rPk−1, j + rPk−1, j−1 + 2(α (k ) + u − β )τ Pk, j −
2(α (k ) + u(k ))τ 2 Pk, j , PA
(3.2)
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where r = 2Dτ /h2 is the mesh ratio of the discretization process, h stands for the space step and τ means the time step. In order to compute easily, we assume τ = 1/2, D = h2 . Then as the numerical process for system (3.2), we have
α+u PA
α+u ,
PA
α+u PA
2 P22 + (1 − α − u + β )P2,2 + P1,2 − P1,3 − P1,1 = 0, 2 P32 + (1 − α − u + β )P3,2 + P2,2 − P2,3 − P2,1 = 0,
2 Pn,n −1 + (1 − α − u + β )Pn,n−1 + Pn−1,n−1 − Pn−1,n − Pn−1,n−2 = 0.
When we define the following symbols as
⎛
−1 ⎜0 M=⎜ . ⎝ .. 0
··· , ··· , .. . 0
0 −1 .. . ··· ,
0 0 .. . −1
(1 − α − u + β )
1 0 .. . 0
1 .. . 0
··· ··· , .. . 1 (1 − α − u + β )
0 0 .. . 0
−1 0 .. . ··· ,
0 −1 .. . 0
··· ··· , .. . −1
⎞
0 0⎟ .. ⎟ ⎠, . 0
P = [P1,1 , P2,1 , · · · , Pn,1 , P1,2 , P2,2 , · · · , Pn,2 , . . . , P1,n , P2,n , · · · , Pn,n ]T , 2 2 2 2 2 2 2 T F = [P22,2 , P32,2 , · · · , Pn, 2 , P2,3 , P3,3 , · · · , Pn,3 , · · · , P2,n−1 , P3,n−1 , Pn,n−1 ] ,
the numerical process of the system (3.2) can be substituted as follows
MP =
α+u PA
F.
(3.3)
Next we will discuss the different effects of the controller u on the growth process of the causative species. Theorem 1. The norm of the growth density ||P||2 will be controlled by the controller u according to the following relationship:
0 ≤ ||P ||2
PA n[4 − α − u + β ] PA ≤ + 2 (α + u )
n2 [4 − α − u + β ]2 + 4c (α + u )2 /PA2 2 (α + u )
where
c=
n
|Pi,1 |2 +
n
i=1
|Pi,n |2 +
i=1
n
|P1, j |2 .
j=1
Proof. For (3.3), we choose the norms of
P 1 =
n n
|Pi j |,
i=1 j=1
P 2 =
n n
|Pi j |2 ,
i=1 j=1
and
M1 = max1≤i≤n
n
|Mi j |.
j=1
In accordance with Lemma 1 and the system (3.3), we obtain
M P 1 ≤ M 1 P 1 , MP1 =
α+u PA
F 1 ,
and
P 1 ≤ nP 2 . Later we have
M1 P 1 ≥ MP 1 =
α+u PA
F 1 =
n n−1 α+u
PA
i=2 j=2
|Pi, j |2 ,
,
(3.4)
L. Zhang et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 185–201
which means that
M 1 P 1 + ≥
n α+u PA
n n−1 α+u PA
|Pi,1 | + 2
i=1
n
|Pi,n | + 2
i=1
|Pi, j |2 +
n
i=2 j=2
n
|P1, j |
2
j=1
|Pi,1 |2 +
i=1
n
189
|Pi,n |2 +
n
i=1
|P1, j |2 .
j=1
Subsequently we get
M 1 P 1 +
α+u PA
c≥
n n α+u
PA
|Pi, j |2 =
i=1 j=1
α+u PA
P22 .
(3.5)
According to Lemma 1 and the inequality of (3.5), we gain
α+u
n||M||1 ||P ||2 +
PA
c ≥ M 1
P 1 +
α+u PA
c≥
α+u PA
P22 ,
which can be written as the following form:
α+u PA
||P||22 − n||M||1 ||P||2 −
α+u PA
c ≤ 0.
(3.6)
Soon afterwards we have the inequality as the relationship of (3.6):
0 ≤ ||P ||2
PA n[4 − α − u + β ] PA ≤ + 2 (α + u )
n2 [4 − α − u + β ]2 + 4c (α + u )2 /PA2 2 (α + u )
,
(3.7)
where
c=
n
|Pi,1 |2 +
i=1
n i=1
|Pi,n |2 +
n
|P1, j |2 .
j=1
Remark 1. In the light of Theorem 1, it is clear that α , u and β play important role in the boundedness states of ||P||2 because of (3.7). Remark 2. Moreover, based on the boundedness effect of Theorem 1, we next further consider other control effect as the following two theorems and the following system. When we use the similar method in Ref. [28] for system (3.3) as follows:
Ci (k, j ) = 2Pk−i, j − Pk−i, j−i − Pk−i, j+i − Pk−i, j ,
(3.8)
and
Di (k, j ) =
P (1 − α − u + β ) 4α ( u ) (Ci (k, j ) − A ), 2 (α − u ) PA [(1 − α − u + β )2 − 2(1 − α − u + β )]
(3.9)
system (3.3) becomes a special example for the following system as i = 1:
Di (k, j ) = 1 − = 1−
PA2 [
PA2 [(1 − α − u + β )2 − 2(1 − α − u + β )]2 4 ( α + u )2 Di−1 (k, j )2 2 ( 4 ( α + u )2 ( 1 − α − u + β ) − 2 ( 1 − α − u + β )]
( α + u − β )2 − 1
4 = 1 − μDi−1 (k, j )2 .
Di−1 (k, j )2 (3.10)
On the basis of the similar theory like (3.10) in Refs.[28,29], the system (3.10) can display complex nonlinear behavior such as spatial chaotic behavior shown in Fig. 1. Moreover, the increasing effect of the periodic term on system (3.3) is obtained by the following two theorems. Theorem 2. The cell density P of the causative species in system (3.3) increases as time k increases when
⎧ ∂ D (k, j ) 0 > 0; ⎪ ∂k ⎪ ⎪ ⎪ ⎪ ∂ ( α ( k ) + u ( k )) ⎪ < 0; ⎨ ∂k
α ( k ) + u ( k ) > β ( k ); ⎪ ⎪ ⎪ ⎪ D0 (k, j )(α + u(k ) − β (k )) > 1; ⎪ ⎪ ⎩ (α (k ) + u(k )) ∂β∂(kk) > ∂ (α (k∂)+k u(k)) (β (k ) + 1 ).
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Fig. 1. The spatial chaotic behavior of Di (k) as α + u = 0.7e−t + 2.5, β = 0.4e−t , t ∈ (0, 1), N = 151.
Proof. According to the conditions (3.8) and (3.9), we have −α (k )−u(k )+β (k ))] Pk, j = −D0 (k, j ) PA [(1−α (k )−u(k )+4β((αk())k )+−2u((k1)) − 2
PA (1−α (k )−u(k )+β (k )) . 2(α (k )+u(k ))
(3.11)
So we get
∂ P (k, j ) ∂ D0 (k, j ) ∂δ ∂δ =− δ1 − D0 (k, j ) 1 − 2 ∂k ∂k ∂k ∂k owning to (3.11), where
δ1 =
PA (α (k ) + u(k )) 2 PA δ0 − δ0 , 4 2
δ2 =
Pa 1 − α (k ) − u(k ) + β (k ) , 2 α (k ) + u (k )
δ0 =
1 − α (k ) − u (k ) + β (k ) , α (k ) + u (k )
P ∂δ ∂δ1 PA ∂ (α (k ) + u(k )) 2 2PA ∂δ = δ0 + (α (k ) + u(k ))δ0 0 − A 0 , ∂k 4 ∂k 4 ∂k 2 ∂k
∂δ2 PA δ0 = , ∂k 2 ∂k ∂δ0 (− ∂ (α (k∂)+k u(k)) + = ∂k
∂β (k ) ∂ (α (k )+u(k )) ∂ k )(α (k ) + u (k )) − (1 − α (k ) − u (k ) + β (k )) ∂k .
