CN113239611B - Population balance system and method based on Taylor expansion optimization - Google Patents

Population balance system and method based on Taylor expansion optimization Download PDF

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CN113239611B
CN113239611B CN202110333221.0A CN202110333221A CN113239611B CN 113239611 B CN113239611 B CN 113239611B CN 202110333221 A CN202110333221 A CN 202110333221A CN 113239611 B CN113239611 B CN 113239611B
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balance
host
circle
parasite
population
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CN113239611A (en
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何美霖
朱铭觉
滕旭阳
胡志蕊
卢从慧
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Hangzhou Dianzi University
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Abstract

The invention discloses a population balance system and method based on Taylor expansion optimization, which comprises the following steps: step 1, selecting a functional reaction function and establishing a mathematical model; step 2, solving the balance point of the host and parasite species models; step 3, drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion; step 4, solving the radius of the circle, and analyzing and optimizing the circle; and 5, obtaining a species balance area through a local progressive stability theorem. The invention establishes a mathematical model related to population quantity on the basis of the negative binomial distribution relationship between the host and the parasite, solves the range of the host-parasite population tending to ecological balance by using a Taylor expansion optimization and local gradual stabilization method, and verifies the correctness of the method through numerical calculation.

Description

Population balance system and method based on Taylor expansion optimization
Technical Field
The invention belongs to the technical field of biology, and relates to a technical scheme for estimating the number of host-parasite populations, in particular to a method for evaluating the accuracy of the method by Taylor expansion optimization and applying local gradual stable theorem to solve the range of the populations tending to ecological balance and then verifying the accuracy of the method by numerical calculation.
Background
Interactions between animal populations are a major factor affecting the development of ecosystems. In ecological mathematics, models of a single population have been extensively studied, and interspecies interaction models remain to be explored, mainly including patterns of lotterka waltera (Lotka-Volterra) interspecies competition and nicolson bery (Nicholson-Bailey) interspecies parasitism. Because the population quantity of the Nicholson-Bailey model is unstable, any factor has deviation, and the whole ecological system loses balance. Therefore, the ecological experts deeply explore the model and put forward a functional response function f (X)n,Yn). For differentThe functional response function has different manifestations in response to the survival status of the host and the parasitism status of the parasite, so that the host-parasite population tends towards different equilibrium areas. There are many documents that have been studied about different functional response functions and have been discussed about the influence of different factors on population dynamics systems. However, the prior art has shortcomings in solving the ecological balance of the host and verifying the correctness.
Disclosure of Invention
Based on the current situation, the invention researches the parasitic relationship in the interspecies relationship based on the Nicholson-Bailey model, and provides a population balance system and method based on Taylor expansion optimization.
The invention conjectures the situation that the number of the host-parasite population tends to ecological balance by mathematical modeling and using a Taylor expansion optimization method. General population models of hosts and parasites are:
Figure BDA0002997112630000011
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, and lambda is the birth rate of the host (lambda)>1)。
Figure BDA0002997112630000012
I.e. the probability of the host escaping the parasite.
To let the host XnAnd parasite YnThe population quantity of the method tends to ecological balance along with the generation, namely, the two populations reach local gradual stability, and a closed expression of a balance point is solved firstly. And then, carrying out Taylor expansion optimization near the balance point, and solving a region of species balance by using a local progressive stability theorem to obtain the range of the related parameters. And finally, verifying the correctness of the method through numerical calculation.
The invention adopts the following technical scheme:
a population balance method based on Taylor expansion optimization comprises the following specific steps:
step 1, selecting a functional reaction function and establishing a mathematical model;
step 2, solving the balance point of the host and parasite species models;
step 3, drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion;
step 4, solving the radius of the circle, and analyzing and optimizing the circle;
and 5, obtaining a species balance area through a local gradual stable theorem.
Preferably, step 1 is specifically mathematical modeling: in this mathematical model, the host is the victim and the parasite is the beneficiary, so a general host-parasitic model was drawn:
Figure BDA0002997112630000021
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, and lambda is the birth rate of the host (lambda)>1)，
Figure BDA0002997112630000022
I.e. the probability of the host escaping the parasite. Substituting the functional reaction function into the formula (2.1) to obtain
Figure BDA0002997112630000023
Wherein the concentration index is
Figure BDA0002997112630000024
When in use
Figure BDA0002997112630000025
When the sum is close to 0, the above formula becomes a nicolson-belley model.
