CN108256624B

CN108256624B - Root branch prediction method based on group interaction environment influence

Info

Publication number: CN108256624B
Application number: CN201810022160.4A
Authority: CN
Inventors: 李松阳; 李华; 郭小波; 孟鹏来; 郭新亚
Original assignee: Henan Institute of Engineering
Current assignee: Henan Institute of Engineering
Priority date: 2018-01-10
Filing date: 2018-01-10
Publication date: 2020-05-15
Anticipated expiration: 2038-01-10
Also published as: CN108256624A

Abstract

The invention provides a root branch prediction method based on group interaction environment influence, which is used for solving the problem that the influence of root primordium on the root branch action is not considered in the conventional root system modeling method; the method comprises the following steps: constructing a root primordium group taking root primordia as individuals; traversing the root primordium group to obtain a neighbor root primordium set of the root primordium; calculating the influence factor of the perception environmental factors and the environmental change intensity of the root primordium; calculating the win-loss of the interaction of the root primordium and the neighbor root primordium individuals in the neighbor root primordium set by adopting a Parrondo game model; calculating the auxin content and the growth time of the root primordium; and predicting the root branches according to the relationship between the auxin content and the maximum auxin demand. The invention can realize the modulation of the process of developing the root primordium into the root branch according to the change of environmental factors and based on the interaction among the root primordium groups, provides a model support and analysis means for the whole research of plant modeling and simulation, and promotes the application of the plant modeling and simulation to precise agriculture.

Description

Root branch prediction method based on group interaction environment influence

Technical Field

The invention relates to the technical field of computer simulation, belongs to the field of plant simulation by using a computer, and particularly relates to a root branch prediction method based on group interaction environment influence.

Background

Plant modeling and simulation are the growth and development processes of plants in a three-dimensional space under a virtual environment simulated on a computer by applying a virtual reality technology, are the cross research fields of various subjects including botany, ecology, agriculture, computer graphics, mathematics and the like, are the key technologies of precise agriculture, and are also important means for quantitatively researching the growth rule of plants. However, in the past, due to the invisibility of the growing environment of the plant root system and the growing complexity of the plant root system, the modeling and simulation research on the root system has considerable limitation and lags behind the overground part. Root modeling and simulation research opens up a new way for overall cognition of complex life activities of plants, and has important significance for system research on plasticity phenotype quantification of root structures such as root length, surface area and spatial distribution characteristics under dynamically changing soil environment conditions, root function influences such as storage, transmission, space change resource acquisition space, water and nutrition competition and the like, and understanding of the relationship between plant individuals and the adjustment root systems and the overground parts to ensure plant survival and population continuation.

Currently, much related work has been done with respect to root modeling and simulation. One method is a method for constructing a virtual root system based on clear description of the growth of the whole root system, so as to realize simulation of the change of a topological structure and a geometric shape of the root system in space and time, but the method does not consider the physiological process of the root system, such as root elongation, diameter growth, branching and death, and the influence of absorption and transmission of nutrients such as water, nitrogen and the like on the growth of the root system; one method is a method for constructing a virtual root system based on a continuous variation function of root density in space and time, so that root system simulation of different scales from a single root to a root group is realized, but the method does not consider the function of the ecological physiology aspect of the root system; the other method is to combine the root system function and the structure, and simultaneously consider the root system structure, the physiological process and the soil environment condition to construct a virtual root system method, but when the root branches are simulated by the model, one method is to control the generation positions of branch points by defining the branch intervals according to the observation of the root system branches and sequentially generate the root branches; another method controls the generation of branches by defining the branch density. The above method has a great disadvantage: the existing root system modeling and simulation analysis show that the root branch and the elongation behavior determine the root system structure, the elongation behavior is determined by the growth and development of the root tip, the branch behavior is determined by the growth and development of the root primordium, but the root branch effect of the root primordium is not considered in the method. Although the root primordia appear sequentially, the development of root primordia into root branches is a modulatable process that is influenced by the environment as well as the root primordia themselves, and is not completely sequential.

Disclosure of Invention

The invention provides a root branch prediction method based on group interaction environment influence, which aims at solving the technical problem that the influence of a root primordium on the action of a root branch is not considered in the conventional root system modeling method, fully considers the environment and the modulatable property of the root primordium on the formation of the root branch, takes the root primordium as an intelligent individual, and performs root branch prediction according to the modulatable environment of the root primordium on the basis of group interaction environment interaction and environment perception of the root primordium.

