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Research article

Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen

  • Gonyostomum semen is a bloom-forming freshwater raphidophyte that is currently on the increase, which concerns water managers and ecologists alike. Much indicates that the recent success of G. semen is linked to its diel vertical migration (DVM), which helps to overcome the spatial separation of optimal light conditions for photosynthesis at the surface of a lake and the high concentration of phosphate in the hypolimnion. I here present data from a field study conducted in Lake Lundebyvannet (Norway) in 2017–2019 that are consistent with the idea that the DVM of G. semen also allows for a hypolimnetic uptake of ammonium. As expected, microbial mineralization of organic matter in a low-oxygen environment led to an accumulation of ammonium in the hypolimnion as long as G. semen was absent. In contrast, a decreasing or constantly lower concentration of hypolimnetic ammonium was found in presence of a migrating G. semen population. In summer of 2019, a short break in the DVM of G. semen coincided with a rapid accumulation of hypolimnetic ammonium, which was equally rapidly decimated when G. semen resumed its DVM. Taken together, these data support the idea that G. semen can exploit the hypolimnetic pool of ammonium, which may be one reason for the recent success of the species and its significant impact on the structure of the aquatic food web.

    Citation: Thomas Rohrlack. Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen[J]. AIMS Microbiology, 2020, 6(2): 92-105. doi: 10.3934/microbiol.2020006

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  • Gonyostomum semen is a bloom-forming freshwater raphidophyte that is currently on the increase, which concerns water managers and ecologists alike. Much indicates that the recent success of G. semen is linked to its diel vertical migration (DVM), which helps to overcome the spatial separation of optimal light conditions for photosynthesis at the surface of a lake and the high concentration of phosphate in the hypolimnion. I here present data from a field study conducted in Lake Lundebyvannet (Norway) in 2017–2019 that are consistent with the idea that the DVM of G. semen also allows for a hypolimnetic uptake of ammonium. As expected, microbial mineralization of organic matter in a low-oxygen environment led to an accumulation of ammonium in the hypolimnion as long as G. semen was absent. In contrast, a decreasing or constantly lower concentration of hypolimnetic ammonium was found in presence of a migrating G. semen population. In summer of 2019, a short break in the DVM of G. semen coincided with a rapid accumulation of hypolimnetic ammonium, which was equally rapidly decimated when G. semen resumed its DVM. Taken together, these data support the idea that G. semen can exploit the hypolimnetic pool of ammonium, which may be one reason for the recent success of the species and its significant impact on the structure of the aquatic food web.



    Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders [6,9,11,17]. Recently, fractional partial differential equations play an important role in interpretation and modeling of many of realism matters appear in applied mathematics and physics including fluid mechanics, electrical circuits, diffusion, damping laws, relaxation processes, optimal control theory, chemistry, biology, and so on [7,13,14,15,16]. Therefore, the search of the solutions for fractional partial differential equations is an important aspect of scientific research.

    Many powerful and efficient methods have been proposed to obtain numerical solutions and analytical solutions of fractional partial differential equations. The most commonly used ones are: Adomian decomposition method (ADM) [5], variational iteration method (VIM) [18], new iterative method (NIM) [8], fractional difference method (FDM) [11], reduced differential transform method (RDTM) [1], homotopy analysis method (HAM) [3], homotopy perturbation method (HPM) [4].

    The main objective of this paper is to present a new numerical technique called modified generalized Taylor fractional series method (MGTFSM) to obtain the approximate and exact solutions of Caputo time-fractional biological population equation. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. The main advantage of the proposed method compare with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction.

    Consider the following Caputo time-fractional biological population equation

    Dαtu=2u2x2+2u2y2+F(u), (1.1)

    with the initial condition

    u(x,y,0)=u0(x,y), (1.2)

    where Dαt=αtα is the Caputo fractional derivative operator of order α, 0<α1,u=u(x,y,t),(x,y)R2,t>0 denotes the population density and F represents the population supply due to birth and death, α is a parameter describing the order of the fractional derivative.

    The plan of our paper is as follows: In Section 2, we present some necessary definitions and properties of the fractional calculus theory. In Section 3, we will propose an analysis of the modified generalized Taylor fractional series method (MGTFSM) for solving the Caputo time-fractional biological population equation (1.1) subject to the initial condition (1.2). In Section 4, we present three numerical examples to show the efficiency and effectiveness of this method. In Section 5, we discuss our obtained results represented by figures and tables. These results were verified with Matlab (version R2016a). Section 6, is devoted to the conclusions on the work.

    In this section, we present some basic definitions and properties of the fractional calculus theory which are used further in this paper. For more details see, [9,11].

