Ecological Modelling, 34 (1986) 191-196 191 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

PARAMETER ESTIMATION OF A NONLINEAR POPULATION MODEL WITH TWO PARAMETERS, GROWTH OF A YEAST POPULATION AS AN EXAMPLE

CHENG ZHI-BIN 1, LI BAI-LIAN 2, ZOU CHAO-SHUN 1 and QUE TAI-LIN 2

1 Department of A utomatic Control, Huazhong University of Science and Technology, Wuhan, Hubei (People's Republic of China) 2 Department of Environmental Science, Wuhan University, Wuhan, Hubei (People's Republic of China)

(Accepted 14 October 1985)

ABSTRACT

Cheng Zhi-Bin, Li Bai-Lian, Zou Chao-Shun and Que Tai-Lin, 1986. Parameter estimation of a nonlinear population model with two parameters, growth of a yeast population as an example, Ecol. Modelling, 34: 191-196.

Quasilinearization is a powerful numerical method used to obtain the solution of system identification problems, Its major advantages are that it is extremely accurate and quadrati- cally convergent. In this paper, the two parameters of single logistic population are estimated using this method.

INTRODUCTION

In population ecology, Lotka (1925) generalized the behavior of a popula- tion using the following equation:

d N / d t = f ( U ) (1)

Using Taylor's Theorem, for a nonlinear population model with two param- eters we may write:

d N / d t = b N + c N 2 (2)

o r

d N / d t = r N ( 1 - N / K ) (3)

w h e r e r = b, c = - r / K , o r t h e i n t e g r a l f o r m

L n ( ( K - N ) / N ) = a - rt (4)

w h e r e a = L n ( K / N o - 1), N o is t h e p o p u l a t i o n a t t i m e t = 0.

0304-3800/86/$03.50 © 1986 Elsevier Science Publishers B.V.

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In later discussions we shall use r rather than b for the Malthusian parameter or unrestricted rate of increase. K, which represents the maxi- mum population density allowed by the limiting resource, is often called the carrying capacity. Equation (4) is the well-known logistic equation.

There are many methods to fit the logistic curve in ecological problems (Verhulst, 1838; Pearl and Reed, 1920; Pearl, 1927; Andrewartha and Birch, 1954; Marquardt, 1963; Gao, 1965; Watt, 1968; Pielou, 1977; Wan and Liang, 1983; Katz et al., 1981). However, we have not seen references to the quasilinearization method used to study ecological problems in the past, in spite of its extreme accuracy and quadratical convergence.

This paper deals with the method of estimation of biological parameters via logistic differential equation (4) by means of quasilinearization method (Bellman and Kalaba, 1965)

IDENTIFICATION OF NONLINEAR MODEL WITH TWO PARAMETERS

Consider the nonlinear dynamic system:

d x / d t = f ( x , y, a, b, t), x (O)- -d (5)

where the unknown system parameters a and b a r e to be estimated by minimizing the sum Q:

N Q= Y' (x( t i , a, b ) - c i ) 2 (6)

i=1

and where x( t ) is a scalar state variable, y(t) the control function or input, d a known initial condition, and c the observed value of x(ti, a, b) at time ti, i = 1, 2 . . . . . N. The method of quasilinearization is utilized to minimize Q.

The differential equation (5) is first linearized around the k th approxima- tion by expanding the function f in a power series and retaining only the linear terms:

d x k / d t =fk + Of/~a [ kdak + ~f/~b[ kdbk + ~f/~x[ kdxk

xk(0 ) = d (7)

Then the k + 1 approximation is defined as:

dXk+l/dt =fk + (ak+l - ak)Of/Ba ]k + (bk+l - bk)af/Ob]k

+ (xk+ - x k ) O f / O x l k

= a (8)

The general solution of (8) is:

xk+,(t ) = p ( t ) + 8k+ 1 h ( t ) --b bk+ 1 g ( / ) (9)

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The function p(t) is the solution of the initial value equation:

dp( t ) /d t =fk - ak ~)f/,Salk - bk Of/i)blk + (p - Xk)Of/Ox [k p(0) =d (10) The function h(t) is the solution of the initial value equation:

dh(t) /dt=3f /OalK +Of/Ox[kh(t) h(O)=O (11)

and the function g(t) is the solution of the initial value equation:

dg(t) /dt=Of/Oblk+Of/Oxlkg(t) g ( 0 ) = 0 (12)

Assuming that the linearized equation (9) is the approximate solution of (5), (6) becomes:

