Phosphorus (P) plays a vital role in global crop production and food security. The aim of this paper is to describe mathematically the rhizosphere phosphorus cycle in order to predict and to estimate the concentrations of diferent phosphorus forms. Our model is a system of first order diferential equations for which we show the existence, uniqueness, positivity and boundedness of solution. We also derive one equilibrium point of the model and we show its asymptotic stability. Furthermore numerical simulations are done to show the behavior of soil phosphorus content in time and to confirm the stability of the equilibrium point. Finally, we investigate which plant uptake rate can maintain the amount of available phosphorus in rhizosphere.
The phosphorus cycle shows the forms of phosphorus in rhizosphere and the pathway by which phosphorus may taken up by plants. It can be summarised as follows dead plants and animals constitute soil organic matter, which can be degraded and transformed into organic phosphorus by soil microorganisms. Organic phosphorus is mineralized by soil microorganisms to release assimilable phosphorus (orthophosphate ions). Assimilable phosphorus with immobilization returns to organic form, and is converted into mineral phosphorus by adsorption or precipitation. By desorption or solubilization, mineral phosphorus become assimilable phosphorus. Plants explore soil through their roots and absorb assimilable phosphorus. Animals eat phosphorus in plant leaves and fruits. After death, these become sources of organic phosphorus in the rhizosphere.
Phosphorus occurs in soil in both organic and inorganic forms [
The availability of phosphorus to plants depends on the mechanisms that control its concentration in solution. These mechanisms are physico-chemical, biochemical and biological in nature, they are degradation of organic matter, mineralization of organic phosphorus and absorption, adsorption, immobilization, precipitation of assimilable with desorption and solubilization of mineral phosphorus. ∙ ∙ ∙ ∙
Degradation of organic matter is a process by which microorganisms fungi and bacteria break down dead plants and animals to release organic phosphorus.
Mineralization of organic phosphorus is a process by which microorganisms release enzymes
like phosphatase or phosphohydrolase, phytases, phosphonatase to convert organic phosphorus
into assimilable phosphorus [
Absorption of available P is a process by which the plants explore soil through its roots and uptake an available phosphorus for their growth, health and development.
Adsorption of available P is a chemical fixation of available phosphorus by soil components such
as iron (Fe) and aluminum (Al), which makes phosphorus unavailable to plants. Throughout
adsorption available phosphorus can be converted to mineral phosphorus.
∙
∙
∙
∙ Immobilization of available P is a process by which available phosphorus is converted into organic
phosphorus by certain soil microorganisms. It occurs when microorganisms consume available
phosphorus , these microorganisms die later and produce organic phosphorus which is unavailable
for plant uptake [
Precipitation of assimilable phosphorus is a process by which metal ions such as Al3+ and
Fe3+ (acidic soils) and Ca2+ (neutral to alkaline soils) react with phosphate ions in the soil
solution to form phosphate minerals such as Ca phosphate dicalcium or octacalcium phosphate,
hydroxyapatite, Fe and Al phosphate such as strengite, vivianite, variscite and plumbogummite
group minerals [
Desorption of mineral phosphorus is a process by which mineral phosphorus is converted to assimilable phosphorus for plants uptake, it is a reverse process of adsorption.
Solubilization of mineral phosphorus is a process by which mineral phosphorus is converted
to assimilable phosphorus by microorganisms actions. It results from mechanisms such as the
production of mineral-dissolving compounds and inorganic acids as well as the release of enzymes
or enzymolyses by microorganisms [
Diferent pools of phosphorus are generally distinguished in soil. This study focuses on the phosphorus
cycle that was developed in [
The parameters include mineralization of organic phosphorus rate , immobilization of assimilable phosphorus rate , degradation of organic matter rate , desorption of mineral phosphorus rate , solubilization of mineral phosphorus , adsorption of assimilable , precipitation of assimilable and uptake of assimilable phosphorus rate , all parameters range between 0 and 1. To estimate the concentrations of diferent pools of phosphorus, we build four diferential equations described by the following compartmental model:
[] ()
︀] () + ︀[ ]︀ () + = ( + ) ︀[ ]︀ () − ( + )︀[
︀] () = [︀ ]︀ () − ︀[ ]︀ (). (1) (2) (4) By combining the four diferential equations, we obtain the following compartmental system : We set () = (︀ [](), [](), [](), []())︀ so that the system (5) can be written as the following linear homogeneous diferential equation of the first order :
︀] () + ︀[ ]︀ () + = − ( + )︀[ ︀] () + ( + )
︀[ ]︀ () = [︀ ]︀ () − ︀[ ]︀ () with (5) (6) (7) (8) (9) 0 0
︂{
Let Ω be an open subset of R4. We consider the Cauchy problem defined by: ⎛
− = ⎜⎜⎝ 0 0 ⎛⎞ 0 0 0 ⎟ ⎟ ⎠ ()
(0) = = ( ()) 0 where: • 0 ∈ Ω and 0 ∈ R+ • : Ω →− • : R+ →−
R
4
R4 is defined by (6)
Proposition 4.1. [
The solution : R+
→− and is generaly defined by :
Since the function is 1 on R4. Thereby, for any non-negative initial condition the Cauchy problem
(9) associated to the diferential equation (5) admits a unique solution [
R4 of the diferential equation (9) with initial condition (0) = 0 is unique ∫︁
0 () = (− 0)0 + (− ), ∀ ∈ [0; +∞[ (10)
Since system (5) represents an amount system, it is important that the solution remains non-negative values. Thus, we must prove the positivity of the solution.