(α (k ) + u(k ))2
The relationship (3.12) means that
PA ∂ P (k, j ) ∂ D0 (k, j ) PA ∂ D0 (k, j ) D0 (k, j )PA 2 ∂ (α (k ) + u(k )) = − (α (k ) + u(k ))δ02 + δ0 − δ0 ∂k 4 ∂k 2 ∂k 4 ∂k PA D0 (k, j ) PA D0 (k, j ) ∂ (α (k ) + u(k )) ∂β + (α (k ) + u(k ))δ0 − (α (k ) + u(k ))δ0 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k )) ∂k PA D0 (k, j ) ∂ (α (k ) + u(k )) + (α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))δ0 ∂k 2(α (k ) + u(k ))2 PA D0 (k, j ) ∂ (α (k ) + u(k )) PA D0 (k, j ) ∂β (k ) − + 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k )) ∂ k PA D0 (k, j ) ∂ (α (k ) + u(k )) PA ∂ (α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) + ∂k 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k ))2 PA ∂β (k ) PA ) ∂ (α (k ) + u(k )) − + (1 − α (k ) − u(k ) + β (k )) , 2(α (k ) + u(k )) ∂ k ∂k 2(α (k ) + u(k ))2
(3.12)
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191
and then
∂ P (k, j ) ∂ D0 (k, j ) PA (α (k ) + u(k )) 2 PA =− δ0 − δ0 ∂k ∂k 4 2 PA ∂δ0 PA ∂ (α (k ) + u(k )) 2 2PA ∂δ0 PA ∂δ0 −D0 (k, j ) δ0 + (α (k ) + u(k ))δ0 − − 4 ∂k 4 ∂k 2 ∂k 2 ∂k 2 PA (α (k ) + u(k )) 1 − α (k ) − u(k ) + β (k ) ∂ D0 (k, j ) PA 1 − α (k ) − u(k ) + β (k ) ∂ D0 (k, j ) =− + 4 (α (k ) + u(k )) ∂k 2 (α (k ) + u(k )) ∂k 2 PA 1 − α (k ) − u(k ) + β (k ) ∂ (α (k ) + u(k )) 2PA D0 (k, j ) +D0 (k, j ) − (α (k ) + u(k )) 4 (α (k ) + u(k )) ∂k 4 ×
1 − α (k ) − u (k ) + β (k ) (− α (k ) + u (k )
∂ (α (k )+u(k )) u(k )) + ∂β∂(kk ) )(α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) ∂ (α (k∂)+ ∂k k
(α (k ) + u(k ))2
∂ (α (k )+u(k )) u(k )) + ∂β∂(kk ) )(α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) ∂ (α (k∂)+ PA D0 (k, j ) (− ∂ k k + −
PA 2
2
(− ∂ (α (k∂)+k u(k)) +
(α (k ) + u(k ))2
∂β (k ) ∂ (α (k )+u(k )) ∂ k )(α (k ) + u (k )) − (1 − α (k ) − u (k ) + β (k )) ∂k .
(α (k ) + u(k ))2
j) At last, we obtain the following conditions for ∂ P∂(k, > 0, : k
⎧ ∂ D (k, j ) 0 > 0; ⎪ ⎪ ⎪ ∂k ⎪ ⎪ ∂ ( α ( k ) + u ( k )) ⎪ < 0; ⎪ ∂k ⎨ (α (k ) + u(k )) > β (k ); ⎪ ⎪ ⎪ ⎪ D0 (k, j )(α (k ) + u(k ) − β (k )) > 1; ⎪ ⎪ ⎪ ⎩ (α (k ) + u(k )) ∂β∂(kk) > ∂ (α (k∂)+k u(k)) (β (k ) + 1 ),
which can represent the increasing effect of the environmental factors on the cell density.
The growth rate defined as follows in Ref. [30]:
Rg =
l nPt1 − l nPt0 , t1 − t0
is generally assumed to be constant. So we consider the more reasonable condition that the growth rate varies with time and then assume the following generalized growth rate
Rg (t, z ) =
lnP (t1 , z ) − lnP (t0 , z ) . t1 − t0
Next theorem is about the control effect of the periodic factor on the growth rate in the discretized processas Rg (k, j ) = lnP (k, j ) − lnP (k − 1, j ). Theorem 3. The growth rate Rg (k, j) increases when the following conditions are satisfied:
⎧ 2 ∂ D(k, j ) ⎪ > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β (k ) ⎪ > − 12 ; ⎪ ⎪ ⎨ ∂k ∂ 2 (α (k )+u(k ))
u (k )) u (k )) 2 [(α (k )+u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ −β (k ) ∂ (α (k∂)+ ] k k
<− ∂ k2 [(α (k )+u(k ))(1−α (k )−u(k )+β (k ))]2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k ))) > 0; ⎪ ⎪ ⎪ ⎩ ∂ (α (k )+u(k )) (α (k ) + u(k )) ∂β . ∂ k > (β + 1 ) ∂k Proof. Due to the conditions (3.8) and (3.9), we have
Rg (k, j ) = lnP (k, j ) − lnP (k − 1, j ) = ln
P (k, j ) P ( k − 1, j )
;
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= ln
P (k, j ) − P (k − 1, j ) + P (k − 1, j ) P ( k − 1, j )
≈ ln
∂ P (k, j ) ∂k
P ( k − 1, j )
+1 .
(3.13)
Then according to (3.13), it is clear that
∂ Rg (k, i ) P (k, j ) ≈ j) ∂k P (k, j ) + ∂ P∂(k, k
∂ 2 P (k, j ) ∂ P (k, j ) 2 ∂ k2 P (k, j ) − ( ∂ k ) .
P (k, j )2
(3.14)
Because
∂ 2 P (k, j ) ∂ 2 D(k, j ) ∂ D(k, j ) ∂δ1 ∂ 2 δ1 ∂ 2 δ2 =− δ1 − 2 − D(k, j ) 2 − 2 2 2 ∂k ∂k ∂k ∂k ∂ k ∂ k ∂ 2 D(k, j ) PA (α (k ) + u(k )) 2 PA ∂ D(k, j ) PA δ02 ∂ (α (k ) + u(k )) PA (α (k ) + u(k )) ∂δ0 =− δ0 − δ0 − 2 + δ0 4 2 ∂k 4 ∂k 2 ∂k ∂ k2 2 PA ∂δ0 ∂δ0 ∂ (α (k ) + u(k )) PA 2 ∂ (α (k ) + u(k )) PA ∂δ0 2 − − D(k, j ) PA δ0 + δ0 + (α (k ) + u(k ))( ) 2 ∂k ∂k ∂k 4 2 ∂k ∂ k2 ∂δ 2 PA ∂δ02 PA ∂δ02 PA + (α (k ) + u(k ))δ0 02 − − 2 2 2 ∂k 2 ∂ k2 ∂k and after substituting it into (3.14), we get that
∂ Rg (k, j ) > 0, ∂k when the following conditions are satisfied:
⎧ ∂ 2 D(k, j ) > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β ⎪ 1 ⎪ ⎪ ∂k > − 2 ; ⎨ ∂ 2 δ0
< 0;
∂ k2 ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k )) > 0; ⎪ ⎪ ⎪ ⎩ 2 2 ∂ (α (k )+u(k )) 0 2 < 0, 2(α (k ) + u(k ))( ∂δ ∂k ) + δ ∂ k2
and
∂ 2 P (k, j ) ∂ P (k, j ) 2 P (k, j ) > ( ) , ∂k ∂ k2
(3.15)
which means that
⎧ 2 ∂ D(k, j ) ⎪ > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β ⎪ > − 12 ; ⎪ ⎪ ⎨ ∂k ∂ 2 (α (k )+u(k ))
∂ (α (k )+u(k )) u (k )) 2 [(α (k )+u(k )) ∂β −β ∂ (α (k∂)+ ] ∂k − ∂k k
<− ∂ k2 [(α (k )+u(k ))(1−α (k )−u(k )+β )]2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k )) > 0; ⎪ ⎪ ⎪ ⎩ ∂ (α (k )+u(k )) (α (k ) + u(k )) ∂β . ∂ k > (β + 1 ) ∂k
;
Remark 3. Theorems 2 and 3 show the conclusion that the cell density and the growth rate of the causative species can be affected by the functional form of the birth rate, the controller and the death rate, meaning that if the conditions of Theorems 2 and 3 have been satisfied, the cell density and the growth rate will monotone increase. Furthermore, depending on the fact that the real growth process shows different characteristics for different algaes under different environments, we can conclude that our results play vital role in further understanding the complex characteristics of the growth process of the causative species.
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Fig. 2. The variation of δ as the periodic factor changes.
4. Simulation examples Exponent growth is the famous form in the causative species growth phenomena so that we take the following form for the birth rate and the death rate as examples: t
t
α + u = A1 e− A2 + u = A1 e− A2 + A3 sinωt, t
β = A4 e− A5 , where A1 > 0, A2 > 0, A4 > 0, A5 > 0, A3 and ω are constant. In the light of Theorem 1, it is clear that the functional forms of α , u and β play important role in the variation of ||P||2 , where
δ=
n (4 − α − u + β ) , α+u
and the variation of δ is shown in Fig. 2. Besides this, the behavior of system (3.10) shows complex behavior as periodic factor changes illustrated in Fig. 3(a), (c) and (e) and the chaotic characteristic which is calculated by Lyapunov exponent shown in Fig. 3(b), (d) and (f). In addition, the cell density P and growth rate Rg can be controlled as increasing monotone by the variation of the periodic factor because of the relationships in Theorems 2 and 3. It is easy to obtain in terms of Theorem 2 that when we choose the time t ∈ (0.1770, π /10), the increase of P can be controlled by the following condition:
∂ (α + u ) = −0.7e−t + cos5t < 0, ∂t
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Fig. 3. The variation of D(k, j) as the periodic factor changes.