Preferably, step 2 specifically calculates the balance point: to let host XnAnd parasite YnThe population quantity ofThe ecological balance is approached in the process of generation, namely, the two populations reach local gradual stabilization, and the balance points of the host and the parasite are respectively XeAnd YeIs obtained by
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
By means of the united type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure BDA0002997112630000026
Preferably, step 3 is to draw a circle with the balance point as the center: let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation can be obtained by calculating the deviation,
Figure BDA0002997112630000031
order to
Figure BDA0002997112630000032
At Taylor expansion near the equilibrium point, one can obtain
Figure BDA0002997112630000033
Wherein 0<θ<1. For F (x)n,yn) And G (x)n,yn) Partial derivation is carried out, and then the balance point is substituted to obtain
Figure BDA0002997112630000034
Figure BDA0002997112630000035
Figure BDA0002997112630000036
Figure BDA0002997112630000037
Figure BDA0002997112630000038
Figure BDA0002997112630000039
Figure BDA00029971126300000310
Figure BDA00029971126300000311
Figure BDA00029971126300000312
Figure BDA00029971126300000313
Preferably, in step 4, the radius of the circle is solved: with the balance point as the center and the radius
Figure BDA00029971126300000314
Calculating Euclidean distance V between the balance point and the balance point2，
Figure BDA00029971126300000315
Order to
Figure BDA0002997112630000041
The characteristic value of the B is calculated,
i.e. | B- α I | ═ 0, to give
Figure BDA0002997112630000042
Figure BDA0002997112630000043
Let N1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure BDA0002997112630000044
Thereby solving for the radius of the circle
Figure BDA0002997112630000045
Preferably, step 5 consists in finding the zone of host-parasite species equilibrium: when the initial number of populations of hosts and parasites is within a circle of radius V centered at the point of equilibrium, the two populations will reach ecological equilibrium. Otherwise, the ecosystem is out of balance.
According to the local gradual stabilization theorem, substituting the matrix A into | trA | < detA +1<2 can obtain the sufficient condition that the species reach ecological balance
Figure BDA0002997112630000046
The invention also discloses a population balance system based on Taylor expansion optimization, which comprises the following modules:
a modeling module: the system is used for selecting a functional reaction function and establishing a mathematical model;
a solving module: an equilibrium point for solving models of host and parasite species;
circle drawing module: drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion;
an analysis and optimization module: the method is used for solving the radius of the circle, analyzing and optimizing;
a result acquisition module: the method is used for obtaining a species equilibrium region through a local progressive stability theorem.
Preferably, in the modeling module, in the mathematical model, a general host-parasitic model is drawn as follows:
Figure BDA0002997112630000047
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, and lambda is the birth rate of the host (lambda)>1)，
Figure BDA0002997112630000051
I.e. the probability of the host escaping the parasite; substituting the functional reaction function into the formula (2.1) to obtain
Figure BDA0002997112630000052
Wherein the concentration index is
Figure BDA0002997112630000053
When in use
Figure BDA0002997112630000054
When the sum is close to 0, the above formula becomes a nicolson-belley model.
Preferably, in the solving module, the balance points of the host and the parasite are respectively XeAnd YeObtained by
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
Through the united vertical type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure BDA0002997112630000055
Preferably, in the circle drawing module, let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation can be obtained by calculating the deviation,
Figure BDA0002997112630000056
order to
Figure BDA0002997112630000057
Taylor expansion near the equilibrium point can be obtained
Figure BDA0002997112630000058
Wherein, 0<θ<1; for F (x)n,yn) And G (x)n,yn) Partial derivation is carried out, and then the balance point is substituted to obtain
Figure BDA0002997112630000059
Figure BDA00029971126300000510
Figure BDA00029971126300000511
Figure BDA00029971126300000512
Figure BDA00029971126300000513
Figure BDA00029971126300000514
Figure BDA0002997112630000061
Figure BDA0002997112630000062
Figure BDA0002997112630000063
Figure BDA0002997112630000064
Preferably, in the result acquisition module, the radius is set with the balance point as the center
Figure BDA0002997112630000065
Calculating Euclidean distance V between the balance point and the balance point2：
Figure BDA0002997112630000066
Order to
Figure BDA0002997112630000067
The characteristic value of the B is calculated,
i.e. | B- α I | ═ 0, to give
Figure BDA0002997112630000068
Figure BDA0002997112630000069
Let N be1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure BDA00029971126300000610
The radius of the circle is thus solved:
Figure BDA00029971126300000611
when the initial number of populations of hosts and parasites is within a circle of radius V centered on the point of equilibrium, the two populations will reach ecological equilibrium; otherwise, the ecosystem is out of balance;
according to the local asymptotic stability theorem, substituting the matrix A into | trA | < detA +1<2 can obtain the sufficient condition that the species reach ecological balance:
Figure BDA0002997112630000071
the invention establishes a mathematical model related to population quantity on the basis of the negative binomial distribution relationship between the host and the parasite, solves the range of the host-parasite population tending to ecological balance by using a Taylor expansion optimization and local gradual stabilization method, and verifies the correctness of the method through numerical calculation.