    Definition 2.1. A real function u(X,t), X=(x1,x2,...,xn)RN,NN,tR+, is considered to be in the space Cμ(RN×R+), μ R, if there exists a real number p>μ, so that u(X,t)=tpv(X,t), where v C(RN×R+), and it is said to be in the space Cnμ if u(n)Cμ(RN×R+),nN.

    Definition 2.2. The Riemann-Liouville fractional integral operator of order α0 of uCμ(RN×R+),μ1, is defined as follows

    Iαtu(X,t)={1Γ(α)t0(tξ)α1u(X,ξ)dξ,α>0,t>ξ>0,u(X,t),                                α=0,                 (2.1)

    where Γ(.) is the well-known Gamma function.

    Definition 2.3. The Caputo time-fractional derivative operator of order α>0 of  uCn1(RN×R+),nN, is defined as follows

    Dαtu(X,t)={1Γ(nα)t0(tξ)nα1u(n)(X,ξ)dξ,n1<α<n,u(n)(X,t),                                         α=n.               (2.2)

    For this definition we have the following properties

    (1)

    Dαt(c)=0, where c is a constant.

    (2)

    Dαttβ={Γ(β+1)Γ(βα+1)tβα if β>n1,0,               if βn1.

    Definition 2.3. The Mittag-Leffler function is defined as follows

    Eα(z)=n=0znΓ(nα+1),αC,Re(α)>0. (2.3)

    For α=1, Eα(z) reduces to ez.

    Theorem 3.1. Consider the Caputo time-fractional biological population equation of the form (1.1) with the initial condition (1.2).

    Then, by MGTFSM the solution of equations (1.1)-(1.2) is given in the form of infinite series which converges rapidly to the exact solution as follows

    u(x,y,t)=i=0ui(x,y)tiαΓ(iα+1),(x,y)R2,t[0,R),

    where ui(x,y) the coefficients of the series and R is the radius of convergence.

    Proof. In order to achieve our goal, we consider the following Caputo time-fractional biological population equation of the form (1.1) with the initial condition (1.2).

    Assume that the solution takes the following infinite series form

    u(x,y,t)=i=0ui(x,y)tiαΓ(iα+1). (3.1)

    Consequently, the approximate solution of equations (1.1)-(1.2), can be written in the form of

    un(x,y,t)=ni=0ui(x,y)tiαΓ(iα+1)=u0(x,y)+ni=1ui(x,y)tiαΓ(iα+1). (3.2)

    By applying the operator Dαt on equation (3.2), and using the properties (1) and (2), we obtain the formula

    Dαtun(x,y,t)=n1i=0ui+1(x,y)tiαΓ(iα+1). (3.3)

    Next, we substitute both (3.2) and (3.3) in (1.1). Therefore, we have the following recurrence relations

    0=n1i=0ui+1(x,y)tiαΓ(iα+1)2x2(ni=0ui(x,y)tiαΓ(iα+1))22y2(ni=0ui(x,y)tiαΓ(iα+1))2F(ni=0ui(x,y)tiαΓ(iα+1)).

    We follow the same analogue used in obtaining the Taylor series coefficients. In particular, to calculate the function un(x,y),n=1,2,3,.., we have to solve the following

    D(n1)αt{G(x,y,t,α,n)}t=0=0,

    where

    G(x,y,t,α,n)=n1i=0ui+1(x,y)tiαΓ(iα+1)2x2(ni=0ui(x,y)tiαΓ(iα+1))22y2(ni=0ui(x,y)tiαΓ(iα+1))2F(ni=0ui(x,y)tiαΓ(iα+1)).

    Now, we calculate the first terms of the sequence {un(x,y)}N1.

    For n=1 we have

    G(x,y,t,α,1)=u1(x,y)2x2(u0(x,y)+u1(x,y)tαΓ(α+1))22y2(u0(x,y)+u1(x,y)tαΓ(α+1))2F(u0(x,y)+u1(x,y)tαΓ(α+1)).

    Solving G(x,y,0,α,1)=0, yields

    u1(x,y)=2x2u20(x,y)+2y2u20(x,y)+F(u0(x,y)).

    For n=2 we have

    G(x,y,t,α,2)=u1(x,y)+u2(x,y)tαΓ(α+1)2x2(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1))22y2(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1))2F(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1)). (3.4)

    Applying Dαt on both sides of equation (3.4) gives

    DαtG(x,y,t,α,2)=u2(x,y)22x2[(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1))×(u1(x,y)+u2(x,y)tαΓ(α+1))]22y2[(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1))×(u1(x,y)+u2(x,y)tαΓ(α+1))](u1(x,y)+u2(x,y)tαΓ(α+1))×F(u0(x,y)+u1(x,y)tαΓ(α+1)+u2(x,y)t2αΓ(2α+1)).