Definition. .
A square real matrix = ( )1≤ ,≤ is called a Metzler-matrix if all its of-diagonal entries are nonnegative, ≥ 0 for ̸= , = 1, · · · , .
Lemma 1 (Metzler [
Let ∈ R× , then ≥ 0 for ≥
0 if and only if is a Metzler-matrix.
Theorem 1. Let (0) be a positive constant vector and be a Metzler matrix. The solution () of system (9), defined by (10) remains positive forall ≥ 0.
Proof. of Theorem(1)
According to Lemma (1), the matrix is a Metzler-matrix, then for 0 ≥ 0 0, ∀ ≥ 0. Since ≥ 0 and (− ) ≥ 0, ∀ ≥ then ∫︀0 (− ) ≥ we have (− 0)0 ≥
0. we deduce that ∫︁
0 (− 0)0 + (− ) = () ≥ 0.
We find equilibrium points of the system (5) by making its right-hand side equal zero. () = 0 ⇐⇒ ⎧ ⎪− []() + []() = − ⎪ ⎪ ⎪ ⎨− ( + )[]() + ( + )[]() = 0 ⎪[]() + ( + )[]() − ( + + + )[]() = 0 ⎪ ⎪ ⎪⎩[]() − []() = 0 ⇐⇒ ⎧ ⎪− []() + []() + = 0 ⎪ ⎪ ⎪ ⎨− ( + )[]() + ( + )[]() = 0 ⎪[]() = ⎪ ⎪ ⎪⎩2
[]() = We obtain the equilibrium point of our model (5), denoted by where * = (︀ []* , []* , []* , []* )︀ ⎧⎪[]* = ⎪⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪[]* = ⎪⎨ ⎪⎪⎪[]* = (+) ⎪⎪⎪ (+) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩[]* = (︀ + 1)︀ .
(11)
The stability of * = 0 of × linear homogeneous system ˙ () = () depends on the sign of the eigenvalues of matrix .
Lemma 2. [
The origin * = 0 is asymptotically stable equilibrium point if the real parts of all eigenvalues of the matrix are less than zero.
We use the Routh-Hurwitz criterion to find out the sign of the real parts of matrix eigenvalues, given the following characteristic polynomial + 1 − 1 + · · · + where ∈ R, = 1, · · · , .
The coeficients are arranged in descending order of degrees. Thus, the Routh-Hurwitz criterion stated
as follows:
Lemma 3. .[
⃒ 1 3 ⃒⃒ > 0, 1 2 ⃒ Theorem 2. The equilibrium point * = (︀ []* , []* , []* , []* )︀ is asymptotically stable. Proof. From (6), we write the system as : . with
˙ () = () + ⎛−
0 = ⎜⎜⎝ 0
0 − ( + ) ( + ) 0
( + )
− ( + + + )
1
det() = 2( + ) , the determinant of matrix does not equal zero, then this matrix
is invertible. We use the results from [
˙ () = ˙ (), ˙ () = () ( ) = det( − ) then we obtain the following (12) that is equivalent system of (6): In order to use Routh-Hurwitz criterion, we define the characteristic polynome of matrix as : (12) ( ) = ⃒⃒ ⃒ ⃒ ⃒ ⃒⃒ + ⃒ − 0 0 0 0 + ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ( ) = ( + )( 3 + 1 2 + 2 + 3) thus, we obtain where 1 = + + + + + + , 2 = ( + )( + ) +
According to Routh Hurwitz’s criterion, the polynomial ( ) has all its roots with negative real parts. Then the origin * = 0 of the system (12) is asymptotically stable, we deduce that the equilibrium point * = ([]* , []* , []* , []* ) is asymptotically stable. Thus, the theorem has been proved.