Drawings
FIG. 1 is a graph of the initial population numbers of hosts and parasites within the ecological balance.
FIG. 2 is a graph of the variation of the initial population of hosts and parasites outside the ecological balance.
Fig. 3 is a flow chart of a preferred embodiment of the present invention.
Fig. 4 is a system block diagram of another preferred embodiment of the present invention.
Detailed Description
The objects and advantages of the present invention will become more apparent from the following detailed description of the population quantity variation chart with reference to the accompanying drawings. In the figure, the parameter θ is 0.65.
As shown in fig. 1 to 3, the population balancing method based on taylor expansion optimization in this embodiment includes the following specific steps:
step 1, selecting a functional reaction function and establishing a mathematical model; specifically, mathematical modeling: in this mathematical model, the host is the victim and the parasite is the beneficiary, so a general host-parasitic model was drawn:
Figure BDA0002997112630000072
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, and lambda is the birth rate of the host (lambda)>1)，
Figure BDA0002997112630000073
I.e. the probability of the host escaping the parasite. Substituting the functional reaction function into the formula (2.1) to obtain
Figure BDA0002997112630000074
Wherein the concentration index is
Figure BDA0002997112630000075
When in use
Figure BDA0002997112630000076
When the sum is close to 0, the above formula becomes a nicolson-belley model.
Step 2, solving the hostAnd a balance point of the parasite species model; specifically, a balance point is calculated: to let the host XnAnd parasite YnThe population quantity of (2) tends to ecological balance along with the passage of generation, namely, both populations reach local gradual stabilization, and the balance points of the host and the parasite are respectively XeAnd YeIs obtained by
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
By means of the united type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure BDA0002997112630000081
Step 3, drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion; specifically, a circle is drawn by taking a balance point as a circle center:
let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation can be obtained by calculating the deviation,
Figure BDA0002997112630000082
order to
Figure BDA0002997112630000083
At Taylor expansion near the equilibrium point, one can obtain
Figure BDA0002997112630000084
Wherein 0<θ<1. For F (x)n,yn) And G (x)n,yn) Partial derivation is carried out, and then the balance point is substituted to obtain
Figure BDA0002997112630000085
Figure BDA0002997112630000086
Figure BDA0002997112630000087
Figure BDA0002997112630000088
Figure BDA0002997112630000089
Figure BDA00029971126300000810
Figure BDA00029971126300000811
Figure BDA00029971126300000812
Figure BDA00029971126300000813
Figure BDA0002997112630000091
Step 4, solving the radius of the circle, and analyzing and optimizing the circle; specifically, the radius of the solution circle is: with the balance point as the center and the radius
Figure BDA0002997112630000092
Calculating Euclidean distance V between the balance point and the balance point2，
Figure BDA0002997112630000093
Order to
Figure BDA0002997112630000094
The characteristic value of the B is calculated,
i.e. | B- α I | ═ 0, to give
Figure BDA0002997112630000095
Figure BDA0002997112630000096
Let N1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure BDA0002997112630000097
Thereby solving for the radius of the circle
Figure BDA0002997112630000098
And 5, obtaining a species balance area through a local gradual stable theorem. Specifically, the areas of host-parasite species balance were found: when the initial population numbers of hosts and parasites are within a circle of radius V centered on the equilibrium point, the two populations will reach ecological equilibrium. Otherwise, the ecosystem is out of balance.
According to the local gradual stabilization theorem, substituting the matrix A into | trA | < detA +1<2 can obtain the sufficient condition that the species reach ecological balance
Figure BDA0002997112630000099
FIG. 1 is a graph showing the initial numbers of host and parasite populations in an ecologically balanced range, and it can be seen that the numbers of host and parasite populations tend to the point of equilibrium and tend to be stable as generations increase.
FIG. 2 is a graph of the initial number of populations of hosts and parasites outside the range of ecological balance, and as generations increase, it can be seen that the parasite population is nearing extinction, meaning that the balance of the ecosystem is disrupted.
Through the numerical calculation of the embodiment, the correctness of the method is verified.