    Solving Dαt{G(x,y,t,α,2)}t=0=0, yields

    u2(x,y)=22x2[u0(x,y)u1(x,y)]+22y2[u0(x,y)u1(x,y)]+u1(x,y)F(u0(x,y)).

    To calculate u3(x,y), we consider G(x,y,t,α,3) and we solve

    D2αt{G(x,y,t,α,3)}t=0=0,

    we have

    u3(x,y)=22x2[3u1(x,y)u2(x,y)+u0(x,y)u3(x,y)]+22y2[3u1(x,y)u2(x,y)+u0(x,y)u3(x,y)]+u2(x,y)F(u0(x,y))+u21(x,y)F(u0(x,y)),

    and so on.

    In general, to obtain the coefficient function uk(x,y) we solve

    D(k1)αt{G(x,y,t,α,k)}t=0=0.

    Finally, the solution of equations (1.1)-(1.2), can be expressed by

    u(x,y,t)=limnun(x,y,t)=limnni=0ui(x,y)tiαΓ(iα+1)=i=0ui(x,y)tiαΓ(iα+1).

    The proof is complete.

    In this section, we test the validity and efficiency of the proposed method to solve three numerical examples of Caputo time-fractional biological population equation.

    We define En to be the absolute error between the exact solution u and the approximate solution un, as follows

    En(x,y,t)=|u(x,y,t)un(x,y,t)|,n=0,1,2,3,...

    Example 4.1. Consider the Caputo time-fractional biological population equation in the form

    Dαtu=2u2x2+2u2y2+hu, (4.1)

    with the initial condition

    u(x,y,0)=u0(x,y)=xy. (4.2)

    By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.1)-(4.2) in the form

    u(x,y,t)=i=0ui(x,y)tiαΓ(iα+1),

    and

    ui(x,y)=hixy, for i=0,1,2,3,...

    So, the solution of equations (4.1)-(4.2), can be expressed by

    u(x,y,t)=xy(1+htαΓ(α+1)+h2t2αΓ(2α+1)+h3t3αΓ(3α+1)+...)=xyi=0(htα)iΓ(iα+1)=xyEα(htα), (4.3)

    where Eα(htα) is the Mittag-Leffler function, defined by (2.3).

    Taking α=1 in (4.3), we have

    u(x,y,t)=xy(1+ht+(ht)22!+(ht)33!+...)=xyexp(ht),

    which is an exact solution to the standard form biological population equation [10].

    Example 4.2. Consider the Caputo time-fractional biological population equation in the form

    Dαtu=2u2x2+2u2y2+u, (4.4)

    with the initial condition

    u(x,y,0)=u0(x,y)=sinxsinhy. (4.5)

    By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.4)-(4.5) in the form

    u(x,y,t)=i=0ui(x,y)tiαΓ(iα+1),

    and

    ui(x,y)=sinxsinhy, for i=0,1,2,3,...

    So, the solution of equations (4.4)-(4.5), can be expressed by

    u(x,y,t)=sinxsinhy(1+tαΓ(α+1)+t2αΓ(2α+1)+t3αΓ(3α+1)+...)=sinxsinhyi=0tiαΓ(iα+1)=sinxsinhyEα(tα), (4.6)

    where Eα(tα) is the Mittag-Leffler function, defined by (2.3).

    Taking α=1 in (4.6), we have

    u(x,y,t)=sinxsinhy(1+t+t22!+t33!+...)=(sinxsinhy)exp(t),

    which is an exact solution to the standard form biological population equation [12].

    Example 4.3 Consider the Caputo time-fractional biological population equation in the form

    Dαtu=2u2x2+2u2y2+hu(1ru), (4.7)

    with the initial condition

    u(x,y,0)=u0(x,y)=exp(hr8(x+y)). (4.8)

    By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.7)-(4.8) in the form

    u(x,y,t)=i=0ui(x,y)tiαΓ(iα+1),

    and

    ui(x,y)=hiexp(hr8(x+y)), for i=0,1,2,3,...

    So, the solution of equations (4.7)-(4.8), can be expressed by

    u(x,y,t)=exp(hr8(x+y))(1+htαΓ(α+1)+h2t2αΓ(2α+1)+h3t3αΓ(3α+1)+...)=exp(hr8(x+y))i=0(htα)iΓ(iα+1)=exp(hr8(x+y))Eα(htα), (4.9)

    where Eα(htα) is the Mittag-Leffler function, defined by (2.3).