In order to achieve the purpose, the technical scheme of the invention is realized as follows: a root branch prediction method based on group interaction environment influence comprises the following steps:

the method comprises the following steps: setting a maximum simulation period t_maxConstructing a root primordium group RS taking the root primordium as an individual, wherein the simulation period t is 0;

step two: traversing the root primordium group RS to obtain the ith root primordium rp_iI is a natural number of 1-n,

step three: calculating the influence factor of the perception environmental factors and the environmental change intensity of the ith root primordium;

step four: calculating the win-loss of the interaction between the ith root primordium and the neighbor root primordium individuals in the neighbor root primordium set by adopting a Parrondo game model;

step five: calculating the auxin content and the growth time of the ith root primordium at the moment t + delta t;

step six: predicting root branches according to the relationship between the auxin content and the maximum auxin demand; if t<t_maxReturning to the step one; otherwise, ending.

The method for constructing the root primordium group RS taking the root primordium as an individual comprises the following steps: dividing the root into a top non-branching region n _ la, a base non-branching region n _ lb and a branching region lb according to the self-similar structure of the root; with the growth of the root, when the length of the root is greater than the sum of n _ la and n _ lb, sequentially dividing the branch regions lb according to the sequence generation mode of the root primordium and the size of a development window dw in the direction towards the top, and giving a label i to each divided region; each partition i corresponds to a root primordium rp_iThereby constructing a root primordial population RS comprising a plurality of root primordial individuals.

Obtaining the ith root primordium rp_iThe neighbor root primordium set comprises the following steps: traversing the root primordium group RS for the ith root primordium rp_iAcquiring a neighbor root primordium set rp _ Neg ═ rp_i-1，rp_i+1}; if the set rp _ Neg is not empty, the set rp _ Neg is traversed, and the jth root primordium rp in the set rp _ Neg is_jIs root primordium rp_iThe neighbor root primordium of (a); if the set rp _ Neg is empty, the root primordium rp_iNo interaction with other root primordia.

The steps of calculating the perception environmental factor influence factor and the environmental change intensity of the ith root primordium are as follows: with root primordium rp_iConstructing a cylindrical area by taking the environmental change intensity gamma as a rotation radius as a rotation axis, and acquiring the total amount E of the environmental factors in the area at the current time t_{i_upt}(t); setting root primordium rp_iMinimum resource requirement is E_{i_min}The lower limit of the optimal resource demand is E_{i_opt1}The upper limit of the optimal resource demand is E_{i_opt2}The maximum resource requirement is E_{i_max}Calculating the influence factor E of the perception environmental factors at the moment t_i(t) and the environmental change intensity γ are:

wherein h is the lower limit of significance change.

The method for calculating the win-win of the interaction between the ith root primordium and the neighbor root primordium in the neighbor root primordium set by adopting a Parrondo game model comprises the following steps: root primordium. rp_iWith the neighbor root primordium rp_jRandomly selecting with a probability p' ═ p epsilon E_iPerforming an A game or performing a B game by using the probability 1-p', wherein p is the probability of selecting the A game under the condition of no environmental influence, epsilon is a slope, and the value of epsilon is a positive decimal number; if the game A is carried out, the root primordium rp is taken as the basis of the current time t_iWith the neighbor root primordium rp_jContent of auxin C_i(t) and C_j(t), initial auxin content C_{i_0}And C_{j_0}And a growth time T_iAnd T_jCalculating the root primordium rp at the current time t_iWith the neighbor root primordium rp_jGrowth factor yield W of_i(t) and W_j(t)：

W_i(t)＝C_i(t)-C_{i_0}；

W_j(t)＝C_j(t)-C_{j_0}；

According to the growth factor yield W_i(t) and W_j(t) calculating the root primordium rp at the current time t_iRelative to the neighbor root primordium rp_jProbability of winning p_ij；

If B-game is played, the root primordium rp is played at time t_iContent of auxin C_i(t) when divisible by M, the computing environment affects the root primordium rp_iProbability of winning is p₂'＝p₂-ε*E_iRoot primordium rp when at time t_iAuxin content C_i(t) when not evenly divisible by M, the computing environment affects the root primordium rp_iProbability of winning is p₃'＝p₃-ε*E_i，p₂And p₃Root primordium rp for two cases respectively_iThe maximum probability of winning, wherein M is the modulus of auxin content depended on by the B game.

