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In Arbia and Paelinck (2003a, b), a Lotka-Volterra model (LVM) is applied to the convergence-divergence problem of European regions in terms of incomes per capita. As the latter have to be non-negative, a double logarithmic version may be substituted for the original specification, a modification that removes at least part of the non-linearity of LVMs; this chapter introduces this non-linearity again. Discussion begins with a general section on LVMs, to go on with a mixed linear-logarithmic specification, of which the positivity of the (possible) equilibrium solution is proved, and for which a (sufficient) stability condition is derived. Section 11.3 presents an application of the model to the classical four macro-regions in which the Netherlands is subdivided.

1 Lotka-Volterra Models

In this section, generalized Lotka-Volterra models are introduced, and examples given of some applications, including estimation aspects of the latter.

1.1 A General Specification

A generalized LVM can be written in matrix-vector notation as

$${\dot{\textbf{u}}} = {\hat{\textbf{u}}}{\textrm{ }}\left( {{\textbf{A}}{\textrm{ }}{\textbf{u}}{\textrm{ }} + {\textrm{ }}{\textbf{a}}} \right),$$
((11.1))

where u is a column-vector of (endogenous) variables, û its diagonal matrix version, A a square matrix, and a is a column-vector of fixed coefficients; the •-notation denotes the time derivative, ∂/∂t.

Given equation (11.1), the variables u describe a time path that can take all the characteristics of general continuous dynamic processes (e.g., convergence, divergence, limit circles; see Braun, 1975, §4.9; Gandolfo, 1996, in particular §24.4; Peschel and Mende, 1986). What can be said about equation (11.1) to converge to its focus, –A –1 a? Constructing a Lyapunov-function (Hahn, 1963)

$${\textrm{v = }}( {{\textbf{u}}{\textrm{ }} + {\textrm{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} ){\textrm{}}\hbox{'}({\textbf{u}}{\textrm{ }} + {\textrm{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}})$$
((11.2))

gives

$${\textrm{v = 2}}( {\textbf{{u}}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} )\hbox{'}{\boldsymbol{\hat{\textbf{u}}}\,\textbf{A}} ({\hat{\textbf{u}}} {\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a)}}$$
((11.3a))
$$= ( {\textbf{{u}}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} )\hbox{'}{\textbf{ }}( {\hat{\textbf{u}} {\textbf{A}} {\textrm{ + }}{\textbf{A}\,\boldsymbol{\hat{\textbf{u}}}}} ){\textbf{ }}( {{\textbf{u }}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} ){\textbf{.}}$$
((11.3b))

In the purely linear case, if the real parts of A’s eigenvalues are negative, v is negative definite (Hahn, 1965, p. 26; La Salle and Lefschetz, 1961, p. 48), and the sufficient conditions for asymptotic stability are satisfied. In the LVM case the problem is more involved; the proof of the above sufficiency conditions still being satisfied is given in Paelinck (1992, pp. 142–143).

1.2 Applications

Originally, special versions of the LVM have been applied to the field of bio-mathematics, i.a., to build so-called “predator-prey” models. An example is the following model (all parameters strictly positive):

$${\dot{\textrm{x}} = \textrm{x (a}} - {\textrm{b y),}}$$
((11.4a))
$${\dot{\textrm{y}} = \textrm{y} ( -\, \textrm{c} + \textrm{d} \textrm{x})}{\textrm{.}}$$
((11.4b))

Here x is the prey, developing at a constant rate a, but preyed upon by the predators y; the latter, in the absence of prey animals, fade out at a rate c, but are kept alive by x.

The resulting state diagram in the x-y plane shows a “pseudo-elliptic” closed curve, and the time-explicit graph shows sinusoidal lagged curves of different amplitudes.

The model just described was proposed by Samuelson (1971) as a candidate for dynamic economic analysis, and applied by Dendrinos and Mullaly (1981) to the evolution of urban populations, although no explicit econometric estimation was performed. Before presenting some econometric results, an appropriate estimation method is unfolded here.

The flexibility of the LVM specification is shown by the various time-paths and singular points resulting from various parameter combinations (presence or absence, signs); Braun (1975, pp. 590–599) gives examples of this. For instance, if a term –ex is added to equation (11.4a), and a term–fy to equation (11.4b), both terms representing competition for limited resources, within the prey and the predator group, the solution becomes [x o = a/e; y o =0] for c/d > a/e.