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Fig. 4. The variation of D(k, j) and its Lyapunov exponent as the periodic factor changes.
where
cos5t > 0, sin5t > 0,
α + u = 0.7e−t − 0.2sin5t, β = 0.4sin5t, (α + u )
∂β ∂ (α + u ) − (β + 1 ) = (0.7e−t − 0.2sin5t )(−0.4e−t ) − (−0.7e−t + cos5t )(0.4e−t + 1 ) ∂t ∂t = 0.08e−t sin5t + 0.7e−t + 0.4e−t cos5t + cos5t > 0.
For example, as the periodic factor increases, D(k, j) decreases shown in Figs. 3(a)–(c) and 4. Meanwhile, the chaotic characteristic calculated by Lyapunov exponent of D(k, j) increases illustrated by Fig. 3(b), (d) and (f). In order to simulate the control effect of the periodic factor on the growth rate Rg , we choose the time t ∈ (π /10, π /5). After that, the growth rate Rg increases as the following conditions are satisfied:
cos5t < 0, sin5t > 0,
α1 + u = 0.7e−t + 2.1sin5t, α2 + u = 0.7e−t + 2.3sin5t, β = 0.4sin5t, ∂ (α1 + u ) = −0.7e−t + 10.5cos5t, ∂t ∂ 2 (α1 + u ) = 0.7e−t − 52.5sin5t < 0, ∂t2
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Fig. 5. The monotone increasing effect of the periodic factor u on D(k, j).
(α1 + u )
∂β ∂ (α1 + u ) − (β + 1 ) = (0.7e−t + 2.1sin5t )(−0.4e−t ) − (−0.7e−t + 10.5cos5t )(0.4e−t + 1 ) ∂t ∂t = −0.84e−t sin5t + 0.7e−t − 4.2e−t cos5t − 10.5cos5t > 0,
(α2 + u )
∂β ∂ (α2 + u ) − (β + 1 ) = (0.7e−t + 2.3sin5t )(−0.4e−t ) − (−0.7e−t + 11.5cos5t )(0.4e−t + 1 ) ∂t ∂t = −0.92e−t sin5t + 0.7e−t − 4.6e−t cos5t − 11.5cos5t > 0.
Figs. 5 and 6 illustrate the monotone effect of the periodic factor on the growth rate. 5. Experimental example Besides, for the sake of showing the control effectiveness of our method, we use the real experimental data in Ref. [31] to test our model. In the experiment, the conditions satisfying our model are as follows: (1) the temperature, the illumination intensity, the ratio of light from the darkness, the salinity and the concentration of the different nitrogen sources are fixed as (24 ± 1) °C, 120 μmol · m−2 · s−1 , 12L: 12D, 30.5 and 100 μmol · L−1 , respectively. Then the controller of the experiment is the different nitrogen source. The one-time deployment of nitrogen means the density of the different nitrogen sources decreases as time increases shown as ∂∂ut < 0. (2) Because we consider the periodic control effect on the algal growth, we can assume that the monotone decreasing and positive function u(t) in the experiment has the following form:
u(t ) = Asin(ωt + θ ), where
t∈
2n + 0.5
ω
π−
θ 2n + 1 θ , π− , ω ω ω
n ∈ Z as an integer set. (3) There are four nitrogen sources as two kinds of inorganic nitrogen: nitrate and ammonium and two organic nitrogen: urea and mixed amino acids which have twenty basic amino acids and each is 5 μmol · L−1 . Moreover, the diffusion of the controller as the different nitrogen sources can be also regarded as the nitrogen uptake of the algae. (4) Also, the process of the experiment holds on as the growth is the exponential growth so that we can assume that the variation of the birth rate and the death rate are positive constant and the birth rate is larger than the death rate shown as
∂α (k ) ∂β (k ) = = 0, ∂k ∂k α ( k ) > β ( k ) > 0.
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Fig. 6. The monotone effect of the periodic factor u on the growth rate δ g (k, j), where the line type named Control means the cell density without controller u and the other four line types stand for the cell density with different nitrogen source.
Consequently we have that
∂ (α (k ) + u(k )) ∂ u(k ) = < 0, ∂k ∂k α ( k ) + u ( k ) > β ( k ), (α (k ) + u(k ))
∂β (k ) ∂ u (k ) ∂ (α (k ) + u(k )) =0> (β (k ) + 1 ) = (β (k ) + 1 ) ∂k ∂k ∂k
which satisfies the conditions of the Theorem 2. Furthermore, the monotone increasing effect of the cell density P is shown clearly in Fig. 7(a) for 9 controlled days, in Fig. 7(b) for 7 controlled days, in Fig. 7(c) for 8 controlled days and in Fig. 7(d) for 5 controlled days. In Fig. 7, the density of the four algae with controllers as four different nitrogen sources is larger than that without these controllers. Moreover, the velocity of the increasing density of the four different algae and the controlled time are different for four nitrogen sources, respectively. And the growth mean velocity quantity V of the four algae in the controlled time is listed as follows:
VChaetocerossp. > VPhaeocystisglobosa > VKareniasp. > VHeterosigmaakashiwo. In Fig. 7(a), from 1 to 4 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Chaetoceros sp. can be compared as follows:
EA > EU > EN > EACM
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Fig. 7. The monotone effect of the periodic factor u on the cell density P which is simulated by the experimental data in Ref.[31], where the line type named as Control means the cell density without controller u and the other four line types stand for the cell density with different nitrogen sources.
and from 4 to 9 days there is
EA > EN > EU > EACM . In Fig. 7(b), from 1 to 5 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Phaeocystis globosa can be compared as follows:
EA > EN > EU > EACM and from 5 to 7 days there is
EN > EU > EA > EACM . In Fig. 7(c), from 1 to 5 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Karenia sp. can be compared as follows:
EA > EN > EU > EACM , from 5 to 7 days there is
EA > EN > EACM > EU and from 7 to 8 days there exists
EN > EA > EACM > EU .
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Fig. 8. The monotone effect of the periodic factor u on the growth rate Rg (k).
In Fig. 7(d), from 1 to 4 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Heterosigma akashiwo can be compared as follows:
EA > EACM > EN > EU and from 4 to 5 days there is
EA > EN > EACM > EU . Besides this, we consider the control effect of the controller u on the growth rate Rg (k). Also because that the process of the experiment holds on as the growth is the exponential growth, we can assume that the variation of the birth rate and the death rate are positive constant and the birth rate is larger than the death rate. Then we have the following inequality:
1 ∂β (k ) =0>− , ∂k 2 u(k )) u(k )) 2 [(α (k ) + u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ − β (k ) ∂ (α (k∂)+ ] ∂ 2 (α (k ) + u(k )) ∂ 2 u(k ) k k = <− , 2 2 2 ∂k ∂k [(α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))]
(α (k ) + u(k ))
∂β ∂ u (k ) ∂ (α (k ) + u(k )) = 0 > (β + 1 ) = (β + 1 ) , ∂k ∂k ∂k
which satisfies the conditions of the Theorem 3. Next, the monotone increasing effect of the growth rate Rg (k) is shown clearly in Fig. 8(a) for 9 controlled days and in Fig. 8(b) for 7 controlled days. As for Fig. 8(c) and (d), the monotone increasing effect of the controller on the growth rate is not obvious. But the control effect of the Ammonium, Amino acid Mixture
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and Urea on the growth rate of the Karenia sp. can be shown distinct from 3 to 6 days in Fig. 8(c). And the control effect of the Ammonium and Amino acid Mixture on the growth rate of the Heterosigma akashiwo can be shown illustrated from 3 to 5 days in Fig. 8(d). Moreover, the variation of the growth rate of Heterosigma akashiwo according to the four different nitrogen sources likes a shape of ‘M’. 6. Conclusions and discussions Due to the fact that the growth process of the causative algae varies by the periodic environmental factor in reality like variation of seasons, light intensity, temperature and some inner cycle clock, etc., we focused on the control effects of the periodic environmental factor on the growth process and the growth rate for the causative algae and also simulate the control effects. Three theorems about the relationships between the periodic factor and the cell density and the growth rate are acquired by norm theory and finite difference method. We also simulate our results for simulated and experimental data in algal growth. Our work explained the monotone effect, chaotic effect of the periodic factor on the cell density and the growth rate of the causative algae, respectively. Furthermore, in the light of the control effect studied by our model for the different nitrogen sources on the four different algae based on the experimental data, there exists a question that why four nitrogen sources have different control effects for the four different algal growth rate. We can explain the question by the conditions of Theorem 3 u(k )) u(k )) 2 − β (k ) ∂ (α (k∂)+ ] [(α (k ) + u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ ∂ 2 (α (k ) + u(k )) ∂ 2 u(k ) k k = <− , 2 2 2 ∂k ∂k [(α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))]
