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Research article Special Issues

Evaluation of plant growth promoting activity and heavy metal tolerance of psychrotrophic bacteria associated with maca (Lepidium meyenii Walp.) rhizosphere

  • The high Andean plateau of Peru is known to suffer harsh environmental conditions. Acidic soils containing high amount of heavy metals due to mining activities and withstanding very low temperatures affect agricultural activities by diminishing crop quality and yield. In this context, plant growth promoting rhizobacteria (PGPR) adapted to low temperatures and tolerant to heavy metals can be considered as an environment-friendly biological alternative for andean crop management. The aim of this work was to select and characterize psychrotrophic PGPR isolated from the rhizosphere of maca (Lepidium meyenii Walp.) a traditional andean food crop. A total of 44 psychrotrophic strains isolated from 3 areas located in the Bombon plateu of Junin-Peru were tested for their PGPR characteristics like indole acetic acid (IAA) production, phosphate solubilization and for their ability to improve seed germination. In addition, their capacity to grow in the presence of heavy metals like cadmium (Cd), lead (Pb), cobalt (Co) and mercury (Hg) was tested. Of the total number of strains tested, 12 were positive for IAA production at 22 °C, 8 at 12 °C and 16 at 6 °C. Phosphate solubilization activities were higher at 12 °C and 6 °C than at 22 °C. Red clover plant assays showed that 16 strains were capable to improve seed germination at 22 °C and 4 at 12 °C. Moreover, 11 strains showed tolerance to Cd and Pb at varying concentrations. This study highlight the importance of obtaining PGPRs to be used in high andean plateu crops that are exposed to low temperatures and presence of heavy metals on soil.

    Citation: Paola Ortiz-Ojeda, Katty Ogata-Gutiérrez, Doris Zúñiga-Dávila. Evaluation of plant growth promoting activity and heavy metal tolerance of psychrotrophic bacteria associated with maca (Lepidium meyenii Walp.) rhizosphere[J]. AIMS Microbiology, 2017, 3(2): 279-292. doi: 10.3934/microbiol.2017.2.279

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  • The high Andean plateau of Peru is known to suffer harsh environmental conditions. Acidic soils containing high amount of heavy metals due to mining activities and withstanding very low temperatures affect agricultural activities by diminishing crop quality and yield. In this context, plant growth promoting rhizobacteria (PGPR) adapted to low temperatures and tolerant to heavy metals can be considered as an environment-friendly biological alternative for andean crop management. The aim of this work was to select and characterize psychrotrophic PGPR isolated from the rhizosphere of maca (Lepidium meyenii Walp.) a traditional andean food crop. A total of 44 psychrotrophic strains isolated from 3 areas located in the Bombon plateu of Junin-Peru were tested for their PGPR characteristics like indole acetic acid (IAA) production, phosphate solubilization and for their ability to improve seed germination. In addition, their capacity to grow in the presence of heavy metals like cadmium (Cd), lead (Pb), cobalt (Co) and mercury (Hg) was tested. Of the total number of strains tested, 12 were positive for IAA production at 22 °C, 8 at 12 °C and 16 at 6 °C. Phosphate solubilization activities were higher at 12 °C and 6 °C than at 22 °C. Red clover plant assays showed that 16 strains were capable to improve seed germination at 22 °C and 4 at 12 °C. Moreover, 11 strains showed tolerance to Cd and Pb at varying concentrations. This study highlight the importance of obtaining PGPRs to be used in high andean plateu crops that are exposed to low temperatures and presence of heavy metals on soil.


    In this paper, we mainly focus on the following stream-population model[1]

    {ut=d2ux2αuxσu+μv,vt=ϵ2vx2+σuμv+f(v), (1.1)

    with the initial data

    u(x,0)=u0(x)0,v(x,0)=v0(x)0,xR. (1.2)

    Here, u(x,t), v(x,t) are the population densities of neuston and benthon respectively; σ represents the per capita drift rate at which the species returns to the benthic population; μ is the per capita rate at which individuals in the benthic population enter the drift; α is the speed of the flow; d, ϵ are the diffusion coefficients of species u, v respectively; the reaction term f(v) is a strictly convex function of second order differentiable satisfying f(0)=f(1)=0, f(0)>0>f(1), and f(v)>0 for v(0,1). σ,μ,α,d,ϵ are positive constants and possess the biological meaning[2,3]. It is worth mentioning that the benthon basically do not move horizontally since ϵ1. Obviously, (0,0) and (μσ,1) are the equilibria and further we can obtain that (0,0) is unstable and (μσ,1) is stable for the corresponding spatially homogeneous system of (1.1).

    In this paper, we are interested in the nonnegative traveling wave, connecting (0,0) and (μσ,1), which possesses the wave profile as

    u(x,t)=¯U(z),v(x,t)=¯V(z),z=xct, (1.3)

    where the wave speed c0. Substituting (1.3) into (1.1), yields the new wave profile as

    {c¯U=d¯Uα¯Uσ¯U+μ¯V,c¯V=ϵ¯V+σ¯Uμ¯V+f(¯V), (1.4)

    with the boundary value condition

    (¯U,¯V)()=(μσ,1),(¯U,¯V)()=(0,0). (1.5)

    It is worth pointing out that the existence of the traveling wave solution of (1.4) was proved, and the determinacy of linear and nonlinear selection of the minimal wave speed was further established by employing the method of upper and lower solutions in [1].

