Inverse problem for the Verhulst equation of limited population growth with discrete experiment data
Anvar Azimov, Syrym Kasenov, Daniyar Nurseitov, and Simon Serovajsky
Citation: AIP Conference Proceedings 1759, 020037 (2016); doi: 10.1063/1.4959651 View online: http://dx.doi.org/10.1063/1.4959651
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1759?ver=pdfcov Published by the AIP Publishing
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Inverse problem for the Verhulst equation of limited population growth with discrete experiment data
Anvar Azimov
∗, Syrym Kasenov
†, Daniyar Nurseitov
∗∗and Simon Serovajsky
∗∗Department of Differential Equations and Control Theory, al-Farabi Kazakh National University, Almaty, Kazakhstan
†Department of Methods of Teaching Mathematics, Physics and Computer Science, Abai Kazakh National Pedagogical University, Almaty, Kazakhstan
∗∗National Open Research Laboratory of Information and Space Technologies, Kazakh National Research Technical University after K.I. Satpayev, Almaty, Kazakhstan
Abstract. Verhulst limited growth model with unknown parameters of growth is considered. These parameters are defined by discrete experiment data. This inverse problem is solved with using gradient method with interpolation of data and without it. Approximation of the delta-function is used for the latter case. As an example the bacteria populationE.coliis considered.
Keywords:Verhulst equation, Inverse problem, Delta-function PACS:02.30.Zz, 02.60.Cb
PROBLEM STATEMENT
Consider the evolution of biological species with bounded habitat. The system is described for easiest case by Verhulst equation [1, Section 1]
x˙= (a−bx)x, t∈(0,T), (1)
with initial condition
x(0) =x0, (2)
wherex=x(t)is the population of the species,x0is its initial population,ais the growth of the species, andbis its sensitiveness to the boundedness of the habitat.
Note that the increase parametersa andb cannot to determine directly by an experiment. It depends from the concrete species and the conditions of its habitat. These parameters can be determined with using experimental data
x(tj) = fj, j=1,...,M, (3)
wheretjare the times of the experiment,xjare results of the experiment. We have the following inverse problem.
Determine the increase parameters such that the solution of Verhulst satisfies the results of the experiment.
Using standard method (see, for example [2, 3]), transform this inverse problem to the problem of minimization of the functional
J=J(a,b) =
∑
Mj=1[x(a,b;tj)−fj]2, (4)
wherex(a,b;tj)is the population of the species at the timetjfor the given values of the parametersaandb.
Commonly inverse problems with discrete experimental data are solved by interpolation of the experimental information. Then the standard optimization algorithms can be applicable. This idea guaranties to avoid the appearance of delta functions that is the result of the discreteness of experimental data. However, we distort the formulation of the problem because we use the information about the state system in the intermediate points in time at which there is no experiment. The aim of this paper is to solve the inverse problem with interpolation of the experimental data and without it, i.e. directly. After working on algorithms for model examples, we solve the inverse problem associated with real experimental data.
ALGORITHM OF SOLVING FOR THE INVERSE PROBLEM
We use two algorithms that is based on the gradient method [4, Section 3]. Determine the derivative of the functional Jdirectly for the first case. This is the vectorJ= (Ja,Jb)with components
Ja=− T
0
p(t)x(t)dt, Jb= T
0
p(t)x2(t)dt, (5)
whereTis a final time. Its value is greater than the time of last experiment. Here, the functionp=p(t)is the solution of the adjoint equation
p˙+ (a−2bx)p=2
∑
M j=1[x(a,b;t)−f(t)]δ(t−tj) (6) with final condition
p(T) =0. (7)
The adjoint equation includes delta-functionδ(t−tj)and the arbitrary function f=f(t)that satisfies the equalities
f(tj) = fj, j=1,...,M. (8)
The next iteration of the unknown parameters is determined by the formulas
ak+1=ak−λkIak, bk+1=bk−λkIbk, (9) whereλk is a parameter of the algorithm, andIak,Ibk are the partial derivatives of the functional at the previous iteration.
Using delta-function which is the distribution [5] gives the serious difficulty for this algorithm. This difficulty is overcome as a rule by the interpolation of the discrete experimental data [2]. We use the linear interpolation
f(t) = fj+fj+1−fj
tj+1−tj (t−tj), t∈(tj,tj+1). (10) Then the unknown parameters can be determine by the minimization of the integral functional
J(a,b) = T
0
[x(a,b;t)−f(t)]2dt. (11)
Its derivative is determined by the previous formula. However, the adjoint equation has the form
p˙+ (a−2bx)p=2(x−f). (12)
We solve the given inverse problem with using both methods, i.e. with interpolation and directly. We use Gauss formula (see [5]).