Calculating the ith primordium rp at the time t + delta t_iThe specific steps of the auxin content and the growth time are as follows: if the game A is played, when the root primordium rp_iAt the time of winning, the neighbor root primordium rp_jPaying α units of auxin to the subject rp_i. When root primordium rp_iDuring transfusion, the root primordium rp_iPaying α units of auxin to the subject rp_j(ii) a If B game is played, if root primordium rp_iAt the time of winning, the root primordium rp_iIncrease of α units of auxin if the root primordium rp_iDuring transfusion, the root primordium rp_iReduction of auxin α units and setting of root primordia rp_iThe winning game time sig is 1, the losing game time sig is-1, and the fixed profit is β units, and the calculation is carried out at the time t +1Root primordium rp_iAuxin content C_i(t +1) is: c_i(t+Δt)＝C_i(T) + sig α + β > α, growth time T_i＝T_i+Δt。

The method of claim 1 or 6, wherein the root branch prediction comprises: traversing the root primordium group RS according to the simulation period t as t + delta t, and aiming at the ith root primordium rp_iObtaining rp at time t_iAuxin content C_i(t); setting root primordium rp_iMaximum auxin requirement C_{i_max}And its development time T_limIf T < T_limAnd C_i(t)≥C_{i_max}Then root primordium rp_iConverting into branches; if T < T_limAnd an auxin content C_i(t)＜C_{i_max}Then root primordium rp_iContinuing to develop; otherwise root primordium rp_iStopping development until new external stimulus exists, and the environmental change intensity gamma is 1; root primordium rp_iAfter a second development, if the auxin content C is present_i(t)≥C_{i_max}Then root primordium rp_iWill be converted into branches; if root primordium rp_iIf the root primordium is converted into a branch, deleting the root primordium from the root primordium group RS; if root primordium rp_iAnd when the development is stopped, the root primordium does not interact with the neighbor root primordium any more.

The invention has the beneficial effects that: based on the characteristic of the controlled sequence generation of the root primordia, the root primordia are further regarded as intelligent individuals, the group decision of the root primordia is realized through the perception of the root primordia individuals to the environment and the competition or cooperation among root primordia groups, so that the root branch modulatable prediction is realized, the defect of the root branch generation mode in the traditional root modeling and simulation research is avoided, the modulation of the root primordia development process into the root branch process can be realized based on the interaction among the root primordia groups according to the change of environmental factors, so that a model support and analysis means can be provided for the whole plant modeling and simulation research, and the plant modeling and simulation can be promoted to be applied to accurate agriculture.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.

FIG. 1 is a flow chart of the present invention.

FIG. 2 is a schematic representation of the root primordial population of the present invention.

FIG. 3 is a diagram illustrating root branch prediction according to the present invention.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without inventive effort based on the embodiments of the present invention, are within the scope of the present invention.

As shown in fig. 1, a root branch prediction method based on population interaction environment influence includes the following steps:

the method comprises the following steps: setting a maximum simulation period t_maxAnd constructing a root primordium group RS taking the root primordium as an individual with the simulation period t being 0.

The specific method comprises the following steps: the root has a self-similar structure, and comprises the following three parts: a top non-branching region n _ la, a base non-branching region n _ lb, and a branching region lb. Root according to formula

Performing elongation, wherein k is the maximum length of a single root represented by the root tip, and r is the initial growth speed; when the length of the root is larger than the sum of n _ la and n _ lb, sequentially dividing the branch regions lb according to the sequence generation mode of the root primordium and the size of the development window dw in the direction towards the top, and endowing each divided region with a label i, wherein i is a natural number of 1-n,

each partition i corresponds to a root primordium rp_iThereby constructing a root primordial population RS comprising a plurality of root primordial individuals, as shown in fig. 2.

The parameters are set as follows: the development window dw is 0.4 cm, and the value of the n _ la of the top non-branching region is 1.57 cm; the value of n _ lb of the base end non-branching region is 0.07 cm; the maximum length k of a single root is 26.9 cm, and the initial growth speed r is 2.