    On this basis of the existence, we shall investigate the local and the global stability of the monotone traveling waves. In this progress, we need to conform that the solution of (1.1) is infinitely close to (¯U,¯V)(xct) when t is large enough for given initial data u0(x) and v0(x). To begin with, we set

    (u,v)(x,t)=(U,V)(z,t), (1.6)

    and then (1.4) can be transformed into the partial differential model

    {Ut=dUzz+(cα)UzσU+μV,Vt=ϵVzz+cVz+σUμV+f(V), (1.7)

    subject to

    U(z,0)=u0(z),V(z,0)=v0(z),zR. (1.8)

    From the above system (1.7) we know that (¯U,¯V)(z) is also its steady state.

    The investigation of the stabilities of traveling wave solutions could be traced back to the original works [4,5]. Then many works assessing stability are available, but differences among them are frequent. For examples, the stability of the traveling waves with critical speed [6] and noncritical speed [7] in appropriate weighted Banach spaces is proved. The global stability of the forced waves [8,9] is presented. The time-periodic traveling waves are asymptotically stable [10,11]. Mei et. al. [12,13,14] proved the exponential stability of traveling wave fronts, and the exponential stability of stochastic systems is demonstrated [15,16]. By analyzing the location of the spectrum, the local stability of the traveling waves are investigated [17,18]. For more related works, we refer to [19,20,21,22,23,24,25,26,27,28,29,30,31,32], and the references cited therein.

    Despite the success in the study of the existence and the speed determinacy of traveling waves to the model (1.1), the local and global stability of the traveling waves in the monostable still remains unsolved. In this paper, mainly inspired by the ideas in [18], we devote our attention to finishing the issue of stability for the steady state (¯U,¯V)(z). Firstly, the local stability is discussed on the basis of asymptotic behavior of the traveling waves near the equilibrium and the method of spectrum analysis. When analyzing the eigenvalue problem, we need to overcome the great challenge brought by the high-order nonlinear terms. Then by using the result of local stability, we should choose appropriate weighted Banach space. Furthermore, also by constructing upper and lower solutions, we prove that the solution (U,V) in the moving coordinates converges to the steady state. The new results on the global stability is established in the special weighted function space by using the squeeze theorem.

    This paper are organized as follows. In section 2, we present some preliminaries and main results. In section 3, we prove the local stability of the traveling waves. The global stability of the steady state (¯U,¯V)(z) in a special choice on the weighted function space Lω(R) is investigated in section 4.

    First, linearizing the system (1.4) at (0,0) we obtain

    {c¯U=d¯Uα¯Uσ¯U+μ¯V,c¯V=ϵ¯V+σ¯Uμ¯V+f(0)¯V. (2.1)

    The transformation

    (¯U,¯V)(z)(ξ1eλz,ξ2eλz) (2.2)

    with λ>0 and ξi(i=1,2) being positive constants, allows us to transform the discussion of the asymptotic behaviors at infinitely in variable z to the following eigenvalue problem

    cλξ=(dλ2+αλσμσϵλ2μ+f(0))ξ, (2.3)

    where ξ=(ξ1,ξ2)T. Letting

    A1(λ)=dλ2+αλσ,A2(λ)=ϵλ2μ+f(0),A(λ)=(A1(λ)μσA2(λ)), (2.4)

    then the eigenvalue problem (2.3) is equivalent to r(λ)ξ=A(λ)ξ. Further, we can obtain the eigenvalues

    r±(λ)=A1(λ)+A2(λ)±(A1(λ)A2(λ))2+4σμ2. (2.5)

    As we know, the principal eigenvalue is greater than or equal to the real part of matrix eigenvalues. According to [33], the principal eigenvalue of matrix A(λ) is

    r(λ)=r+(λ), (2.6)

    where r+ is real number and for any λ(0,+),

    r+(λ)=12[A1(λ)+A2(λ)+(A1(λ)+A2(λ))24(A1(λ)A2(λ)σμ)]=12[(d+ϵ)λ2+αλσμ+f(0)+[(dϵ)λ2+αλσ+μf(0)]2+4σμ]>0. (2.7)

    Actually, through the classified discussion of the positive and negative of A1and A2, r+(λ)>0 holds true.

    Lemma 2.1. ([1], see Section 2) r(λ), defined in (2.7), is a real, continuous, and convex function with respect to λR. Assume that

    c0=infλ(0,)r(λ)λR+, (2.8)

    where c0 is called the minimal wave speed, then we can obtain that the equation cλ=r(λ) has

    no solution, if c<c0;

    a unique solution λ(c0), if c=c0;

    two solutions λ1(c) and λ2(c) with λ1(c)<λ2(c), if c>c0.

    Based on the above analysis, we could give the asymptotic behaviors as z with the following lemma.