    Taking α=1 in (4.9), we have

    u(x,y,t)=exp(hr8(x+y))(1+ht+(ht)22!+(ht)33!+...)=exp(hr8(x+y)+ht),

    which is an exact solution to the standard form biological population equation [2].

    In this section the numerical results for Examples 4.1, 4.2 and 4.3 are presented. Figures 1, 3 and 5 represents the surface graph of the exact solution and the approximate solution u6(x,y,t) at α=0.6,0.8,1. Figures 2, 4 and 6 represents the behavior of the exact solution and the approximate solution u6(x,y,t) at α=0.7,0.8,0.9,1. These figures affirm that when the order of the fractional derivative α tends to 1, the approximate solutions obtained by MGTFSM tends continuously to the exact solutions. Tables 13 show the absolute errors between the exact solution and the approximate solution u6(x,y,t) at α=1 for different values of x,y and t. These tables clarifies the convergence of the approximate solutions to the exact solutions.

    Figure 1.  The surface graph of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.1 when h=1 and t=1.5.
    Figure 2.  The behavior of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.1 when h=y=1 and t=1.5.
    Figure 3.  The surface graph of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.2 when t=1.5.
    Figure 4.  The behavior of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.2 when y=1 and t=1.5.
    Figure 5.  The surface graph of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.3 when h=1,r=2 and t=1.5.
    Figure 6.  The behavior of the exact solution u and the approximate solution u6 by MGTFSM for different values of α for Example 4.3 when h=y=1,r=2 and t=1.5.
    Table 1.  Comparison of the absolute errors for the obtained results and the exact solution for Example 4.1 when h=1,n=6 and α=1.
    t/x,y 0.1 0.3 0.5 0.7
    0.1 1.4090×1010 4.2269×1010 7.0449×1010 9.8629×1010
    0.3 1.0576×107 3.1727×107 5.2879×107 7.4030×107
    0.5 2.3354×106 7.0062×106 1.1677×105 1.6348×105
    0.7 1.8129×105 5.4387×105 9.0645×105 1.2690×104
    0.9 8.4486×105 2.5346×104 4.2243×104 5.9140×104

     | Show Table
    DownLoad: CSV
    Table 2.  Comparison of the absolute errors for the obtained results and the exact solution for Example 4.2 when n=6 and α=1.
    t/x,y 0.1 0.3 0.5 0.7
    0.1 1.4090×1010 4.2268×1010 7.0425×1010 9.8497×1010
    0.3 1.0576×107 3.1726×107 5.2860×107 7.3932×107
    0.5 2.3354×106 7.0059×106 1.1673×105 1.6326×105
    0.7 1.8129×105 5.4385×105 9.0614×105 1.2673×104
    0.9 8.4486×105 2.5345×104 4.2228×104 5.9061×104

     | Show Table
    DownLoad: CSV
    Table 3.  Comparison of the absolute errors for the obtained results and the exact solution for Example 4.3 when h=1,r=2,n=6 and α=1.
    t/x,y 0.1 0.3 0.5 0.7
    0.1 1.5572×109 1.9019×109 2.3230×109 2.8373×109
    0.3 1.1688×106 1.4276×106 1.7436×106 2.1297×106
    0.5 2.5810×105 3.1525×105 3.8504×105 4.7029×105
    0.7 2.0036×104 2.4472×104 2.9890×104 3.6507×104
    0.9 9.3372×104 1.1404×103 1.3929×103 1.7013×103

     | Show Table
    DownLoad: CSV

    In addition, numerical results have confirmed the theoretical results and high accuracy of the proposed scheme.

    Remark 5.1. In this paper, we only apply Six terms to approximate the solutions, if we apply more terms of the approximate solutions, the accuracy of the approximate solutions will be greatly improved.

    In this paper, a new numerical technique called modified generalized Taylor fractional series method (MGTFSM) has been successfully applied for solving the Caputo time-fractional biological population equation. The method was applied to three numerical examples. The results show that the MGTFSM is an efficient and easy to use technique for finding approximate and exact solutions for these problems. The obtained approximate solutions using the suggested method is in excellent agreement with the exact solutions. This confirms our belief that the effciency of our technique gives it much wider applicability for general classes of fractional problems.

    The authors are very grateful to the guest editors of this special issue and would like to express their sincere thanks to the referees for the careful and noteworthy reading of the paper and for their constructive comments and suggestions which are improved the paper substantially.

    The authors declare that there is no conflict of interest in this paper.


    Acknowledgments



    The author thanks Johnny Kristiansen and Pia Frostad for their assistance in the field and laboratory. The study was supported by NMBU grant 1750051444.

    Conflict of interest



    The author declares no conflict of interests.

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