Step two: traversing the root primordium group RS to obtain the ith root primordium rp_iThe neighbor root primordium set;

the method comprises the following specific steps: traversing the root primordium group RS for the ith root primordium rp_iAnd acquiring a neighbor root primordium set rp _ Neg ═ { rp }_i-1，rp_i+1}; if the set rp _ Neg is not empty, the set rp _ Neg is traversed, and the jth root primordium rp in the set rp _ Neg is_jSubsequent calculation of rp_iAnd rp_jThe interaction result of (1). If the set rp _ Neg is empty, the root primordium rp_iNo interaction with other root primordia.

the method comprises the following specific steps: with root primordium rp_iConstructing a cylindrical area by taking the environmental change intensity gamma as a rotation radius as a rotation axis, and acquiring the total amount E of the environmental factors in the area at the current time t_{i_upt}(t) of (d). Setting root primordium rp_iMinimum resource requirement is E_{i_min}The lower limit of the optimal resource demand is E_{i_opt1}The upper limit of the optimal resource demand is E_{i_opt2}The maximum resource requirement is E_{i_max}Calculating the influence factor E of the perception environmental factors at the moment t_i(t) and the environmental change intensity γ are:

wherein h is the lower limit of significance change.

Wherein the parameters are set as: the rotation radius gamma is 0.01 cm, and the minimum resource demand E _{i_min}30 mu mol/L, the lower limit E of the optimal resource demand_{i_opt1}130 mu mol/L, the upper limit E of the optimal resource demand_{i_opt2}230 mu mol/L, maximum resource requirement E_{i_max}It was 330. mu. mol/L, and h was 0.05.

Step four: and calculating the win-loss of the interaction between the ith root primordium and the neighbor root primordium individuals in the neighbor root primordium set by adopting a Parrondo game model.

Calculating the ith root primordium rp by adopting a Parrondo game model_iWith the neighbor root primordium rp in the neighbor root primordium set_jThe interactive win-loss steps are as follows: when root primordium rp_iIs perceived as an environmental factor influencing factor E_i(t) smaller, indicates abundant environmental resources, root primordium rp_iTendency to take auxin, root primordium, rp from the external environment_iWith the neighbor root primordium rp_jThe competitive strength of (2) is reduced; when root primordium rp_iIs greater in perception environmental factor influence factor E_i(t) indicates lack of environmental resources and root primordium rp_iWith the neighbor root primordium rp_jThe competitive strength of (2) is improved. Root primordium. rp_iWith the neighbor root primordium rp_jRandomly selecting with a probability p' ═ p epsilon E_iPerforming an A game or performing a B game by using the probability 1-p', wherein p is the probability of selecting the A game under the condition of no environmental influence, epsilon is a slope, and the value of epsilon is a positive decimal number; if the game A is carried out, the root primordium rp is taken as the basis of the current time t_iWith the neighbor root primordium rp_jContent of auxin C_i(t) and C_j(t), initial auxin content C_{i_0}And C_{j_0}And a growth time T_iAnd T_jCalculating the root primordium rp at the current time t_iWith the neighbor root primordium rp_jGrowth factor yield W of_i(t) and W_j(t); according to the growth factor yield W_i(t) and W_j(t) calculating the root primordium rp at the current time t_iRelative to the neighbor root primordium rp_jProbability of winning p_ij；

W_i(t)＝C_i(t)-C_{i_0}；

W_j(t)＝C_j(t)-C_{j_0}；

If B-game is played, the root primordium rp is played at time t_iContent of auxin C_i(t) when divisible by M, the computing environment affects the root primordium rp_iProbability of winning is p₂'＝p₂-ε*E_iRoot primordium rp when at time t_iAuxin content C_i(t) when not evenly divisible by M, the computing environment affects the root primordium rp_iProbability of winning is p₃'＝p₃-ε*E_i，p₂，p₃Root primordium rp for two cases respectively_iMaximum probability of winning. Wherein M is the modulus of auxin content depended on by the B game.