    Lemma 2.2. ([1], see Section 2) The asymptotic behaviors of the traveling wave solution (¯U(z),¯V(z)) (see (2.1)) can be represented as follows:

    (¯U¯V)=C1(ξ1(λ1(c))ξ2(λ1(c)))eλ1(c)z+C2(ξ1(λ2(c))ξ2(λ2(c)))eλ2(c)z (2.9)

    with C1>0 or C1=0,C2>0, where the eigenvectors corresponding to eigenvalues λi(c)(i=1,2) are

    (ξ1(λi(c))ξ2(λi(c)))=(μcλi(c)A1(λi(c)))or(cλi(c)A2(λi(c))σ). (2.10)

    The proof of the above lemmas can refer to the proof of Theorem 1 in [18], which will not be repeated here.

    Before stating our main results, let us make the following notation.

    Notation: Throughout the paper, Lp are function spaces defined by using a natural generalization of the p-norm for finite dimensional vector spaces. The weighted Lebesgue space Lpω with 1p< can be expressed by

    Lpω={g(z):g(z)ω(z)Lp(R)} (2.11)

    with the norm

    g(z)Lpω=(|g(z)|p|ω(z)|pdz)1p, (2.12)

    where the weighted function

    ω(z)={eβ(zz0),z>z0,1,zz0 (2.13)

    for some positive constants z0 and β.

    With the introduction above, we state our main conclusions for two theorems.

    Theorem 1. (Local stability) For any c>c0, the wavefront (¯U,¯V)(z) is locally stable in the weighted function space Lpω if β(λ1,λ2) to be chosen.

    Theorem 2. (Global stability) Assume that c>c0, λ1<β<λ2 and the initial data U(z,0)=U0(z) and V(z,0)=V0(z) satisfy

    (0,0)(U0,V0)(z)(μσ,1), zR,lim infz(U0,V0)(z)>(0,0),

    and

    |U0(z)¯U(z)|Lω(R), |V0(z)¯V(z)|Lω(R).

    Then (1.7) with initial data admits a unique solution (U,V)(z,t) satisfying

    (0,0)(U,V)(z,t)(μσ,1), (z,t)R×R+.

    Moreover, the inequalities

    supzR|U(z,t)¯U(z)|keηt, t>0

    and

    supzR|V(z,t)¯V(z)|keηt, t>0

    hold for some positive constants k and η.

    The purpose in this section is to apply the weighted energy method and spectral analysis to prove the local stability of the traveling waves presented in Theorem 1.

    First, we assume that

    U(z,t)=¯U(z)+δψ1(z)eχt,V(z,t)=¯V(z)+δψ2(z)eχt, (3.1)

    where δ1, χ is a parameter, and ψ1(z) and ψ2(z) are real functions. Substituting (U(z,t),V(z,t)) into system (1.7), and linearizing the new system with respect to (¯U,¯V)(z), we obtain the following spectral problem:

    χΨ=LΨ:=DΨ+CΨ+BΨ, (3.2)

    where

    Ψ=(ψ1,ψ2)T,D=(d00ϵ),C=(cα00c),B=(σμσμ+f(¯V)), (3.3)

    where the sign of the maximal real part to the spectrum of the operator L is related to the local stability of the traveling wave solution. To proceed, we set

    Ψ=(ψ1ψ2)=(ωϕ1ωϕ2), (3.4)

    where ϕ1 and ϕ2 are the functions in the space Lp, ω has been defined in (2.13) with λ1<β<λ2. Substitute (3.4) into (3.2) to obtain a new spectral problem as:

    χΦ=LωΦ:=DΦ+HΦ+JΦ, (3.5)

    where Φ=(ϕ1,ϕ2)T, D is defined in (3.3) and

    H=(cα+2dωω00c+2ϵωω),J=(dωω+(cα)ωωσμσϵωω+cωωμ+f(¯V)). (3.6)

    Since H(z) and J(z) are bounded real matrix functions and ω, ω, ω exist as z±, we define

    H±=limz±H(z),J±=limz±J(z). (3.7)

    Further, we analyze the position of the essential spectrum for the operator Lω and give the following lemma.

    Lemma 3.1. ([34], see Theorem A.2) If we define

    S±={χ|det(θ2D+iθH±+J±χI)=0,<θ<}, (3.8)

    then the essential spectrum of Lω is contained in the union of the regions inside or on the curves S+ and S, which are on the left-half complex plane, when the condition λ1<β<λ2 is satisfied.

    Proof. We prove the lemma 3.1 for two cases with respect to z.

    Case 1. When z+ in (3.6), we have

    H+=limz+H(z)=(cα2dβ00c2ϵβ),J+=limz+J(z)=(dβ2(cα)βσμσϵβ2cβμ+f(0)). (3.9)

    Therefore, the equation det(θ2D+iθH++J+χI)=0 has two solutions, i.e.

    χ1,2=12{(E+G)+i(F+K)±[(EG)+i(FK)]2+4σμ} (3.10)

    with E=dθ2+A1(β)cβ, F=(cα2dβ)θ, G=ϵθ2+A2(β)cβ, K=(c2ϵβ)θ, where A1 and A2 have been noted in (2.4). Assume that

    S+,1={χ1|<θ<},S+,2={χ2|<θ<}. (3.11)

    By Euler's formula, we obtain the real parts of the eigenvalues χ1,2 as follows:

    Re(χ1)=12[(E+G)+12(((EG)2+(FK)2+4σμ)216(FK)2σμ+(EG)2(FK)2+4σμ)],Re(χ2)=12[(E+G)12(((EG)2+(FK)2+4σμ)216(FK)2σμ+(EG)2(FK)2+4σμ)]. (3.12)

    Obviously, Re(χ1)>Re(χ2). By a simple calculation, one can obtain that

    Re(χ1)<12[(E+G)+(EG)2+4σμ]=12[(d+ϵ)θ2+A1(β)+A2(β)+(dϵ)θ2+(A1(β)A2(β))2+4σμ]cβ12[A1(β)+A2(β)+(A1(β)A2(β))2+4σμ]cβ=r+(β)cβ<0. (3.13)

    since the formulas (2.7 and 2.8) and λ1<β<λ2 are satisfied for c>c0. That is to say S+=S+,1S+,2 is on the left-half complex plane.