that these conditions must be hold on in the experiment of the algal growth. However, the variation of the diffusion or the absorption process for the different nitrogen sources is complex according to the different algal growth. Thus the control effect of the nitrogen sources on the two algae is not obvious. Moreover, it is interesting and important to further consider the dynamical process of the controller both in our model and in the experiments of algal growth. Hence our future work will focus on the controlled theory about the coupled dynamical system between the controller and the algal growth and also the corresponding experimental theory. Acknowledgments The authors are very grateful to referees for valuable suggestions. This work was supported by the Foundation for the National Natural Science Foundation of China (Nos. 61533011 and 61273088) and the China Postdoctoral Science Foundation funded project (No. 2015M572033), the National Natural Science Foundation of Shandong Province (No. ZR2016GM09), the Soft Science Research Project of Shandong Province (No. 2016RKB01341), the Social Science Planning Project of Shandong Province (No. 15CJJJ34). References [1] Kirkpatrick B, Fleming LE, Squicciarini D, Backer LC, Clark R, Abraham W, et al. Literature review of Florida red tide: implications for human health effects. Harmful Algae 2004;3(2):99–115. [2] Ye NH, Zhang XW, Mao YZ, Liang CW, Xu D, Zou J, et al. Green tides are overwhelming the coastline of our blue planet: taking the worlds largest example. Ecol Res 2011;26:477–85. [3] Hallegraeff GM. Ocean climate change, phytoplankton community responses, and harmful algal blooms: a formidable predictive challenge 1. J Phycol 2010;46(2):220–35. [4] Trimbee AM, Prepas EE. Evaluation of total phosphorus as a predictor of the relative biomass of blue-green algae with emphasis on Alberta lakes. Can J Fish Aquat Sci 2011;44(7):1337–42. [5] Smith Val H. Low nitrogen to phosphorus ratios favor dominance by blue-green algae in lake phytoplankton. Science 1983;221(4611):669–71. [6] Bennett A, Bogorad L. Complementary chromatic adaptation in a filamentous blue-green alga. J Cell Biol 1973;58:419–35. [7] Talebi AF, Tohidfar M, Tabatabaei M. Genetic manipulation, a feasible tool to enhance unique characteristic of Chlorella vulgaris as a feedstock for biodiesel production. Mol Biol Rep 2013;40(7):4421–8. [8] Reiter MA. Colonization time and substratum relief as factors in the uptake of algae from the water column. J Freshwater Ecol 2011;16:57–65. [9] Schilling S, Oesterhelt C. Structurally reduced monosaccharide transporters in an evolutionarily conserved red alga. Biochem J 2007;406:325–31. [10] Pisman TI. Experimental and mathematical model of the interactions in the mixed culture of links in the producer-consumer cycle. Adv Space Res 2009;44:177–83. [11] Wang J, Jiang W. Impulsive perturbations in a predator-prey model with dormancy of predators. Appl Math Model 2014;38:2533–42. [12] Serizawa H, Amemiya T, Enomoto T, Itoh K. Mathematical modeling of colony formation in algal blooms: phenotypic plasticity in cyanobacteria. Ecol Res 2008;23(5):841–50. [13] Mukhopadhyay B, Bhattacharyya R. Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity. Ecol Model 2006;198:163–73. [14] Yamasaki Y, Zou Y, Go J, Shikata T, Matsuyama Y, Nagai K, et al. Cell contact-dependent lethal effect of the dinoflagellate Heterocapsa circularisquama on phytoplankton-phytoplankton interactions. J Sea Res 2011;65:76–83. [15] Chakraborty S, Roy S, Chattopadhyay J. Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model. Ecol Model 2008;213:191–201. [16] Shukla JB, Misra AK, Chandra P. Mathematical modeling and analysis of the depletion of dissolved oxygen in eutrophied water bodies affected by organic pollutants. Nonlinear Anal Real 2008;9:1851–65. [17] Beran B, Kargi F. A dynamic mathematical model for wastewater stabilization ponds. Ecol Model 2005;181:39–57. [18] Sugiura K, Kawasaki Y, Kinoshita M, Murakami A, Yoshida H, Ishikawa Y. A mathematical model for microcosms: formation of the colonies and coupled oscillation in population densities of bacteria. Ecol Model 2003;168:173–201. [19] Fergola P, Cerasuolo M, Pollio A, Pinto G, Dellagreca M. Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: experiments and mathematical model. Ecol Model 2007;208:205–14.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Control effect of periodic variation on the growth of harmful algal bloom causative species L. Zhang a,b,∗, S.T. Liu a, T. Liu b, C. Yu c,d, Z. Hu e,f a
College of Control Science and Engineering, Shandong University, Jinan 250061, PR China Business School, Shandong University of Political Science and Law, Jinan, 250014, PR China c Mind and Brain Theme, South Australian Health and Medical Research Institute, North Terrace, Adelaide, SA 5000, Australia d School of Medicine, Flinders University, Bedford Park, SA 5042, Australia e CAS Key Laboratory of Marine Ecology and Environmental Sciences, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, 266071, PR China f Laboratory for Marine Ecology and Environmental Science, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266071, PR China b
a r t i c l e
i n f o
Article history: Received 30 December 2016 Revised 21 April 2017 Accepted 25 May 2017 Available online 30 May 2017 Keywords: Periodic variation Causative species Control Harmful algae blooms
a b s t r a c t Blue-green algae and Dinoflagellate etc. are common types of phytoplankton as causative species which cause the harmful algal blooms (HABs). The growth process of causative species is complex according to the variation of the environmental disturbance such as the periodic factor in reality and recent studies have not revealed the secret of the growth complexity yet. Based on the empirical and theoretical results of the growth of causative species, a nonlinear controlled system with periodic factor was obtained and the different effects of the periodic factor on the control of the cell density and the growth rate of causative species were studied by three theorems using the norm theory and finite difference method. Simulations and experimental data were also used to assess the effectiveness of the controlled results. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Harmful algal blooms (HABs) are mainly produced by rapid growth of algal species named as the causative species with certain conditions and have severe impacts on public health and coastal ecosystems [1,2]. One of the most famous reproduction mode of the causative species for the unicellular algae [3,4] is the well-known fact that one cell divided into two cells by asexual reproduction. In addition, the rate of the growth is complicatedly and nonlinearly affected by different environmental factors such as the ratios of nitrogen to phosphorus [5], fluorescent and red light environments [6]. So the inherent regularity between the growth process of the causative species and the complex environmental factors still remains a mystery. Nowadays, many works focus on the control of growth process for these causative species in HABs experimentally. For instance, the higher concentration of kanamycin was found to inhibit the growth velocity of unicellular green algae Chlorella vulgaris [7]. The effect of assemblage age on the uptake rate of algal species was studied for the incorporation of cells and
∗
Corresponding author at: College of Control Science and Engineering, Shandong University, Jinan 250061, PR China. E-mail addresses: [email protected], [email protected] (L. Zhang), [email protected] (S.T. Liu), [email protected] (T. Liu), [email protected] (C. Yu), [email protected] (Z. Hu). http://dx.doi.org/10.1016/j.cnsns.2017.05.023 1007-5704/© 2017 Elsevier B.V. All rights reserved.