1.3 Simultaneous Dynamic Least Squares (SDLS) Estimation

Consider a (e.g., sectorally, spatially, dynamically) interdependent econometric model (Paelinck, 1996b, §2.1)

$${\textbf{A u }}{\textrm{ + }}{\textbf{ B x }}{\textrm{ = }}\,{\boldsymbol{\upvarepsilon}},$$
((11.5))

where u is a column-vector of endogenous variables, x a column-vector of exogenous ones, ɛ being the usual column-vector of random elements. Equation (11.5) always can be rewritten as

$${\textbf{y}} = {\textbf{Z}}\ {\boldsymbol{\upbeta }} + {\boldsymbol{\upvarepsilon}},$$
((11.6))

where Z comprises at the same time endogenous and exogenous variables. The basic idea of SDLS is to minimize the sum of squared deviations between the observed and the endogenously computed (shown by caps) values of the endogenous variables, u; this leads to

$${\textbf{u}} - {\hat{\textbf{u}}} = [{\textbf{u}}-({\textbf{Z}}-{\hat{\textbf{Z}}}{\boldsymbol{\upbeta}}]-{\textbf{Z}}{\boldsymbol{\upbeta}})] ,$$
((11.7))

and minimizing as said before gives

$${\hat{\boldsymbol{\upbeta}}} = {( {{\textbf{Z}}{\textrm{'}}{\hat{\textbf{Z}}}} )^{-{\textbf{1}}}}{\textbf{Z}}{\textrm{ }}{\textbf{u}},$$
((11.8))

where \({\hat {\textbf{Z}}}\) includes the computed values of the endogenous variables; a possible computing process is an iterative one, but Sect. 11.4 presents a specification with endogenously computed u values.

The following properties hold (Paelinck, 1990b, p. 7–8):

  • β is a generalized reduced form estimator;

  • if ɛ ~ N(0, σ 2 I), then β is a maximum likelihood estimator; and,

  • β is a consistent estimator, and plim ββ’ = σ 2 (Z’Z) –1, for homoscedastic ɛ.

The method has been applied to a two-equation full-parameter LVM process for the city of Rotterdam, The Netherlands, and for the time period 1946–1978 (Paelinck, 1996a, §3), the equations being

$$\Delta\hbox{'} {\textrm{ln }}{{\textrm{x}}_{\textrm{t}}} = {\textrm{ a }} + {\textrm{ b }}{{\textrm{x}}_{{\textrm{t}}-{\textrm{1}}}} + {\textrm{ c }}{{\textrm{y}}_{{\textrm{t}}-{\textrm{1}}}},$$
((11.9a))
$$\Delta\hbox{'} {\textrm{ln }}{{\textrm{y}}_{\textrm{t}}}{\textrm{ = d + e }}{{\textrm{x}}}_{{\textrm{t}}-1}{\textrm{ + f }}{{\textrm{y}}}_{{\textrm{t}}-1}{\textrm{,}}$$
((11.9b))

where x represents population and y per capita income. Table 11.1 (taken from Paelinck, 1990a) presents the estimation results, which can be given plausible interpretations.

Table 11.1 Parameter values of the Rotterdam application
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Note also the presence of the two parameters a * and d *, which allow the initial values to be shifted optimally by the computed process with respect to the observed initial values. Moreover, the resulting relevant eigenvalues lie between –1 and 0, so—abstracting from the discretionarity problem (see Gandolfo, 1996, pp. 411–412)—the process would be asymptotically convergent in terms of population and per capita income. According to the divergence criterion (Gandolfo, 1996, p. 456) the system is possibly anti-dissipative along certain stretches of its time-path, although conservative in its non-trivial singular point.

2 Mixed Specification

In this section, a combined linear-logarithmic specification is presented.

2.1 Equations

Instead of equation (11.1), now consider

^

$${\hat{\textbf{u}}^{-{\textbf{1}}}}{\textbf{u = A }}( {{\textbf{u + ln u}}} ){\textbf{ + a,}}$$
((11.10))

for which the equilibrium solution (if it exists) is

$${{\textbf{u}}^{\textbf{o}}}{\textrm{ + }}{\textbf{ ln }}{{\textbf{u}}^{\textbf{o}}} = -{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}.$$
((11.11))

For each variable ui, the equilibrium solution can be written as

$${{\textrm{u}}_{\textrm{i}}}^{\textrm{o}}{\textrm{ = }}{{\textrm{b}}_{\textrm{i}}}-{\textrm{ln(}}{{\textrm{u}}_{\textrm{i}}}^{\textrm{o}}{\textrm{),}}$$
((11.12))

where b i is generated by the i-th row of –A –1 times a. Now while u i o increases linearly starting from zero, -ln(u i o ) decreases monotonically from +∞ to –∞. Thus, equation (11.12) should be satisfied for some strictly positive value of u i o.