    Case 2. When z, it follows that

    H=limzH(z)=(cα00c),J=limzJ(z)=(σμσμ+f(0)). (3.14)

    Similar to Case 1, solving the equation det(θ2D+iθH+JχI)=0, we obtain

    χ3,4=12[(M+P)+i(N+Q)±((MP)+i(NQ))2+4σμ], (3.15)

    where M=dθ2σ, N=(cα)θ, P=ϵθ2μ+f(1), Q=cθ. Further, Euler's formulas allow us to obtain the maximal real part of χ3,4 as

    Re(χ3)=12[(M+P)+12(((MP)2+(NQ)2+4σμ)216(NQ)2σμ+(MP)2(NQ)2+4σμ)]12[(M+P)+(M+P)24(MPσμ)]<0, (3.16)

    because of M+P<0 and MPσμ>0. It means that S={χ3|<θ<} {χ4|<θ<} is also on the left-half complex plane.

    By the above analysis, the essential spectrum of the operator Lω is on the left-half complex plane.

    If the sign of the real part of the principal eigenvalue in the point spectrum (3.2) is negative, the traveling wave solutions is locally stable. So, we need the following step to check the sign of the principal eigenvalue to ensure the result of local stability.

    Finally, we judge the sign of the principal eigenvalue in the point spectrum to prove the local stability. We first discuss the asymptotic behavior of the system (1.4) as z. By linearizing the system (1.4) at (μσ,1), we have

    {c¯U=d¯Uα¯Uσ¯U+μ¯V,c¯V=ϵ¯V+σ¯Uμ¯V+f(1)(¯V1). (3.17)

    Let

    (¯U,¯V)(z)=(μσξ3eλz,1ξ4eλz) (3.18)

    with λ>0 and ξ3, ξ4 being positive constants. Substituting (3.18) into (3.17), it follows that

    (dλ2+(cα)λσμσϵλ2+cλμ+f(1))ξ=0, (3.19)

    where ξ=(ξ3,ξ4)T. Suppose that λi(i=3,4,5,6) are the eigenvalues to the left matrix of Eq (3.19). According to the Vieta's theorem, we know these four eigenvalues are two positive numbers and two negative numbers. Without loss of generality, we assume that λ3>λ4>0>λ5>λ6 for c>0. Then the wave profile has the following asymptotic behaviors:

    (¯U¯V)=(μσ1)C3(ξ3(λ4)ξ4(λ4))eλ4zC4(ξ3(λ3)ξ4(λ3))eλ3z,asz (3.20)

    with C3>0 or C3=0,C4>0.

    To associate with (3.2), we consider the following system

    ut=Duzz+Cuz+Bu, (3.21)

    where u(z,t)=(u1(z,t),u2(z,t))T and D, C, B are defined in (3.3). For a given solution semiflow Qt=u(z,t,ψ) of (3.21) with any given initial data ϕLp, we denote by eχtΨ the solution of (3.21). It is easy to see that Qt is compact and strongly positive. By the Krein-Rutman theorem (see, e.g., [35]), Qt has a simple principal eigenvalue χmax with a strongly positive eigenvector, and all other eigenvalues must satisfy eχtΨ<eχmaxtΨ.

    Next, we shall discuss the eigenvalue χ for two cases as χ=0 and χ>0.

    Case 1. When χ=0, it can be directly calculated that (¯U,¯V)(z) is the corresponding positive eigenvector derived form (3.2), where the asymptotic behavior of the solution of (2.1), i.e. (¯U,¯V)(z)(C1eλ1z,C1eλ1z), C1>0, as z. Although it could be check that the positive eigenvector (¯U,¯V)(z) is not belong to the weighted space Lpω since λ1<β<λ2.

    Case 2. In this case, we discuss the eigenvalue χ>0 to produce a contrary for two cases with respect to z, namely, z and z. Assume that χ>0 and ΨLpω. Then we know that the minimal positive eigenvalue of the Ψ is larger than β. Since ¯Ψ(z)=(¯U,¯V)(z) is a positive solution of (3.21), it follows that ¯Ψ(z)>Ψ as z.