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eliminating differences in substrate roughness [8]. When three proteins named GsSPT1 (G. sulphuraria sugar and polyol transproter 1), GsSPT2 and GsSPT4 were studied, results were shown that the deletion of N-terminus of the protein in SPT1 affected the maximal transport velocity and released the dependency of the pH on sugar uptake, while the uptake by other two proteins was not active for the pH-dependence [9]. Moreover, according to the fact that the growth process of the causative species in reality becomes much more complex than that of growth under laboratory, the growth process can be described mathematically by dynamical systems and many interesting and important works have been obtained. For example, the dynamical behavior of algal growth with impulsive disturbance [10], such as chaos [11] and stability of attractors [12,13], was studied in colony formation, in the interaction growth and in the allelopathic of phytoplankton. In the light of the experimental results of the growth interaction such as toxin or non-toxic algae, organic pollutants, nutrients and bacteria, the coupled dynamical models were used to simulate the interactions [14–18]. The effect of allelopathic competition between two algal species on the growth rate of algae was also reported by coupled nonlinear models [19–21]. However, there exist few results about the effect of the environmental factor on the growth process of the causative species from the point view of control theory in term of the experimental results, data collection in field and the current mathematical models of causative species. In this paper, we concentrate on the control effect of the periodic environmental factor on the cell density and the growth rate of the causative species, in which the periodic environmental factor include variation of seasons, light intensity, temperature and some inner cycle clock, etc. [22,23]. Our work can play vital roles for revealing the secret of the HABs. The paper is organized as follows. In Section 2, the nonlinear controlled system for the causative species growth is established and the stable condition of the system is presented. Three theorems about the relationships between the periodic factor and the cell density and the growth rate of the causative species are studied in Section 3, respectively. Simulations and experimental data in Section 4 and in Section 5 also illustrate the control effect of the periodic factor on the cell density and the growth rate of causative species theoretically and experimentally. Conclusions and discussions are given in Section 6. 2. Problem assumption This section revolves around the conditions like exist and unique solution and stable theory for the nonlinear reactiondiffusion system with initial and boundary conditions based on the experimental results of growth of the causative species. In general, the growth process of the causative species dynamically satisfies the following system:
∂ P (x, y, z, t ) = f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )), ∂t
(2.1)
where P(x, y, z, t) is the cell density of the causative species,
P ∈ Hilbert space H ([ξ , η], Rn ),
(x, y, z, t ) ∈ R3 × (0, +∞ ), f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )) is the factor including the birth rate α (x, y, z, t), the death rate β (x, y, z, t) and the controller as u(x, y, z, t), like gene modulating factors and the strength of sunlight, etc. Based on the models in Ref.[24,25] and the fact that the main reproductive mode of single cell species in the causative species is binary fission and the growth velocity varies with different spatial location and the different environmental factors, the dynamical system (2.1) for the growth process of the causative species can be represented as follows:
∂ P (x, y, z, t ) = f (P (x, y, z, t ), α (x, y, z, t ), β (x, y, z, t ), u(x, y, z, t )) ∂t P 2 (x, y, z, t ) = (α + u(t ) − β (t ))P (x, y, z, t ) − (α + u(t )) + DP (x, y, z, t ), PA
(2.2)
where
(α + u(t ) − β (t ))P (x, y, z, t ) − (α + u(t ))
P 2 (x, y, z, t ) PA
means the binary fission term,
∂ 2 P (x, y, z, t ) ∂ 2 P (x, y, z, t ) ∂ 2 P (x, y, z, t ) DP (x, y, z, t ) = D + + ∂ x2 ∂ y2 ∂ z2
stands for the spatial diffusion term. Since the velocity of the cell division of the causative species is affected by different periodic factors like strength of sunlight, temperature and inner cycle clock, etc., we introduce the controller as
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u = Asin(ωt + θ ), where A is the strength of the periodic factors. And ω and θ represent the amplitude and phase in periodic environmental factors like strength of sunlight, temperature and inner cycle clock, etc., respectively. D indicates the diffusion coefficient. In order to find the control effect of the controller u on the growth process of the causative species easily, we assume the following initial and boundary conditions for system (2.2):
P (x, y, z, 0 ) = (φ (x, y, z ), 0, 0, · · · , 0 ), P (0, y, z, t ) = P (x, 0, z, t ) = P (x, y, 0, t ) = (0, 0, · · · , 0, a0 ), P (l, y, z, t ) = P (x, l, z, t ) = P (x, y, l, t ) = (1, 1, · · · , 1, a1 ),
where φ (x, y, z) is any continuous function, a0 and a1 are constant parameters and a1 > a0 ≥ 0. The stable condition will be discussed for the system (2.2) due to the exponential stability in this paragraph. On the basis of the condition of (2.2) and the reason is clear that the function
f1 (P, α , β , u ) = (α + u − β )P − (α + u ) PPA
2
satisfies
f1 (P0 , α , β , u ) = (0, 0, · · · , 0 )T , where P0 = (0, 0, · · · , 0 )T is the initial growth density. Then we assume f1 (P, α , β , u) is locally Lipschitz continuous, i.e. there exist positive real numbers K, K0 for any P1 , P2 ∈ H satisfying the following condition:
max{ P1 , P2
} ≤ K.
It means that
f1 (P1 , α , β , u ) − f1 (P2 , α , β , u ) ≤ K0 P1 − P2 , where P1 , P2 ∈ Hilbert space H ([ξ , η], Rn ). In addition, the parameters of (2.2) satisfy the following condition:
(α + u ) + P ) K0 ≥ (α + u − β ) − . φ
(2.3)
Because the norm theory will be used in the proof of the following theorems in next section, the following lemma is presented: Lemma 1. For different norm definitions:
P 1 =
n n
|Pi, j |,
i=1 j=1
and
P 2 =
n n
|Pi, j |2 ,
i=1 j=1
the two norms have the following inequality relationship:
P 1 ≤ nP 2 . 3. Main theorem Without loss of generality and in order to calculate easily, we omit the panel effect and consider only the water depth so that the system (2.2) becomes the following form
∂ P (z, t ) = f (P (z, t ), α (z, t ), β (z, t ), u(z, t )) ∂t = (α (z, t ) + u(z, t )) − β (z, t ))P (z, t ) − (α (z, t ) + u(z, t ))
P 2 (z, t ) ∂ 2 P (z, t ) +D , PA ∂ z2
(3.1)
with the initial and boundary conditions shown below as:
P (z, 0 ) = (0, 0, · · · , 0, φ (z )), P ( 0, t ) = ( 0, 0, · · · , 0, a0 ), P ( 1, t ) = ( 1, 1, · · · , 0, a1 ).
As the complexity of the form of f in system (3.1), it is difficult to solve the solution analytically. So we next use the finite difference method for the system of (3.1), which is the famous difference method for PDEs [26,27]. Then system (3.1) is discretized as follows:
Pk, j − Pk−1, j = rPk−1, j+1 − 2rPk−1, j + rPk−1, j−1 + 2(α (k ) + u − β )τ Pk, j −
2(α (k ) + u(k ))τ 2 Pk, j , PA
(3.2)
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where r = 2Dτ /h2 is the mesh ratio of the discretization process, h stands for the space step and τ means the time step. In order to compute easily, we assume τ = 1/2, D = h2 . Then as the numerical process for system (3.2), we have
α+u PA
α+u ,
PA
α+u PA
2 P22 + (1 − α − u + β )P2,2 + P1,2 − P1,3 − P1,1 = 0, 2 P32 + (1 − α − u + β )P3,2 + P2,2 − P2,3 − P2,1 = 0,
2 Pn,n −1 + (1 − α − u + β )Pn,n−1 + Pn−1,n−1 − Pn−1,n − Pn−1,n−2 = 0.
When we define the following symbols as
⎛
−1 ⎜0 M=⎜ . ⎝ .. 0
··· , ··· , .. . 0
0 −1 .. . ··· ,
0 0 .. . −1
(1 − α − u + β )
1 0 .. . 0
1 .. . 0
··· ··· , .. . 1 (1 − α − u + β )
0 0 .. . 0
−1 0 .. . ··· ,
0 −1 .. . 0
··· ··· , .. . −1
⎞
0 0⎟ .. ⎟ ⎠, . 0
P = [P1,1 , P2,1 , · · · , Pn,1 , P1,2 , P2,2 , · · · , Pn,2 , . . . , P1,n , P2,n , · · · , Pn,n ]T , 2 2 2 2 2 2 2 T F = [P22,2 , P32,2 , · · · , Pn, 2 , P2,3 , P3,3 , · · · , Pn,3 , · · · , P2,n−1 , P3,n−1 , Pn,n−1 ] ,
the numerical process of the system (3.2) can be substituted as follows
MP =
α+u PA
F.
(3.3)
Next we will discuss the different effects of the controller u on the growth process of the causative species. Theorem 1. The norm of the growth density ||P||2 will be controlled by the controller u according to the following relationship:
0 ≤ ||P ||2
PA n[4 − α − u + β ] PA ≤ + 2 (α + u )
n2 [4 − α − u + β ]2 + 4c (α + u )2 /PA2 2 (α + u )
where
c=
n
|Pi,1 |2 +
n
i=1
|Pi,n |2 +
i=1
n
|P1, j |2 .
j=1
Proof. For (3.3), we choose the norms of
P 1 =
n n
|Pi j |,
i=1 j=1
P 2 =
n n
|Pi j |2 ,
i=1 j=1
and
M1 = max1≤i≤n
n
|Mi j |.
j=1
In accordance with Lemma 1 and the system (3.3), we obtain
M P 1 ≤ M 1 P 1 , MP1 =
α+u PA
F 1 ,
and
P 1 ≤ nP 2 . Later we have
M1 P 1 ≥ MP 1 =
α+u PA
F 1 =
n n−1 α+u
PA
i=2 j=2
|Pi, j |2 ,
,
(3.4)
L. Zhang et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 185–201
which means that
M 1 P 1 + ≥
n α+u PA
n n−1 α+u PA
|Pi,1 | + 2
i=1
n
|Pi,n | + 2
i=1
|Pi, j |2 +
n
i=2 j=2
n
|P1, j |
2
j=1
|Pi,1 |2 +
i=1
n
189
|Pi,n |2 +
n
i=1
|P1, j |2 .
j=1
Subsequently we get
M 1 P 1 +
α+u PA
c≥
n n α+u
PA
|Pi, j |2 =
i=1 j=1
α+u PA
P22 .
(3.5)
According to Lemma 1 and the inequality of (3.5), we gain
α+u
n||M||1 ||P ||2 +
PA
c ≥ M 1
P 1 +
α+u PA
c≥
α+u PA
P22 ,
which can be written as the following form:
α+u PA
||P||22 − n||M||1 ||P||2 −
α+u PA
c ≤ 0.