N

S= ~'~ (p(ti)+ak+ 1 h(ti)+bk+ , g ( t i ) - - C i ) 2 (13) i = l

The necessary condition for minimum S is given by: N

OS/Oak+l=2Y'~h(t i )(p( t i )+ak+lh(t i )+bk+lg(t i )-ci)=O (14) i = 1

N

OS/Obk+l=2 Y'~g(ti)(p(ti)+ak+ 1 h(ti)=bk+ 1 g(t~)-c~)=O (15) /=1

Solving for ak+ a, bk+ 1 to minimize S: N N N N

a k + 1 =

Y' (ci-P(ti))h(ti) ~_~g2(ti)- ~_~ (ci-P(ti))g(ti) Y'~h(ti)g(ti) i = 1 i = 1 i = 1 i = 1

y" h2(ti) y" g2(t i )- y' h(ti)g(t i i = 1 i = 1 i = 1

(16)

N N N N

~., (ci-P(ti))g(ti) Y'~ h2(t i)- y' (ci--P(ti))h(tl) Y" h(ti)g(ti) i = 1 i = 1 i = 1 i = 1

b k + 1 -=

E h2(ti) E g2(t i )- h(ti)g(ti i = 1 i = 1

E X A M P L E F O R D E T E R M I N I N G P A R A M E T E R S , r , K

Consider the logistic equation:

O N / O t = r ( 1 - N / K ) N ; O<<.t<~ T; N(O)=c

(17)

(18)

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TABLE 1

Growth of a yeast population

t i (h) Amount of yeast, b, t, (h) Amount of yeast, b~

0 9.6 10 513.3 1 18.3 11 559.7 2 29.0 12 594.8 3 47.2 13 629.4 4 71.1 14 640.8 5 119.1 15 651.1 6 174.6 16 655.9 7 257.3 17 659.6 8 350.7 18 661.8 9 441.0

Source: Data of Carlson, 1913 (after Pearl, 1927).

and observation equation:

B(t)=N(t)+W(t); O<~t~T (19)

where V(t) typically is a zero-mean Gaussian white noise process, and B(t) an observation; the unknown parameters r, K can be estimated using (16) and (17).

Numerical results were obtained using a fourth-order Runge-Kutta in- tegration method with grid intervals equal to 1 /10. Nineteen observations were made over a period t = 18. The observations including the initial condition at time t = 0 are shown in Table 1. The initial estimates of the parameters r, K were assumed to be:

P= (1, 0.41, 0.41); / £ = (2000, 1000, 840) T h e es t imates of r, K for the success ive a p p r o x i m a t i o n s are g iven in

Tables 2 - 4 for the integrat ion o f the l inearized e q u a t i o n s (10), (11) a n d (12).

TABLE 2

Estimates of the parameters as a function of iteration number

Iteration number

Parameter estimates r and K

0 1 2000 1 0.7914 620.7979 2 0.5226 623.7809 3 0.5503 663.0148 4 0.5498 664.0493 5 0.5497 664.0428

T A B L E 3

Est imates of the parameters as a funct ion of i terat ion n u m b e r

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I tera t ion n u m b e r

Parameter estimates r and K

0 0.41 1000 1 0.4865 533.4796 2 0.5585 659.8988 3 0.5498 664.0021 4 0.5497 664.0431

T A B L E 4

Est imates of the parameters as a funct ion of i terat ion n u m b e r

I tera t ion n u m b e r

Parameter estimates r and K

0 0.41 840 1 0.5072 558.0405 2 0.5538 662.6367 3 0.5498 664.0566 4 0.5497 664.0432

The results show that in all cases the final estimates of the parameters r and K are almost the same. It should be noted that the region of conver- gence is larger than from other methods. Pearl (1927) calculated logistic curves for this data and obtained the results P = 0.55 and /£ = 665.

D I S C U S S I O N

It is well-known that quasilinearization is a powerful numerical method used to obtain the solution of dynamic system identification problems (Bellman and Kalaba 1965; Lee, 1968 pp. 179-212; Miele et al, 1970; Sage and Melsa, 1971 pp. 90-157; Kalaba and Spingarn, 1982). Its major advantages are that it is extremely accurate and quadratically convergent. But we have not seen references on its applications to ecological problems in the past. The only drawback in quasilinearization is that for large dimen- sional problems, i.e. with many state variables and parameters, a lot of partial derivatives must be computed by hand. This aspect has reduced the applicability of quasilinearization. We suggest that this method, in many cases, may be more appropriate for small-dimensional problems than other methods for identifying biological parameters of nonlinear dynamic ecosys- tem models.

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