    When z, assume that Ψ(z) has the asymptotic behavior as keλz for some positive k and λ. Substituting it into the spectral problem (3.5), we obtain the characteristic equation in eigenvalue χ as follows:

    |dλ2+(cα)λσχμσϵλ2+cλμ+f(1)χ|=0. (3.22)

    Next, we shall show the relationship between the four roots of (3.22) denoted by ˆλi(i=3,4,5,6) and λi(i=3,4,5,6), namely, ˆλ3>λ3, ˆλ4>λ4, ˆλ5<λ5, ˆλ6<λ6. In fact, when χ>0, the parabolic

    p1:dλ2+(cα)λσχ=0,p2:dλ2+(cα)λσ=0, (3.23)

    have two roots r±p1 and r±p2 with one positive and one negative, respectively. It is easy to see that rp1<rp2 and r+p1<r+p2. Similar discussion to

    p3:ϵλ2+cλμ+f(1)χ=0,p4:ϵλ2+cλμ+f(1)=0, (3.24)

    we also obtain that rp3<rp4 and r+p3<r+p4. Further, we know that rp1, r+p1, rp3 and r+p3 are also the roots of

    p5:(dλ2+(cα)λσχ)(ϵλ2+cλμ+f(1)χ)=0 (3.25)

    and rp2, r+p2, rp4 and r+p4 are the roots of

    p6:(dλ2+(cα)λσ)(ϵλ2+cλμ+f(1))=0. (3.26)

    That is to say that the positive roots of p5 related to p6 are moving right. Therefore, when the curves of p5 and p6 intersects with the same line y=μσ, the points are denoted by ˆλi(i=3,4,5,6) and λi(i=3,4,5,6), respectively. And the points of intersection satisfy ˆλ3>λ3, ˆλ4>λ4, ˆλ5<λ5, ˆλ6<λ6. In other words, the positive roots of λ are increasing with respect to χ. This indicates that ¯Ψ(z)k1eλ3z and Ψ(z)k2eˆλ3z as z.

    Thus, we choose ¯k sufficient large such that ¯k¯Ψ|Ψ|. By the comparison principal for the system (3.21), it must be ¯k¯Ψ(z)|Ψ|eχt. Hence the assumption χ>0 is incorrect. This implies that the real parts of all eigenvalues χ of (3.5) should not be positive for ΨLpω.

    The proof is complete.

    In this section, we will prove Theorem 2. In order to realize it, we need the following conclusion.

    (Comparison principle) Let (U+,V+)(z,t) and (U,V)(z,t) be the solutions of (1.7) with respect to the initial values

    U+0(z)=max{U0(z),¯U(z)},V+0(z)=max{V0(z),¯V(z)},U0(z)=min{U0(z),¯U(z)},V0(z)=min{V0(z),¯V(z)} (4.1)

    respectively; namely

    {U±t=dU±zz+(cα)U±zσU±+μV±,V±t=ϵV±zz+cV±z+σU±μV±+f(V±),(U±,V±)(z,0)=(U±0,V±0)(z). (4.2)

    It holds that

    (0,0)(U0,V0)(z)(U0,V0)(z)(U+0,V+0)(z)(μσ,1),(0,0)(U0,V0)(z)(¯U,¯V)(z)(U+0,V+0)(z)(μσ,1). (4.3)

    By comparison principle, we have

    (0,0)(U,V)(z,t)(¯U,¯V)(z,t)(U+,V+)(z,t)(μσ,1),(z,t)R×R+,(0,0)(U,V)(z,t)(¯U,¯V)(z)(U+,V+)(z,t)(μσ,1),(z,t)R×R+. (4.4)

    If both (U+,V+)(z,t) and (U,V)(z,t) converge to (¯U,¯V)(z), then the squeezing theorem ensures the global stability of system (1.7) stated in (2.2).

    Lemma 4.1. Under the conditions given in Theorem 2, (U+,V+)(z,t) converges to (¯U,¯V)(z).

    Proof. To begin with, we suppose that

    R(z,t)=U+(z,t)¯U(z),S(z,t)=V+(z,t)¯V(z),(z,t)R×R+, (4.5)

    which satisfies the initial values

    R(z,0)=U+0(z)¯U(z),S(z,0)=V+0(z)¯V(z). (4.6)

    By (4.2) and (4.4), we have

    (0,0)(R,S)(z,t)(μσ,1),(z,t)R×R+. (4.7)

    Through (1.4) and (4.3), the following inequality

    (RS)tD(RS)zz+C(RS)z+B0(RS) (4.8)

    holds for the matrices C and D defined in (3.3), and

    B0=(σμσμ+f(0)). (4.9)

    Then we consider the convergence of (R,S)(z,t) for two cases with respect to z.

    Case 1. When z>z0, we first define

    (RS)(z,t)=eβ(zz0)(¯R¯S)(z,t), (4.10)

    so as to investigate the stability in weighted function space Lω for all (z,t)R×R+, where ¯R and ¯S are the functions in L(R), the weighted function ω(z) is defined in (3.3). Substituting (4.10) into (4.8), we obtain

    (¯R¯S)tD(¯R¯S)zz+(cα2dβ00c2ϵβ)(¯R¯S)z+(A1(β)cβμσA2(β)cβ)(¯R¯S):=(L1(¯R,¯S)L2(¯R,¯S)), (4.11)

    where A1(β) and A2(β) are defined in (2.4).

    Further, we choose

    ¯R1(z,t)=k1ζ1eη1t,¯S1(z,t)=k1ζ2eη1t,(z,t)R×R+, (4.12)

    for some positive constants k1 and η1, where (ζ1,ζ2)T=(ζ1(β),ζ2(β))T is the eigenvector of matrix M1. In fact,

    M1=(A1(β)cβμσA2(β)cβ) (4.13)

    which has the eigenvalue

    ¯λ1=12(A1(β)+A2(β)+(A1(β)A2(β))2+4σμ)cβ.