we obtain the equivalent system We consider the system (5) and we set : () = () − * , where * is the equilibrium point (11), defined as follow :
Thus, it resuts that :
Lemma 4. ([
From the results in [
(0) = = () 0. () = 0(− 0)
∀ ∈ [0, +∞[ . () − * = 0(− 0) ∀ ∈ [0, +∞[ (13) (14) (15)
The boundedness of the solution (), is given in the next theorem: Theorem 1. If every eigenvalue of has negative real part, then () is uniformly bounded. Therefore the solution () of model (5) is bounded.
If 1, · · · , are the eigenvalues of a × square matrix , the set of associated eigenvectors {1, · · · , } = [1, · · ·
, ] − 1 = diag[ 1, · · · , ].
In this work, the matrix defined by (7) is a square matrix of order 4, with real coeficients. So,
is
diagonalizable because it has 4 distinct eigenvalues. Then, there is an invertible matrix such that
= − 1 where is the diagonal matrix. According to results from [
We deduce that , the solution of the system (13) verify
Given triangle inquality in [
Since < 0 then ‖ ()‖ ≤ ‖0‖. Which implies
‖() − * ‖ ≤ ‖0‖
According to the consequence of the triangle inequality in [
Note that for a vector = (1, · · · , ) ∈ R and a square real matrix = ( )1≤ ,≤ : | ‖()‖ − ‖ * ‖ | ≤ ‖0‖ ∀ ∈ [0, +∞[ ,
‖()‖ ≤ ‖ * ‖ + ‖0‖ ‖‖ = ⎷⎯⎸⎸∑︁ 2 and ‖ ‖ = =1
max ∑︁ | |. 1≤ ≤ =1
In this section, we justify our theoretical results by considering the concentrations of diferent
phosphorus forms in Imeko soil, a region in Nigeria. This region is located less than 30 kilometers from
Ketou, one of a town in Benin where agriculture is significant. According to the study in [
We use transformation rates between stable and labile phosphorus pools presented by [
To identify which uptake rate allows us to maintain an available phosphorus concentration in rhizosphere, we evaluated the percentage of available phosphorus as a function of the uptake rate. Depending on the diferent types of soil considered, the soil achieves some sustainability in phosphorus (P), as follows : • Figure 4: * = (2240.19, 840.07, 70.00, 175.01) • Figure 3: * = (1400.12, 700.06, 140.01, 700.05) • Figure 5: * = (437.53, 87.50, 70.00, 175.01) • Figure 6: * = (295.24, 83.84, 60.87, 132.33) It can be seen that to achieve significant sustainability in phosphorus (P), a significant number of days are also required, specifically : • Figure 4: > 500 days. • Figure 3: > 200 days. • Figure 5: ≈ 70 days.
• Figure 6: ≈ 40 days.
During the time, we observe that the concentrations of organic phosphorus [] and inorganic phosphorus content [] follow a strong increase, and concentrations of diferent phosphorus forms converge to the equilibrium point * = ([]* , []* , []* , []* ) of model (5) which is asymptotically stable.
As the rate of mineralization increases, the soil becomes less fertile in organic phosphorus []. Moreover, the rates of immobilization , precipitation and adsorption do not directly influence the sustainability of the mineral phosphorus [] since the origin which is [] has a small fraction of the total phosphorus in the soil.
Figure 7 shows the efect of the uptake rate on the availability of []. Note that if the uptake rate increases,then the availability of phosphorus [] decreases. To maintain available phosphorus in the soil, it suggest that the uptake rate must be between 0.2 and 0.3.
. In this work, we described a phosphorus dynamics in the rhizosphere using a compartmental mathematical model of the phosphorus cycle with constant transformation rates of phosphorus cycle phenomena. This model is a system of first order diferential equations, a qualitative study was conducted to establish an uniqueness, positivity and boundedness of the solution, and we show that the equilibrium point of the model is asymptotically stable. Computer simulations were performed to justify theoretical results. Finally, we suggest suitable uptake rate values which can not remove assimilable phosphorus in the rhizosphere.
Future research could consider using Artificial Intelligence in particulary Machine Learning to predict the concentration of assimilable phosphorus in soil.
The authors have not employed any Generative AI tools.