2.2 Stability

Instead of equations (11.3), consider

$${\textrm{v = }}( {{\textbf{u }}{\textrm{ + }}{\textbf{ ln u }}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} ){\textbf{}}\hbox{'}( {{\textbf{u }}{\textrm{ + }}{\textbf{ ln u }}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} ){\textbf{,}}$$
((11.13))

from which one can derive

$$\begin{array}{ll}&\dot{\textrm{v}} = 2( {{\textbf{u }}{\textrm{ + }}{\textbf{ln}}\ {{\textbf{u} }}{\textrm{ + }}{\textbf{ }}{{\textbf{A}}^{-{\textbf{1}}}}{\textbf{a}}} )\hbox{'}{\textbf{}}{( {{\textbf{I }}{\textrm{ + }}}\hat{{\textbf{u}}} )^{{\textbf{1/2}}}}[ {{{( {{\textbf{I }}{\textrm{ + }}}\hat{{\textbf{u}}} )}^{{\textbf{1/2}}}}{\textbf{A }}{{( {{{{\textbf{I}}}}{\textrm{ + }}\boldsymbol{\hat{\textbf{u}}}} )}^{-{\textbf{1/2}}}}} ]{( {{{{\textbf{I}}}}{\textrm{ + }}\boldsymbol{\hat{\textbf{u}}}} )^{{\textbf{1/2}}}}\\&\quad{\textbf{(u }}{\textrm{ + }}{\textbf{ ln u }}{\textrm{ + }}{{\textbf{A}}^{{\textrm{-}}{\textbf{1}}}}{\textbf{a)}}.\end{array}$$
((11.14))

Matrix A has undergone a similarity transformation which keeps the eigenvalues unchanged (Allen, 1956, p. 468). Again Hahn’s argument quoted at the end of Sect. 11.1.1 can be invoked here, which completes the proof of the fact that a sufficient condition for the mixed LVM to be stable is the negativity of the real parts of A’s eigenvalues.

3 Application

Model (11.10) has been applied to the relative GDPs of the four classical Dutch macro-regions; Table 11.2 lists the numbers (N = North; S = South; E = East; W = West, the latter region being known as the “Rimcity”); see Fig. 11.1. The numbers are percentages and relate to the years 1988–2000.

Fig. 11.1
figure 11_1_214082_1_En

The four Dutch regions

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Table 11.2 Numbers used in the Dutch application
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SDLS estimates using the numbers in Table 11.2 are obtained by introducing the following equations (discrete versions of equations (11.10); tildes relate to the computed SDLS endogenous variables)

$${\tilde{{\textbf{u}}}_{\textbf{t}}}{\textrm{ = }}{\textbf{ }}\left( {{\textbf{I }}{\textrm{ + }}{\textbf{ A}}} \right){\textbf{ }}{\tilde{{\textbf{u}}}_{{\textbf{t}}-{\textbf{1}}}}{\textrm{ + }}{\textbf{ A ln }}{\tilde{{{\textbf{u}}}}_{{\textbf{t}}-{\textbf{1}}}}{\textrm{ + }}{\textbf{ a,}}$$
((11.15))

into a mathematical programming model, minimizing the squared residuals of (11.5) or (11.6). As only the fourth observation (relating to 1991) produces a sum diverging significantly from 100 (see also Figs. 11.1,Footnote 1 11.2, 11.3, and 11.4), no extra constraint was introduced. Furthermore, optimal starting points were computed by optimizing simultaneously over the starting vector of computed SDLS variables.

Fig. 11.2
figure 11_2_214082_1_En

Shares GDP time-series plot: northern Netherlands

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Fig. 11.3
figure 11_3_214082_1_En

Shares GDP time-series plot: southern Netherlands

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Fig. 11.4
figure 11_4_214082_1_En

GDP time-series plot: eastern Netherlands

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Table 11.3 presents the main econometric results.

Table 11.3 Econometric results of the Dutch application
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Every region has been assigned only three parameters: its own influence (a 1 ), that of the other three regions (a 2), and a constant (a 3).

A 4x4 matrix can be constructed, dividing each a 2 by 3. From the trace (5.5192) this matrix appears to be non-negative definite (two out of the four a i s are positive), so no mathematical convergence toward the right hand side of equation (11.11) is present.

Figs. 11.2, 11.3, 11.4, and 11.5 present the observed (series 1) and the SDLS-computed (series 2) series.

Fig. 11.5
figure 11_5_214082_1_En

GDP time-series plot: western Netherlands

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Table 11.4 presents simulation results over 20 periods, starting from the value of the year 2000. One notices a progressive decline in the share of the West (“Rimcity”) in favor of all other regions; whether this should be taken at its face value is a problem related to what will be said in the conclusions.

Table 11.4 Simulation results of the Dutch application
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4 Conclusion

The method has proven itself to be workable and could be combined with an appropriate estimation method (SDLS); it has moreover the nice property that, if convergence is present, it will lead to economically acceptable (positive) equilibrium values. This means that discrete LVMs, adapted in the way described, could be an ever more useful tool for future research in multiregional dynamics.

Inspection of Figs. 11.2, 11.3, 11.4, and 11.5 shows some local discrepancies between observed and SDLS computed values. Though the specification chosen is already very flexible, it is not the only one. Other candidates (min-algebraic specifications, finite automata; see Chap. 13) are available, and should be tested against the present specification. Tools are also available (see Chap. 12) and will be used in future research.