    The associated eigenvectors can be found by direct calculation as follows:

    ζ1(β)=μ,ζ2(β)=12(A1(β)+A2(β)+(A1(β)A2(β))2+4σμ). (4.14)

    It is not hard to check that, for λ1<β<λ2, ζ1(β), ζ2(β) are positive and ¯λ1 is negative. Moreover, one can obtain

    L1(¯R1,¯S1)=¯λ1¯R1<0,L2(¯R1,¯S1)=¯λ1¯S1<0. (4.15)

    Thus, we can choose η1¯λ1 such that

    (¯R1¯S1)t=η1k1(ζ1ζ2)eη1t(L1(¯R1,¯S1)L2(¯R1,¯S1)). (4.16)

    Once we choose k1maxzR{¯R(z,0)ζ1,¯S(z,0)ζ2} to make

    (¯R1,¯S1)(z,0)=(k1ζ1,k2ζ2)(¯R,¯S)(z,0),

    then by comparison principle on unbounded domain, (z,t)R×R+, we obtain

    (R,S)(z,t)=eβ(zz0)(¯R,¯S)(z,t)k1(ζ1,ζ2)eβ(zz0)η1t. (4.17)

    Therefore, the above inequality holds for any fixed point z0. That is to say, for z>z0, (R,S)(z,t) converges to (0,0).

    Case 2. When zz0, the system marked (R,S)(z,t) can be shown as

    (RS)t=D(RS)zz+C(RS)z+(σμσμ)(RS)+(0f(S+¯V)f(¯V)). (4.18)

    Here, f(S+¯V)f(¯V) is a second-order differentiable function that can be expressed as f(1)S+h(S) (h(S) also is second-order differentiable and h(0)=0) as (¯U,¯V)(z)(μσ,1). Consequently, we need to select z0, so that

    (RS)tD(RS)zz+C(RS)z+(σμσμ+f(1)+ε1)(RS)+(0h(S)) (4.19)

    for some given enough small positive ε1 and ε1μf(1).

    From the following autonomous system

    (ˆRˆS)t=(σμσμ+f(1)+ε1)(ˆRˆS)+(0h(ˆS)) (4.20)

    with the initial data

    ˆR(0)¯R(z,0),ˆS(0)¯S(z,0),zR, (4.21)

    we know that the solution (ˆR,ˆS)(t) of (4.20) is an upper solution of system (4.18).

    Next, for the convergence of (R,S)(z,t), it is sufficient to prove that (ˆR,ˆS)(t) converges to (0,0) as t tends to . A direct calculation shows that the eigenvalues ˆλ1 and ˆλ2 of the Jacobian matrix of system (4.20) at the fixed point (0,0) are less than zero. Therefore, (0,0) is a stable node. From the phase plane analysis, any manifold in ˆRˆS-space for any initial value (ˆR,ˆS)(0) in region [0,μσ]×[0,1] converges to the origin (0,0). Then, as t, we define

    (ˆR,ˆS)=ˆk1(ˆC1,ˆC2)eˆλ1t (4.22)

    with ˆk1>0 and (ˆC1,ˆC2)T is the eigenvector of the Jacobian matrix respect to ˆλ1. We can choose ˆk1 large enough and ˜λ1=min{η1,ˆλ1} to be used that we have

    (R,S)(z0,t)k1(ζ1,ζ2)eη1tˆk1(ζ1,ζ2)e˜λ1t (4.23)

    at the boundary z=z0. As a result, by comparison on the domain (,z0]×[0,), see Lemma 3.2 in [36], for all (z,t)(,z0]×R+,

    (R,S)(z,t)ˆk1(ζ1,ζ2)e˜λ1t, (4.24)

    then (R,S)(z,t) converges to (0,0) for z(,z0].

    Lemma 4.2. Under the conditions given in Theorem 2.2, (U,V)(z,t) converges to (¯U,¯V)(z).

    Proof. Let

    X(z,t)=¯U(z)U(z,t),Y(z,t)=¯V(z)V(z,t),(z,t)R×R+ (4.25)

    to satisfy the initial values

    X(z,0)=¯U(z)U0(z),Y(z,0)=¯V(z)V0(z). (4.26)

    Similar discussion as lemma 4.1, (z,t)R×R+, we have

    (0,0)(X,Y)(z,t)(μσ,1), (4.27)

    and

    (XY)t=D(XY)zz+C(XY)z+B1(XY)+(0f(V+Y)f(V)), (4.28)

    where the second-order matrices C and D are consistent with those in (3.3) and

    B1=(σμσμ). (4.29)

    Now we also consider the convergence for the following two cases with respect to z.

    Case 1. (z>z0). By using the fact X<¯U and Y<¯V, we can verify that η2>0 and k2eβ(zz0)maxzR{X(z,0)ζ1,Y(z,0)ζ2} bring

    (X,Y)(z,t)k2(ζ1,ζ2)eη2t (4.30)

    for all (z,t)R×R+.