Wherein the parameters are set as: initial auxin content C_{i_0}Is 1 and C_{j_0}Is 1, the probability p value is 0.5, the probability p₂Value 0.15, probability p₃The value is 0.75, the slope ε is 0.01, and M is 3.

calculating the ith primordium rp at the time t + delta t_iThe specific steps of the auxin content and the growth time are as follows: if the game A is played, when the root primordium rp_iAt the time of winning, the neighbor root primordium rp_jPaying α units of auxin to the subject rp_i. When root primordium rp_iDuring transfusion, the root primordium rp_iPaying α units of auxin to the subject rp_j(ii) a If B game is played, if root primordium rp_iAt the time of winning, the root primordium rp_iIncrease of α units of auxin if the root primordium rp_iDuring transfusion, the root primordium rp_iReduction of auxin α units root primordium rp is set_iThe game winning time sig is 1, the game losing time sig is-1 and the fixed income β units, and the root primordium rp at the time t +1 is calculated_iAuxin content C_i(t +1) is: c_i(t+Δt)＝C_i(T) + sig α + β > α, growth time T_i＝T_i+Δt。

Wherein the value of the parameter α is 1, the value of the fixed gain β is 2, and the value of the time interval Δ t is 1.

Step six: and predicting the root branches according to the relationship between the auxin content and the maximum auxin demand.

The method comprises the following specific steps: traversing the root primordium group RS according to the simulation period t as t + delta t, and aiming at the ith root primordium rp_iObtaining rp at time t_iAuxin content C_i(t) of (d). Setting root primordium rp_iMaximum auxin requirement C_{i_max}And its development time T_lim. If T < T_limAnd C_i(t)≥C_{i_max}Then root primordium rp_iWill be converted into branches; if T < T_limAnd an auxin content C_i(t)＜C_{i_max}Then root primordium rp_iContinuing to develop; otherwise root primordium rp_iThe development is stopped until there is a new external stimulus, i.e. an environmental change intensity γ of 1. Root primordium rp_iAfter a second development, if the auxin content C is present_i(t)≥C_{i_max}Then root primordium rp_iWill be converted into branches. If root primordium rp_iAnd converting into branches, and deleting the root primordia from the root primordia group RS. If root primordium rp_iAnd when the development is stopped, the root primordium does not interact with the neighbor root primordium any more. If t<t_maxReturning to the step one; otherwise, ending.

Wherein, the parameter C_{i_max}A value of 9, T_limA value of 3, t_maxIt was 20 days.

FIG. 3 is a schematic diagram of a root branch obtained by a root branch prediction method based on group interaction environment influence. Wherein the line 1 is a branch predicted value of each root primordium obtained according to the method under the uniform distribution of 50 mu mol/L environment; the root primordial environment was set as follows: the first 10 root primordia are distributed in 50 mu mol/L environment, the 10 th to 20 th root primordia are distributed in 80 mu mol/L environment, and the remaining root primordia are distributed in 50 mu mol/L environment. Line 2 is the predicted branch value of each root primordium obtained according to the method of the invention under the above-mentioned situation distribution; the line 3 is a branch predicted value of each root primordium obtained according to the method under the uniform distribution of 90 mu mol/L environment; the root primordial environment was set as follows: the root primordia were distributed in a 50. mu. mol/L environment 15 days prior to the simulation period. 5 days after the simulation period, the 10 th to 20 th root primordia are distributed in an environment of 80 μmol/L, and the line 4 is the branch prediction value of each root primordia obtained by the method of the invention under the above environment distribution. Therefore, the root branch prediction obtained by the method provided by the invention can realize the modulation of the influence of the environment and the root primordia on the process of converting the root primordia into the root branch.

The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims

1. A root branch prediction method based on group interaction environment influence is characterized by comprising the following steps:

wherein lb is the branch region and dw is the development window size;

step six: according to the content and the requirement of auxinCarrying out root branch prediction on the relation of the maximum values; if t<t_maxReturning to the step one; otherwise, ending;

wherein h is the lower limit of significance change.

2. The method for predicting root branches based on group interaction environment influence according to claim 1, wherein the method for constructing the root primordia group RS with root primordia as individuals comprises the following steps: dividing the root into a top non-branching region n _ la, a base non-branching region n _ lb and a branching region lb according to the self-similar structure of the root; with the growth of the root, when the length of the root is greater than the sum of n _ la and n _ lb, sequentially dividing the branch regions lb according to the sequence generation mode of the root primordium and the size of a development window dw in the direction towards the top, and giving a label i to each divided region; each partition i corresponds to a root primordium rp_iThereby constructing a root primordial population RS comprising a plurality of root primordial individuals.