δ(t)≈√σ
πe−(σt)2 (13)
for the approximation of the delta-function, whereσ is a parameter of the algorithm.
TABLE 1. Results of computation with determined data
Algorithm k ak bk Δa Δb δa δb J(ak,bk)
1 18532 2.998 1.998 0.002 0.002 0.0007 0.001 3·10−8
2 11187 2.976 1.983 0.024 0.017 0.008 0.009 2·10−7
020037-2
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
0 0.2 0.4 0.6 0.8 1
x(t)
t
x(t) (alg.1)f(t) x(t) (alg.2)
FIGURE 1. The functionxwith determined data. Algorithm 1 and 2.
TABLE 2. Results of computation the data with noise
Algorithm k ak bk Δa Δb δa δb J(ak,bk)
1 18021 2.484 1.577 0.516 0.423 0.172 0.211 0.171
2 10623 2.502 1.592 0.498 0.408 0.166 0.204 0.0115
RESULTS
At first we solve the problem with exact values of the increase parametersa=3,b=2. SupposeT=1,x0=1. We calculation solution of differential equations with time step 10−4, algorithm parametersλk=10−2,σ=105, and the zero values of the initial approximation of the unknown parameters.
The results of the computer calculation are given in the Table 1. The algorithm 1 uses delta-function directly, the algorithms applies the interpolation,kis the quantity of the iterations,akandbkare the final values of the increase parameters,ΔaandΔbare absolute error,δaandδbare relative error, andJ(ak, bk)is the value of functional at the final iteration.
The state functionxfor the computation values of the increase parameters and its experimental values are given at the Figure 1.
The exactness of the results is good enough for both algorithms. Note that the experimental date of the practical situation is non-exact. Therefore, we consider our numerical methods for the case the experimental data is known with a noise. The computing results for the experimental data for the 10% noise are given at the Table 2.
The state functionxfor the computation values of the increase parameters and its experimental values for the data with noise are given at the Figure 2.
The exactness now is less that its value for the data without noise. However, both algorithms guaranty finding the approximate solution of the inverse problem with crudely exactness.
Finally, we solve an inverse problem with natural data. We consider a bacteria population E.coli with using experiments of Scientific Centre of Anti-infectious Drugs (Almaty). The experiments were staged every hour with an optical density measurement of the substrate (see Table 3).
Results of computing are given at the Table 4.
The state function x for the computation values of the increase parameters and its experimental values for the natural data are given at the Figure 3.
The obtained results demonstrate the high efficiency of these methods.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0 0.2 0.4 0.6 0.8 1
x(t)
t
f(t) with errorf(t) x(t) (alg.1) x(t) (alg.2)
FIGURE 2. The functionxfor the data with 10% noise. Algorithm 1 and 2.
TABLE 3. Results of the natural ex- periment
tk 0 1 2 3 4
fk 63 83 136 217 313
TABLE 4. Results of computing
Algorithm k ak bk J(ak,bk)
1 98 0.403 0 194.2
2 552 0.407 0 115.2
50 100 150 200 250 300 350
0 0.5 1 1.5 2 2.5 3 3.5 4
x(t)
t
fj x(t) (alg.1) x(t) (alg.2)
FIGURE 3. The functionxfor the natural data. Algorithm 1 and 2.
020037-4
ACKNOWLEDGMENTS
We would like to thank A. I. Ilyin, R. A. Islamov and M. V. Lankina of the Scientific Center of Anti-Infectious Drugs for the natural experiment and discussion of these results.
This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan under the grant number 1746/GF4 and by the Ministry for Investments and Development under the project number 006/156.
REFERENCES
1. N. T. J. Bailey,The Mathematical Approach to Biology and Medicine, J. Wiley and Sons, New York, 1970, (p. 326).
2. S. I. Kabanikhin,Inverse and Ill-posed Problems,Theory and Applications, De Gruyter, Berlin, Boston, 2011, (p. 459).
3. S. I. Kabanikhin, M. A. Shishlenin, D. B. Nurseitov, A. T. Nurseitova, and S. Y. Kasenov,Journal of Applied Mathematics 2014, 1-7 (2014), (Article ID 786326).
4. J. Sea,Inverse and Ill-posed Problems, Theory and Applications, Mir, Moscow, 1973, (p. 244).
5. I. M. Gel’fand, and G. E. Shilov,Generalized Functions and Operations on Them, Dobrosvet, Moscow, 2000, (p. 400).