    Case 2. (zz0). Define (ˆX,ˆY)(z,t) as the solution of the following autonomous system

    (ˆXˆY)t=(σμσμ+f(1)+ε1)(ˆXˆY)+(0h(ˆY)) (4.31)

    with the initial data

    ˆX(0)¯X(z,0),ˆY(0)¯Y(z,0),zR. (4.32)

    Then (ˆX,ˆY) is an upper solution to the system

    (XY)t=D(XY)zz+C(XY)z+(σμσμ)(XY)+(0f(¯V)f(¯VY)). (4.33)

    Phase plane analysis shows that for any initial values in region [0,μσ]×[0,1], (ˆX,ˆY)(t) always converges to (μσ,1). Similar to the proof of case 2 in Lemma 4.1, for some positive numbers ˆk2 and ˜λ2, we obtain

    (X,Y)(z,t)ˆk2(ζ1,ζ2)eˆλ2t,(z,t)(,z0]×R+. (4.34)

    Now we will use the squeezing theorem to obtain the conclusion of the Theorem (2.2).

    By the inequalities (4.4), for any (z,t)R×R+, it gives

    |X(z,t)||U(z,t)¯U(z)||R(z,t)|,|Y(z,t)||V(z,t)¯V(z)||S(z,t)|. (4.35)

    From Lemmas 4.1 and 4.2 and the squeezing theorem, k>0,η>0 so that (z,t)R×R+,

    |U(z,t)¯U(z)|keηt,|V(z,t)¯V(z)|keηt. (4.36)

    To sum up, Theorem 2 is proved completely.

    Y. Tang and C. Pan: Methodology; H. Wang and C. Pan: Validation; Y. Tang and C. Pan: Formal analysis; C. Pan: Writing-original draft preparation; Y. Tang and H. Wang: Writing-review and editing; C.Pan and H. Wang: Supervision; C. Pan: Project administration; C. Pan: Funding acquisition. All authors have read and agreed to the published version of the manuscript.

    This work is supported by the National Natural Science Foundation of China grant (No.11801263).

    The authors declare no conflict of interest.