As shown in fig. 4, the population balance system based on taylor expansion optimization in this embodiment includes:
the modeling module is used for selecting a functional reaction function and establishing a mathematical model; specifically, mathematical modeling: in this mathematical model, the host is the victim and the parasite is the beneficiary, so a general host-parasitic model was drawn:
Figure BDA0002997112630000101
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, and lambda is the birth rate of the host (lambda)>1)，
Figure BDA0002997112630000102
I.e. the probability of the host escaping the parasite. Substituting the functional reaction function into the formula (2.1) to obtain
Figure BDA0002997112630000103
Wherein the concentration index is
Figure BDA0002997112630000104
When in use
Figure BDA0002997112630000105
When the sum is close to 0, the above formula becomes a nicolson-belley model.
A solving module for solving equilibrium points of the host and parasite species models; specifically, a balance point is calculated: to let host XnAnd parasite YnThe population quantity of (2) tends to ecological balance along with the passage of generation, namely, both populations reach local gradual stabilization, and the balance points of the host and the parasite are respectively XeAnd YeIs obtained by
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
By means of the united type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure BDA0002997112630000106
The circle drawing module is used for drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion; specifically, a circle is drawn with a balance point as a center: let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation can be obtained by calculating the deviation,
Figure BDA0002997112630000107
order to
Figure BDA0002997112630000111
At Taylor expansion near the equilibrium point, one can obtain
Figure BDA0002997112630000112
Wherein, 0<θ<1. For F (x)n,yn) And G (x)n,yn) Partial derivation is carried out, and then the balance point is substituted to obtain
Figure BDA0002997112630000113
Figure BDA0002997112630000114
Figure BDA0002997112630000115
Figure BDA0002997112630000116
Figure BDA0002997112630000117
Figure BDA0002997112630000118
Figure BDA0002997112630000119
Figure BDA00029971126300001110
Figure BDA00029971126300001111
Figure BDA00029971126300001112
The analysis and optimization module is used for solving the radius of the circle and analyzing and optimizing the radius; specifically, the radius of the solution circle is: to balancePoint as center, set radius
Figure BDA00029971126300001113
Calculating Euclidean distance V between the balance point and the balance point2，
Figure BDA00029971126300001114
Order to
Figure BDA0002997112630000121
The characteristic value of the B is calculated,
i.e. | B- α I | ═ 0, to give
Figure BDA0002997112630000122
Figure BDA0002997112630000123
Let N1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure BDA0002997112630000124
Thereby solving for the radius of the circle
Figure BDA0002997112630000125
And the result acquisition module is used for obtaining a species balance region through a local gradual stable theorem. Specifically, the areas of host-parasite species balance were found: when the initial number of populations of hosts and parasites is within a circle of radius V centered at the point of equilibrium, the two populations will reach ecological equilibrium. Otherwise, the ecosystem is out of balance.
According to the local gradual stabilization theorem, substituting the matrix A into | trA | < detA +1<2 can obtain the sufficient condition that the species reach ecological balance
Figure BDA0002997112630000126
The invention carries out Taylor expansion optimization by taking a balance point as a center, solves the radius of a circular area, and finds the range of ecological balance reached by a host-parasite population by applying a local gradual stable theorem. The invention is not limited to the specific mathematical model described above, and the skilled person can modify or adjust the expression of the functional response function within the scope of the claims without affecting the essence of the invention.

Claims (4)

1. A population balance method based on Taylor expansion optimization is characterized by comprising the following specific steps:
step 1, selecting a functional reaction function and establishing a mathematical model;
step 2, solving the balance point of the host and parasite species models;
step 3, drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion;
step 4, solving the radius of the circle, and analyzing and optimizing the circle;
step 5, obtaining a species balance area through a local gradual stable theorem;
the step 1 is as follows: in the mathematical model, a general host-parasitic model is drawn up as follows:
Figure FDA0003545322850000011
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, lambda is the birth rate of the host (lambda is more than 1),
Figure FDA0003545322850000012
i.e. the probability of the host escaping the parasite; substituting the functional reaction function into the formula (2.1) to obtain
Figure FDA0003545322850000013
Wherein the concentration index is
Figure FDA0003545322850000014
When in use
Figure FDA0003545322850000015
When approaching to 0, the above formula becomes Nicolessen Belli model;
the step 2 is as follows: let the balance point of the host and parasite be XeAnd YeTo obtain
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