3. The method of claim 2, wherein an ith root primordium rp is obtained_iThe neighbor root primordium set comprises the following steps: traversing the root primordium group RS for the ith root primordium rp_iAcquiring a neighbor root primordium set rp _ Neg ═ rp_i-1，rp_i+1}; if the set rp _ Neg is not empty, the set rp _ Neg is traversed, and the jth root primordium rp in the set rp _ Neg is_jIs root primordium rp_iThe neighbor root primordium of (a); if the set rp _ Neg is empty, the root primordium rp_iNo interaction with other root primordia.

4. The group interaction environment influence-based root branch prediction method according to claim 1, wherein the step of calculating the win-or-lose of the interaction between the ith root primordium and the neighbor root primordium in the neighbor root primordium set by adopting a Parrondo game model comprises the following steps: root primordium. rp_iWith the neighbor root primordium rp_jRandomly selecting with a probability p' ═ p epsilon E_iPerforming an A game or performing a B game by using the probability 1-p', wherein p is the probability of selecting the A game under the condition of no environmental influence, epsilon is a slope, and the value of epsilon is a positive decimal number; if the game A is carried out, the root primordium rp is taken as the basis of the current time t_iWith the neighbor root primordium rp_jContent of auxin C_i(t) and C_j(t), initial auxin content C_{i_0}And C_{j_0}And a growth time T_iAnd T_jCalculating the root primordium rp at the current time t_iWith the neighbor root primordium rp_jGrowth factor yield W of_i(t) and W_j(t)：

W_i(t)＝C_i(t)-C_{i_0}；

W_j(t)＝C_j(t)-C_{j_0}；

If B-game is played, the root primordium rp is played at time t_iContent of auxin C_i(t) capable of being evenly divided by M, computing environmentInfluencing root primordial rp_iProbability of winning is p₂'＝p₂-ε*E_iRoot primordium rp when at time t_iAuxin content C_i(t) when not evenly divisible by M, the computing environment affects the root primordium rp_iProbability of winning is p₃'＝p₃-ε*E_i，p₂And p₃Root primordium rp for two cases respectively_iThe maximum probability of winning, wherein M is the modulus of auxin content depended on by the B game.

5. The method of claim 4, wherein the ith root primordium rp at time t + Δ t is calculated_iThe specific steps of the auxin content and the growth time are as follows: if the game A is played, when the root primordium rp_iAt the time of winning, the neighbor root primordium rp_jPaying α units of auxin to the subject rp_i(ii) a When root primordium rp_iDuring transfusion, the root primordium rp_iPaying α units of auxin to the subject rp_j(ii) a If B game is played, if root primordium rp_iAt the time of winning, the root primordium rp_iIncrease of α units of auxin if the root primordium rp_iDuring transfusion, the root primordium rp_iReduction of auxin α units and setting of root primordia rp_iThe game winning time sig is 1, the game losing time sig is-1 and the fixed profit is β units, and the root primordium rp at the time t + delta t is calculated_iAuxin content C_i(t + Δ t) is: c_i(t+Δt)＝C_i(T) + sig α + β > α, growth time T_i＝T_i+Δt。

6. The method according to claim 1 or 5, wherein the root branch prediction comprises the steps of: traversing the root primordium group RS according to the simulation period t as t + delta t, and aiming at the ith root primordium rp_iObtaining rp at time t_iAuxin content C_i(t); setting root primordium rp_iMaximum auxin requirement C_{i_max}And its development time T_limIf T < T_limAnd C_i(t)≥C_{i_max}Then, thenRoot primordium rp_iConverting into branches; if T < T_limAnd an auxin content C_i(t)＜C_{i_max}Then root primordium rp_iContinuing to develop; otherwise root primordium rp_iStopping development until new external stimulus exists, and the environmental change intensity gamma is 1; root primordium rp_iAfter a second development, if the auxin content C is present_i(t)≥C_{i_max}Then root primordium rp_iWill be converted into branches; if root primordium rp_iIf the branch is converted into a branch, deleting the root primordium from the root primordium group RS; if root primordium rp_iAnd when the development is stopped, the root primordium does not interact with the neighbor root primordium any more.