    [1] Upadhyay A, Srivastava S (2010) Evaluation of multiple plant growth promoting traits of an isolate of Pseudomonas fluorescens strain Psd. Indian J Exp Biol 48: 601–609.
    [2] Patten CL, Glick BR (2002) Role of Pseudomonas putida indoleacetic acid in development of the host plant root system. Appl Environ Microbiol 68: 3795–3801. doi: 10.1128/AEM.68.8.3795-3801.2002
    [3] Döbereiner J (1992) History and new perspectives of diazotrophs in association with non-leguminous plants. Symbiosis 13: 1–13.
    [4] American Public Health Association (1998) Standard methods for examination of water and waste water. Washington DC, USA.
    [5] Calvo P, Zúñiga D (2010) Caracterización fisiológica de cepas de Bacillus spp. aisladas de la rizósfera de papa (Solanum tuberosum). Ecol Apl 9: 31–39.
    [6] Ehmann A (1977) The Van Urk-Salkowski reagent a sensitive and specific chromogenic reagent for silica gel thin-layer chromatographic detection and identification of indole derivatives. J Chromatogr 132: 267–276. doi: 10.1016/S0021-9673(00)89300-0
    [7] Gordon SA, Weber RP (1951) Colorimetric estimation of indolacetic acid. Plant Physiol 26: 192–195. doi: 10.1104/pp.26.1.192
    [8] Glick BR (1995) The enhancement of plant growth by free-living bacteria. Can J Microbiol 41: 109–117. doi: 10.1139/m95-015
    [9] Alikhani H, Saleh-Rastin N, Antoun H (2006) Phosphate solubilization activity of rhizobia native to Iranian soils. Plant Soil 287: 35–41. doi: 10.1007/s11104-006-9059-6
    [10] Nautiyal CS (1999) An efficient microbiological growth medium for screening phosphate solubilizing microorganisms. FEMS Microbiol Lett 170: 265–270. doi: 10.1111/j.1574-6968.1999.tb13383.x
    [11] Nguyen C, Yan W, Tacon F, et al. (1992) Genetic variability of phosphate solubilizing activity by monocaryotic and dicaryotic mycelia of the ectomycorrhizal fungus Laccaria bicolor (Maire) PD Orton. Plant Soil 143: 193–199. doi: 10.1007/BF00007873
    [12] Ogata K, Zúñiga D (2008) Estudio de la microflora de la rizósfera de Caesalpinia spinosa en la provincia de Huánuco. Zonas Áridas 12: 191–208.
    [13] Lalitha MK (2005) Manual on antimicrobial susceptibility testing. Indian Assoc of Med Microbiol 14–16.
    [14] Sabry SA, Ghozlan HA, Abou-Zeid DM (1997) Metal tolerance and antibiotic resistance patterns of a bacterial population isolated from sea water. J Appl Microbiol 82: 245–252. doi: 10.1111/j.1365-2672.1997.tb02858.x
    [15] Weisburg WG, Barns SM, Pelletier DA, et al. (1991) 16S ribosomal DNA amplification for phylogenetic study. J Bacteriol 173: 697–703. doi: 10.1128/jb.173.2.697-703.1991
    [16] Larkin MA, Blackshields G, Brown NP, et al. (2007) Clustal W and Clustal X version 2.0. Bioinformatics 23: 2947–2948. doi: 10.1093/bioinformatics/btm404
    [17] Tamura K, Stecher G, Peterson D, et al. (2013) MEGA6: Molecular evolutionary genetics analysis version 6.0. Mol Biol Evol 30: 2725–2729. doi: 10.1093/molbev/mst197
    [18] MINAM, Decreto supremo 002-2013 Environmental Quality Standards of agricultural soils, El Peruano, 2012. Available from: http://www.minam.gob.pe/calidadambiental/wp-content/ uploads/sites/22/2013/10/D-S-N-002-2013-MINAM.pdf.
    [19] Bric JM, Bostock RM, Silvestone SE (1991) Rapid in situ assay for indolacetic acid production by bacteria immobilized on a nitrocellulose membrane. Appl Environ Microbio 57: 535–538.
    [20] Spaepen S, Vanderleyden J, Remans R (2007) Indole-3-acetic acid in microbial and microorganism-plant signaling. FEMS Microbiol Rev 31: 425–448. doi: 10.1111/j.1574-6976.2007.00072.x
    [21] Subramanian P, Kim K, Krishnamoorthy R, et al. (2016) Cold stress tolerance in psychrotolerant soil bacteria and their conferred chilling resistance in tomato (Solanum lycopersicum Mill.) under low temperatures. Plos One 11: 1–17.
    [22] Zúñiga-Dávila D, Tolentino J, García M Pérez W, et al. (2011) Characterization of rhizospheric bacteria isolated from maca (Lepidium meyenii Walp.) in the highlands of Junin-Peru, In: Mendez-Vilas A. Editor, Microorganisms in Industry and Environment: From scientific and industrial research to consumer products, Singapore: World Scientific Pub Co, 21–25.
    [23] Trivedi P, Pandey A (2007) Low temperature phosphate solubilization and plant growth promotion by psychrotrophic bacteria, isolated from the Himalayan region. Res J Microbiol 2: 452–461.
    [24] Mishra V, Gupta A, Kaur P, et al. (2016) Synergistic effects of Arbuscular mycorrhizal fungi and plant growth promoting rhizobacterial. Int J Phytoremediation 18: 697–703. doi: 10.1080/15226514.2015.1131231
    [25] Kloepper JW, Zablotowicz RM, Tipping EM, et al. (1991) Plant growth promotion mediated by bacterial rhizosphere colonizers, In: Keister DL, Cregan PB, Editors, The rhizosphere and plant growth, Netherlands: Kluwer Academic Publishers, 315–326.
    [26] Gul B, Hameed A, Weber DJ, et al. (2016) Assessing seed germination responses of great basin halophytes to various exogenous chemical treatments under saline conditions, In: Sabkha Ecosystems Volume V: The Americas Tasks for Vegetation Science, Switzerland: Springer International Publishing, 85–104.
    [27] Acosta JM, Bentivegna DJ, Panigo ES, et al. (2012) Factors affecting seed germination and emergence of Gomphrena perennis. Weed Res 53: 69–75.
    [28] Babich H, Stotizky G (1976) Sensitivity of various bacteria, including actinomycetes, and fungi to cadmium and the influence of pH on sensitivity. Appl Environ Microbiol 33: 681–695.
    [29] Franklin NM, Stauber JL, Markich SJ, et al. (2000) pH-dependent toxicity of copper and uranium to a tropical freshwater alga (Chlorella sp.). Aquat Toxicol 48: 275–289. doi: 10.1016/S0166-445X(99)00042-9
    [30] Sandrin TR, Maier RM (2002). Effect of pH on cadmium toxicity, speciation, and accumulation during naphthalene biodegradation. Environ Toxicol Chem 21: 2075–2079. doi: 10.1002/etc.5620211010
    [31] Rouch DA, Lee TOB, Morby AP (1995) Understanding cellular responses to toxic agents: a model for mechanisms-choice in bacterial resistance. J Ind Microbiol 14: 132–141. doi: 10.1007/BF01569895
    [32] Kumar V, Singh S, Bhadrecha P, et al. (2015). Bioremediation of heavy metals by employing resistant microbial isolates from agricultural soil irrigated with industrial waste water. Orient J Chem 31: 357–361. doi: 10.13005/ojc/310142
    [33] Ochiai EI (1987) General principles of biochemistry of the elements, New York: Plenum Press, 381–382.
    [34] Gadd GM (1992) Metals and microorganism: a problem of definition. FEMS Microbiol Lett 100: 197–204. doi: 10.1111/j.1574-6968.1992.tb05703.x
    [35] Kumar V, Singh S, Singh J, et al. (2015). Potential of plant growth promoting traits by bacteria isolated from heavy metal contaminated soils. Bull Environ Contam Toxicol 94: 807–814. doi: 10.1007/s00128-015-1523-7
    [36] Azario R, Salvarezza S, Ibarra A, et al. (2010) Efecto del cromo hexavalente y trivalente sobre el crecimiento de Escherichia coli ATCC 35218. Información Tecnológica 21: 51–56.
    [37] Zahid M, Abbasi MK, Hameed S, et al. (2015) Isolation and identification of indigenous plant growth promoting rhizobacteria from Himalayan region of Kashmir and their effect on improving growth and nutrient contents of maize (Zea mays L.). Front Microbiol 6: 1–10.
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