(3.6)
Soon afterwards we have the inequality as the relationship of (3.6):
0 ≤ ||P ||2
PA n[4 − α − u + β ] PA ≤ + 2 (α + u )
n2 [4 − α − u + β ]2 + 4c (α + u )2 /PA2 2 (α + u )
,
(3.7)
where
c=
n
|Pi,1 |2 +
i=1
n i=1
|Pi,n |2 +
n
|P1, j |2 .
j=1
Remark 1. In the light of Theorem 1, it is clear that α , u and β play important role in the boundedness states of ||P||2 because of (3.7). Remark 2. Moreover, based on the boundedness effect of Theorem 1, we next further consider other control effect as the following two theorems and the following system. When we use the similar method in Ref. [28] for system (3.3) as follows:
Ci (k, j ) = 2Pk−i, j − Pk−i, j−i − Pk−i, j+i − Pk−i, j ,
(3.8)
and
Di (k, j ) =
P (1 − α − u + β ) 4α ( u ) (Ci (k, j ) − A ), 2 (α − u ) PA [(1 − α − u + β )2 − 2(1 − α − u + β )]
(3.9)
system (3.3) becomes a special example for the following system as i = 1:
Di (k, j ) = 1 − = 1−
PA2 [
PA2 [(1 − α − u + β )2 − 2(1 − α − u + β )]2 4 ( α + u )2 Di−1 (k, j )2 2 ( 4 ( α + u )2 ( 1 − α − u + β ) − 2 ( 1 − α − u + β )]
( α + u − β )2 − 1
4 = 1 − μDi−1 (k, j )2 .
Di−1 (k, j )2 (3.10)
On the basis of the similar theory like (3.10) in Refs.[28,29], the system (3.10) can display complex nonlinear behavior such as spatial chaotic behavior shown in Fig. 1. Moreover, the increasing effect of the periodic term on system (3.3) is obtained by the following two theorems. Theorem 2. The cell density P of the causative species in system (3.3) increases as time k increases when
⎧ ∂ D (k, j ) 0 > 0; ⎪ ∂k ⎪ ⎪ ⎪ ⎪ ∂ ( α ( k ) + u ( k )) ⎪ < 0; ⎨ ∂k
α ( k ) + u ( k ) > β ( k ); ⎪ ⎪ ⎪ ⎪ D0 (k, j )(α + u(k ) − β (k )) > 1; ⎪ ⎪ ⎩ (α (k ) + u(k )) ∂β∂(kk) > ∂ (α (k∂)+k u(k)) (β (k ) + 1 ).
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Fig. 1. The spatial chaotic behavior of Di (k) as α + u = 0.7e−t + 2.5, β = 0.4e−t , t ∈ (0, 1), N = 151.
Proof. According to the conditions (3.8) and (3.9), we have −α (k )−u(k )+β (k ))] Pk, j = −D0 (k, j ) PA [(1−α (k )−u(k )+4β((αk())k )+−2u((k1)) − 2
PA (1−α (k )−u(k )+β (k )) . 2(α (k )+u(k ))
(3.11)
So we get
∂ P (k, j ) ∂ D0 (k, j ) ∂δ ∂δ =− δ1 − D0 (k, j ) 1 − 2 ∂k ∂k ∂k ∂k owning to (3.11), where
δ1 =
PA (α (k ) + u(k )) 2 PA δ0 − δ0 , 4 2
δ2 =
Pa 1 − α (k ) − u(k ) + β (k ) , 2 α (k ) + u (k )
δ0 =
1 − α (k ) − u (k ) + β (k ) , α (k ) + u (k )
P ∂δ ∂δ1 PA ∂ (α (k ) + u(k )) 2 2PA ∂δ = δ0 + (α (k ) + u(k ))δ0 0 − A 0 , ∂k 4 ∂k 4 ∂k 2 ∂k
∂δ2 PA δ0 = , ∂k 2 ∂k ∂δ0 (− ∂ (α (k∂)+k u(k)) + = ∂k
∂β (k ) ∂ (α (k )+u(k )) ∂ k )(α (k ) + u (k )) − (1 − α (k ) − u (k ) + β (k )) ∂k .
(α (k ) + u(k ))2
The relationship (3.12) means that
PA ∂ P (k, j ) ∂ D0 (k, j ) PA ∂ D0 (k, j ) D0 (k, j )PA 2 ∂ (α (k ) + u(k )) = − (α (k ) + u(k ))δ02 + δ0 − δ0 ∂k 4 ∂k 2 ∂k 4 ∂k PA D0 (k, j ) PA D0 (k, j ) ∂ (α (k ) + u(k )) ∂β + (α (k ) + u(k ))δ0 − (α (k ) + u(k ))δ0 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k )) ∂k PA D0 (k, j ) ∂ (α (k ) + u(k )) + (α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))δ0 ∂k 2(α (k ) + u(k ))2 PA D0 (k, j ) ∂ (α (k ) + u(k )) PA D0 (k, j ) ∂β (k ) − + 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k )) ∂ k PA D0 (k, j ) ∂ (α (k ) + u(k )) PA ∂ (α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) + ∂k 2(α (k ) + u(k )) ∂k 2(α (k ) + u(k ))2 PA ∂β (k ) PA ) ∂ (α (k ) + u(k )) − + (1 − α (k ) − u(k ) + β (k )) , 2(α (k ) + u(k )) ∂ k ∂k 2(α (k ) + u(k ))2
(3.12)
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191
and then
∂ P (k, j ) ∂ D0 (k, j ) PA (α (k ) + u(k )) 2 PA =− δ0 − δ0 ∂k ∂k 4 2 PA ∂δ0 PA ∂ (α (k ) + u(k )) 2 2PA ∂δ0 PA ∂δ0 −D0 (k, j ) δ0 + (α (k ) + u(k ))δ0 − − 4 ∂k 4 ∂k 2 ∂k 2 ∂k 2 PA (α (k ) + u(k )) 1 − α (k ) − u(k ) + β (k ) ∂ D0 (k, j ) PA 1 − α (k ) − u(k ) + β (k ) ∂ D0 (k, j ) =− + 4 (α (k ) + u(k )) ∂k 2 (α (k ) + u(k )) ∂k 2 PA 1 − α (k ) − u(k ) + β (k ) ∂ (α (k ) + u(k )) 2PA D0 (k, j ) +D0 (k, j ) − (α (k ) + u(k )) 4 (α (k ) + u(k )) ∂k 4 ×
1 − α (k ) − u (k ) + β (k ) (− α (k ) + u (k )
∂ (α (k )+u(k )) u(k )) + ∂β∂(kk ) )(α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) ∂ (α (k∂)+ ∂k k
(α (k ) + u(k ))2
∂ (α (k )+u(k )) u(k )) + ∂β∂(kk ) )(α (k ) + u(k )) − (1 − α (k ) − u(k ) + β (k )) ∂ (α (k∂)+ PA D0 (k, j ) (− ∂ k k + −
PA 2
2
(− ∂ (α (k∂)+k u(k)) +
(α (k ) + u(k ))2
∂β (k ) ∂ (α (k )+u(k )) ∂ k )(α (k ) + u (k )) − (1 − α (k ) − u (k ) + β (k )) ∂k .
(α (k ) + u(k ))2
j) At last, we obtain the following conditions for ∂ P∂(k, > 0, : k
⎧ ∂ D (k, j ) 0 > 0; ⎪ ⎪ ⎪ ∂k ⎪ ⎪ ∂ ( α ( k ) + u ( k )) ⎪ < 0; ⎪ ∂k ⎨ (α (k ) + u(k )) > β (k ); ⎪ ⎪ ⎪ ⎪ D0 (k, j )(α (k ) + u(k ) − β (k )) > 1; ⎪ ⎪ ⎪ ⎩ (α (k ) + u(k )) ∂β∂(kk) > ∂ (α (k∂)+k u(k)) (β (k ) + 1 ),
which can represent the increasing effect of the environmental factors on the cell density.
The growth rate defined as follows in Ref. [30]:
Rg =
l nPt1 − l nPt0 , t1 − t0
is generally assumed to be constant. So we consider the more reasonable condition that the growth rate varies with time and then assume the following generalized growth rate
Rg (t, z ) =
lnP (t1 , z ) − lnP (t0 , z ) . t1 − t0
Next theorem is about the control effect of the periodic factor on the growth rate in the discretized processas Rg (k, j ) = lnP (k, j ) − lnP (k − 1, j ). Theorem 3. The growth rate Rg (k, j) increases when the following conditions are satisfied:
⎧ 2 ∂ D(k, j ) ⎪ > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β (k ) ⎪ > − 12 ; ⎪ ⎪ ⎨ ∂k ∂ 2 (α (k )+u(k ))
u (k )) u (k )) 2 [(α (k )+u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ −β (k ) ∂ (α (k∂)+ ] k k
<− ∂ k2 [(α (k )+u(k ))(1−α (k )−u(k )+β (k ))]2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k ))) > 0; ⎪ ⎪ ⎪ ⎩ ∂ (α (k )+u(k )) (α (k ) + u(k )) ∂β . ∂ k > (β + 1 ) ∂k Proof. Due to the conditions (3.8) and (3.9), we have
Rg (k, j ) = lnP (k, j ) − lnP (k − 1, j ) = ln
P (k, j ) P ( k − 1, j )
;
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= ln
P (k, j ) − P (k − 1, j ) + P (k − 1, j ) P ( k − 1, j )
≈ ln
∂ P (k, j ) ∂k
P ( k − 1, j )
+1 .