By means of the united type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure FDA0003545322850000016
The step 3 is as follows: let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation is obtained by calculating the deviation,
Figure FDA0003545322850000017
order to
Figure FDA0003545322850000018
Taylor expansion near the equilibrium point, obtaining
Figure FDA0003545322850000021
Wherein theta is more than 0 and less than 1; for F (x)n,yn) And G (x)n,yn) Performing partial derivation, and substituting the balance point to obtain
Figure FDA0003545322850000022
Figure FDA0003545322850000023
Figure FDA0003545322850000024
Figure FDA0003545322850000025
Figure FDA0003545322850000026
Figure FDA0003545322850000027
Figure FDA0003545322850000028
Figure FDA0003545322850000029
Figure FDA00035453228500000210
Figure FDA00035453228500000211
2. The population balancing method based on taylor expansion optimization as claimed in claim 1, wherein: the step 4 is as follows: with the balance point as the center and the radius
Figure FDA00035453228500000212
Calculating Euclidean distance V between the Euclidean distance V and the balance point2：
Figure FDA00035453228500000213
Order to
Figure FDA00035453228500000214
Obtaining a characteristic value of B, namely | B- α I | ═ 0 to obtain
Figure FDA0003545322850000031
Figure FDA0003545322850000032
Let N be1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure FDA0003545322850000033
The radius of the circle is thus solved:
Figure FDA0003545322850000034
the step 5 is as follows: when the initial population number of the host and the parasite is in a circle with a balance point as the center and a radius of V, the two populations reach ecological balance; otherwise, the ecosystem is out of balance;
according to the local gradual stability theorem, substituting the matrix A into | trA | < detA +1<2 to obtain the sufficient condition that the species reach ecological balance:
Figure FDA0003545322850000035
3. a population balance system based on Taylor expansion optimization is characterized by comprising the following modules:
a modeling module: the system is used for selecting a functional reaction function and establishing a mathematical model;
a solving module: an equilibrium point for solving models of host and parasite species;
circle drawing module: drawing a circle by taking the balance point as the center of the circle to obtain a Taylor expansion;
an analysis and optimization module: the method is used for solving the radius of the circle, analyzing and optimizing the circle;
a result acquisition module: obtaining a species balance region through a local gradual stable theorem;
in the modeling module, in the mathematical model, a general host-parasitic model is drawn up as follows:
Figure FDA0003545322850000036
wherein, XnIs the population number of the nth generation host, YnIs the population number of the nth host, lambda is the birth rate of the host (lambda is more than 1),
Figure FDA0003545322850000037
i.e. the probability of the host escaping the parasite; substituting the functional reaction function into the formula (2.1) to obtain
Figure FDA0003545322850000041
Wherein the concentration index is
Figure FDA0003545322850000042
When the temperature is higher than the set temperature
Figure FDA0003545322850000043
When approaching to 0, the above formula becomes Nicolessen Belli model;
in the solving module, the balance points of the host and the parasite are respectively XeAnd YeTo obtain
Xn+1＝Xn＝Xe，Yn+1＝Yn＝Ye (2.3)
By means of the united type (2.2) and (2.3), the balance points of the two populations are solved respectively
Figure FDA0003545322850000044
In the circle drawing module, let F (x)n,yn)＝Xn+1，G(xn,yn)＝Yn+1The deviation is obtained by calculating the deviation,
Figure FDA0003545322850000045
order to
Figure FDA0003545322850000046
Taylor expansion near the equilibrium point, obtaining
Figure FDA0003545322850000047
Wherein theta is more than 0 and less than 1; for F (x)n,yn) And G (x)n,yn) Performing partial derivation, and substituting the balance point to obtain
Figure FDA0003545322850000048
Figure FDA0003545322850000049
Figure FDA00035453228500000410
Figure FDA00035453228500000411
Figure FDA00035453228500000412
Figure FDA00035453228500000413
Figure FDA00035453228500000414
Figure FDA00035453228500000415
Figure FDA0003545322850000051
Figure FDA0003545322850000052
4. The population balance system based on taylor expansion optimization of claim 3, wherein: in the result acquisition module, the balance point is taken as the center and the radius is set
Figure FDA0003545322850000053
Calculating Euclidean distance V between the balance point and the balance point2：
Figure FDA0003545322850000054
Order to
Figure FDA0003545322850000055
Obtaining a characteristic value of B, namely | B- α I | ═ 0 to obtain
Figure FDA0003545322850000056
Figure FDA0003545322850000057
Let N1＝max(n3,n4,n5)＝n4，N2＝max(m3,m4,m5)＝m5Equation (2.16) is simplified to
Figure FDA0003545322850000058
Thus solving for the radius of the circle:
Figure FDA0003545322850000059
when the initial number of populations of hosts and parasites is within a circle of radius V centered on the point of equilibrium, the two populations will reach ecological equilibrium; otherwise, the ecosystem is out of balance;
according to the local gradual stability theorem, substituting the matrix A into | trA | < detA +1<2 to obtain the sufficient condition that the species reach ecological balance:
Figure FDA0003545322850000061
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