(3.13)
Then according to (3.13), it is clear that
∂ Rg (k, i ) P (k, j ) ≈ j) ∂k P (k, j ) + ∂ P∂(k, k
∂ 2 P (k, j ) ∂ P (k, j ) 2 ∂ k2 P (k, j ) − ( ∂ k ) .
P (k, j )2
(3.14)
Because
∂ 2 P (k, j ) ∂ 2 D(k, j ) ∂ D(k, j ) ∂δ1 ∂ 2 δ1 ∂ 2 δ2 =− δ1 − 2 − D(k, j ) 2 − 2 2 2 ∂k ∂k ∂k ∂k ∂ k ∂ k ∂ 2 D(k, j ) PA (α (k ) + u(k )) 2 PA ∂ D(k, j ) PA δ02 ∂ (α (k ) + u(k )) PA (α (k ) + u(k )) ∂δ0 =− δ0 − δ0 − 2 + δ0 4 2 ∂k 4 ∂k 2 ∂k ∂ k2 2 PA ∂δ0 ∂δ0 ∂ (α (k ) + u(k )) PA 2 ∂ (α (k ) + u(k )) PA ∂δ0 2 − − D(k, j ) PA δ0 + δ0 + (α (k ) + u(k ))( ) 2 ∂k ∂k ∂k 4 2 ∂k ∂ k2 ∂δ 2 PA ∂δ02 PA ∂δ02 PA + (α (k ) + u(k ))δ0 02 − − 2 2 2 ∂k 2 ∂ k2 ∂k and after substituting it into (3.14), we get that
∂ Rg (k, j ) > 0, ∂k when the following conditions are satisfied:
⎧ ∂ 2 D(k, j ) > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β ⎪ 1 ⎪ ⎪ ∂k > − 2 ; ⎨ ∂ 2 δ0
< 0;
∂ k2 ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k )) > 0; ⎪ ⎪ ⎪ ⎩ 2 2 ∂ (α (k )+u(k )) 0 2 < 0, 2(α (k ) + u(k ))( ∂δ ∂k ) + δ ∂ k2
and
∂ 2 P (k, j ) ∂ P (k, j ) 2 P (k, j ) > ( ) , ∂k ∂ k2
(3.15)
which means that
⎧ 2 ∂ D(k, j ) ⎪ > 0; ⎪ ∂ k2 ⎪ ⎪ ⎪ ∂β ⎪ > − 12 ; ⎪ ⎪ ⎨ ∂k ∂ 2 (α (k )+u(k ))
∂ (α (k )+u(k )) u (k )) 2 [(α (k )+u(k )) ∂β −β ∂ (α (k∂)+ ] ∂k − ∂k k
<− ∂ k2 [(α (k )+u(k ))(1−α (k )−u(k )+β )]2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 + D(k, j )(β − α (k ) − u(k )) > 0; ⎪ ⎪ ⎪ ⎩ ∂ (α (k )+u(k )) (α (k ) + u(k )) ∂β . ∂ k > (β + 1 ) ∂k
;
Remark 3. Theorems 2 and 3 show the conclusion that the cell density and the growth rate of the causative species can be affected by the functional form of the birth rate, the controller and the death rate, meaning that if the conditions of Theorems 2 and 3 have been satisfied, the cell density and the growth rate will monotone increase. Furthermore, depending on the fact that the real growth process shows different characteristics for different algaes under different environments, we can conclude that our results play vital role in further understanding the complex characteristics of the growth process of the causative species.
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193
Fig. 2. The variation of δ as the periodic factor changes.
4. Simulation examples Exponent growth is the famous form in the causative species growth phenomena so that we take the following form for the birth rate and the death rate as examples: t
t
α + u = A1 e− A2 + u = A1 e− A2 + A3 sinωt, t
β = A4 e− A5 , where A1 > 0, A2 > 0, A4 > 0, A5 > 0, A3 and ω are constant. In the light of Theorem 1, it is clear that the functional forms of α , u and β play important role in the variation of ||P||2 , where
δ=
n (4 − α − u + β ) , α+u
and the variation of δ is shown in Fig. 2. Besides this, the behavior of system (3.10) shows complex behavior as periodic factor changes illustrated in Fig. 3(a), (c) and (e) and the chaotic characteristic which is calculated by Lyapunov exponent shown in Fig. 3(b), (d) and (f). In addition, the cell density P and growth rate Rg can be controlled as increasing monotone by the variation of the periodic factor because of the relationships in Theorems 2 and 3. It is easy to obtain in terms of Theorem 2 that when we choose the time t ∈ (0.1770, π /10), the increase of P can be controlled by the following condition:
∂ (α + u ) = −0.7e−t + cos5t < 0, ∂t
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Fig. 3. The variation of D(k, j) as the periodic factor changes.
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195
Fig. 4. The variation of D(k, j) and its Lyapunov exponent as the periodic factor changes.
where
cos5t > 0, sin5t > 0,
α + u = 0.7e−t − 0.2sin5t, β = 0.4sin5t, (α + u )
∂β ∂ (α + u ) − (β + 1 ) = (0.7e−t − 0.2sin5t )(−0.4e−t ) − (−0.7e−t + cos5t )(0.4e−t + 1 ) ∂t ∂t = 0.08e−t sin5t + 0.7e−t + 0.4e−t cos5t + cos5t > 0.
For example, as the periodic factor increases, D(k, j) decreases shown in Figs. 3(a)–(c) and 4. Meanwhile, the chaotic characteristic calculated by Lyapunov exponent of D(k, j) increases illustrated by Fig. 3(b), (d) and (f). In order to simulate the control effect of the periodic factor on the growth rate Rg , we choose the time t ∈ (π /10, π /5). After that, the growth rate Rg increases as the following conditions are satisfied:
cos5t < 0, sin5t > 0,
α1 + u = 0.7e−t + 2.1sin5t, α2 + u = 0.7e−t + 2.3sin5t, β = 0.4sin5t, ∂ (α1 + u ) = −0.7e−t + 10.5cos5t, ∂t ∂ 2 (α1 + u ) = 0.7e−t − 52.5sin5t < 0, ∂t2
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Fig. 5. The monotone increasing effect of the periodic factor u on D(k, j).
(α1 + u )
∂β ∂ (α1 + u ) − (β + 1 ) = (0.7e−t + 2.1sin5t )(−0.4e−t ) − (−0.7e−t + 10.5cos5t )(0.4e−t + 1 ) ∂t ∂t = −0.84e−t sin5t + 0.7e−t − 4.2e−t cos5t − 10.5cos5t > 0,
(α2 + u )
∂β ∂ (α2 + u ) − (β + 1 ) = (0.7e−t + 2.3sin5t )(−0.4e−t ) − (−0.7e−t + 11.5cos5t )(0.4e−t + 1 ) ∂t ∂t = −0.92e−t sin5t + 0.7e−t − 4.6e−t cos5t − 11.5cos5t > 0.
Figs. 5 and 6 illustrate the monotone effect of the periodic factor on the growth rate. 5. Experimental example Besides, for the sake of showing the control effectiveness of our method, we use the real experimental data in Ref. [31] to test our model. In the experiment, the conditions satisfying our model are as follows: (1) the temperature, the illumination intensity, the ratio of light from the darkness, the salinity and the concentration of the different nitrogen sources are fixed as (24 ± 1) °C, 120 μmol · m−2 · s−1 , 12L: 12D, 30.5 and 100 μmol · L−1 , respectively. Then the controller of the experiment is the different nitrogen source. The one-time deployment of nitrogen means the density of the different nitrogen sources decreases as time increases shown as ∂∂ut < 0. (2) Because we consider the periodic control effect on the algal growth, we can assume that the monotone decreasing and positive function u(t) in the experiment has the following form:
u(t ) = Asin(ωt + θ ), where
t∈
2n + 0.5
ω
π−
θ 2n + 1 θ , π− , ω ω ω
n ∈ Z as an integer set. (3) There are four nitrogen sources as two kinds of inorganic nitrogen: nitrate and ammonium and two organic nitrogen: urea and mixed amino acids which have twenty basic amino acids and each is 5 μmol · L−1 . Moreover, the diffusion of the controller as the different nitrogen sources can be also regarded as the nitrogen uptake of the algae. (4) Also, the process of the experiment holds on as the growth is the exponential growth so that we can assume that the variation of the birth rate and the death rate are positive constant and the birth rate is larger than the death rate shown as
∂α (k ) ∂β (k ) = = 0, ∂k ∂k α ( k ) > β ( k ) > 0.
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197
Fig. 6. The monotone effect of the periodic factor u on the growth rate δ g (k, j), where the line type named Control means the cell density without controller u and the other four line types stand for the cell density with different nitrogen source.
Consequently we have that
∂ (α (k ) + u(k )) ∂ u(k ) = < 0, ∂k ∂k α ( k ) + u ( k ) > β ( k ), (α (k ) + u(k ))
∂β (k ) ∂ u (k ) ∂ (α (k ) + u(k )) =0> (β (k ) + 1 ) = (β (k ) + 1 ) ∂k ∂k ∂k
which satisfies the conditions of the Theorem 2. Furthermore, the monotone increasing effect of the cell density P is shown clearly in Fig. 7(a) for 9 controlled days, in Fig. 7(b) for 7 controlled days, in Fig. 7(c) for 8 controlled days and in Fig. 7(d) for 5 controlled days. In Fig. 7, the density of the four algae with controllers as four different nitrogen sources is larger than that without these controllers. Moreover, the velocity of the increasing density of the four different algae and the controlled time are different for four nitrogen sources, respectively. And the growth mean velocity quantity V of the four algae in the controlled time is listed as follows:
VChaetocerossp. > VPhaeocystisglobosa > VKareniasp. > VHeterosigmaakashiwo. In Fig. 7(a), from 1 to 4 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Chaetoceros sp. can be compared as follows:
EA > EU > EN > EACM
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Fig. 7. The monotone effect of the periodic factor u on the cell density P which is simulated by the experimental data in Ref.[31], where the line type named as Control means the cell density without controller u and the other four line types stand for the cell density with different nitrogen sources.
and from 4 to 9 days there is
EA > EN > EU > EACM . In Fig. 7(b), from 1 to 5 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Phaeocystis globosa can be compared as follows:
EA > EN > EU > EACM and from 5 to 7 days there is
EN > EU > EA > EACM . In Fig. 7(c), from 1 to 5 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Karenia sp. can be compared as follows:
EA > EN > EU > EACM , from 5 to 7 days there is
EA > EN > EACM > EU and from 7 to 8 days there exists
EN > EA > EACM > EU .
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Fig. 8. The monotone effect of the periodic factor u on the growth rate Rg (k).
In Fig. 7(d), from 1 to 4 days in the experiment, the control effect E of the four nitrogen sources on the cell density of Heterosigma akashiwo can be compared as follows:
EA > EACM > EN > EU and from 4 to 5 days there is
EA > EN > EACM > EU . Besides this, we consider the control effect of the controller u on the growth rate Rg (k). Also because that the process of the experiment holds on as the growth is the exponential growth, we can assume that the variation of the birth rate and the death rate are positive constant and the birth rate is larger than the death rate. Then we have the following inequality:
1 ∂β (k ) =0>− , ∂k 2 u(k )) u(k )) 2 [(α (k ) + u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ − β (k ) ∂ (α (k∂)+ ] ∂ 2 (α (k ) + u(k )) ∂ 2 u(k ) k k = <− , 2 2 2 ∂k ∂k [(α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))]
(α (k ) + u(k ))
∂β ∂ u (k ) ∂ (α (k ) + u(k )) = 0 > (β + 1 ) = (β + 1 ) , ∂k ∂k ∂k
which satisfies the conditions of the Theorem 3. Next, the monotone increasing effect of the growth rate Rg (k) is shown clearly in Fig. 8(a) for 9 controlled days and in Fig. 8(b) for 7 controlled days. As for Fig. 8(c) and (d), the monotone increasing effect of the controller on the growth rate is not obvious. But the control effect of the Ammonium, Amino acid Mixture
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and Urea on the growth rate of the Karenia sp. can be shown distinct from 3 to 6 days in Fig. 8(c). And the control effect of the Ammonium and Amino acid Mixture on the growth rate of the Heterosigma akashiwo can be shown illustrated from 3 to 5 days in Fig. 8(d). Moreover, the variation of the growth rate of Heterosigma akashiwo according to the four different nitrogen sources likes a shape of ‘M’. 6. Conclusions and discussions Due to the fact that the growth process of the causative algae varies by the periodic environmental factor in reality like variation of seasons, light intensity, temperature and some inner cycle clock, etc., we focused on the control effects of the periodic environmental factor on the growth process and the growth rate for the causative algae and also simulate the control effects. Three theorems about the relationships between the periodic factor and the cell density and the growth rate are acquired by norm theory and finite difference method. We also simulate our results for simulated and experimental data in algal growth. Our work explained the monotone effect, chaotic effect of the periodic factor on the cell density and the growth rate of the causative algae, respectively. Furthermore, in the light of the control effect studied by our model for the different nitrogen sources on the four different algae based on the experimental data, there exists a question that why four nitrogen sources have different control effects for the four different algal growth rate. We can explain the question by the conditions of Theorem 3 u(k )) u(k )) 2 − β (k ) ∂ (α (k∂)+ ] [(α (k ) + u(k )) ∂β∂(kk ) − ∂ (α (k∂)+ ∂ 2 (α (k ) + u(k )) ∂ 2 u(k ) k k = <− , 2 2 2 ∂k ∂k [(α (k ) + u(k ))(1 − α (k ) − u(k ) + β (k ))]
that these conditions must be hold on in the experiment of the algal growth. However, the variation of the diffusion or the absorption process for the different nitrogen sources is complex according to the different algal growth. Thus the control effect of the nitrogen sources on the two algae is not obvious. Moreover, it is interesting and important to further consider the dynamical process of the controller both in our model and in the experiments of algal growth. Hence our future work will focus on the controlled theory about the coupled dynamical system between the controller and the algal growth and also the corresponding experimental theory. Acknowledgments The authors are very grateful to referees for valuable suggestions. This work was supported by the Foundation for the National Natural Science Foundation of China (Nos. 61533011 and 61273088) and the China Postdoctoral Science Foundation funded project (No. 2015M572033), the National Natural Science Foundation of Shandong Province (No. ZR2016GM09), the Soft Science Research Project of Shandong Province (No. 2016RKB01341), the Social Science Planning Project of Shandong Province (No. 15CJJJ34). References [1] Kirkpatrick B, Fleming LE, Squicciarini D, Backer LC, Clark R, Abraham W, et al. Literature review of Florida red tide: implications for human health effects. Harmful Algae 2004;3(2):99–115. [2] Ye NH, Zhang XW, Mao YZ, Liang CW, Xu D, Zou J, et al. Green tides are overwhelming the coastline of our blue planet: taking the worlds largest example. Ecol Res 2011;26:477–85. [3] Hallegraeff GM. Ocean climate change, phytoplankton community responses, and harmful algal blooms: a formidable predictive challenge 1. J Phycol 2010;46(2):220–35. [4] Trimbee AM, Prepas EE. Evaluation of total phosphorus as a predictor of the relative biomass of blue-green algae with emphasis on Alberta lakes. Can J Fish Aquat Sci 2011;44(7):1337–42. [5] Smith Val H. Low nitrogen to phosphorus ratios favor dominance by blue-green algae in lake phytoplankton. Science 1983;221(4611):669–71. [6] Bennett A, Bogorad L. Complementary chromatic adaptation in a filamentous blue-green alga. J Cell Biol 1973;58:419–35. [7] Talebi AF, Tohidfar M, Tabatabaei M. Genetic manipulation, a feasible tool to enhance unique characteristic of Chlorella vulgaris as a feedstock for biodiesel production. Mol Biol Rep 2013;40(7):4421–8. [8] Reiter MA. Colonization time and substratum relief as factors in the uptake of algae from the water column. J Freshwater Ecol 2011;16:57–65. [9] Schilling S, Oesterhelt C. Structurally reduced monosaccharide transporters in an evolutionarily conserved red alga. Biochem J 2007;406:325–31. [10] Pisman TI. Experimental and mathematical model of the interactions in the mixed culture of links in the producer-consumer cycle. Adv Space Res 2009;44:177–83. [11] Wang J, Jiang W. Impulsive perturbations in a predator-prey model with dormancy of predators. Appl Math Model 2014;38:2533–42. [12] Serizawa H, Amemiya T, Enomoto T, Itoh K. Mathematical modeling of colony formation in algal blooms: phenotypic plasticity in cyanobacteria. Ecol Res 2008;23(5):841–50. [13] Mukhopadhyay B, Bhattacharyya R. Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity. Ecol Model 2006;198:163–73. [14] Yamasaki Y, Zou Y, Go J, Shikata T, Matsuyama Y, Nagai K, et al. Cell contact-dependent lethal effect of the dinoflagellate Heterocapsa circularisquama on phytoplankton-phytoplankton interactions. J Sea Res 2011;65:76–83. [15] Chakraborty S, Roy S, Chattopadhyay J. Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model. Ecol Model 2008;213:191–201. [16] Shukla JB, Misra AK, Chandra P. Mathematical modeling and analysis of the depletion of dissolved oxygen in eutrophied water bodies affected by organic pollutants. Nonlinear Anal Real 2008;9:1851–65. [17] Beran B, Kargi F. A dynamic mathematical model for wastewater stabilization ponds. Ecol Model 2005;181:39–57. [18] Sugiura K, Kawasaki Y, Kinoshita M, Murakami A, Yoshida H, Ishikawa Y. A mathematical model for microcosms: formation of the colonies and coupled oscillation in population densities of bacteria. Ecol Model 2003;168:173–201. [19] Fergola P, Cerasuolo M, Pollio A, Pinto G, Dellagreca M. Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: experiments and mathematical model. Ecol Model 2007;208:205–14.
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