Analysis of the Lotka–Volterra competition equations as a technological substitution model

doi:10.1016/S0040-1625(01)00185-8

Technological Forecasting and Social Change

Volume 70, Issue 2, February 2003, Pages 103-133

https://doi.org/10.1016/S0040-1625(01)00185-8 Get rights and content

Abstract

This paper provides insight into the dynamics of the Lotka–Volterra competition (LVC) equations, a much used competition model, and compares the dynamics of LVC competitive substitution to that of several well-known substitution models. The behavior of the LVC equations is analyzed for the special case of a dominant competitor at equilibrium being replaced after the introduction of a small population of an invading competitor with a competitive advantage. Expressions are derived that describe the growth of the invading competitor and that growth is shown to be of four classes: left asymmetric, logistic, right asymmetric with 1−ε₂ asymptote and right asymmetric with γ asymptote. It is shown that the LVC model reverts to logistic substitution in a market of fixed size, a result with important implications. The LVC equations are fitted to the Gompertz, Bass, Non-Symmetrical Responding Logistic (NSRL) and Sharif–Kabir substitution models and compared using a novel graphical technique. The LVC equations can reasonably mimic the full range of curve shapes exhibited by each of these models.

Introduction

Forecasting technological substitution requires a model that generates intuitive understanding of the factors affecting substitution but that also has good predictive ability. Many first-order substitution models exist [1], [2] that, although often quite effective, model only the invading competitor whose population is increasing and ignore the declining competitor, thus failing to model the fundamental process driving substitution.

The Lotka–Volterra competition (LVC) equations [3], a set of coupled logistic differential equations, model the interaction of biological species competing for the same resources and can also model parasitic and symbiotic relations [4]. The LVC equations model both the emerging and declining competitors, allowing intuitive understanding of the factors driving substitution. The LVC equations have no analytic solution, yet software simulations are simple to construct and are easily modified to dynamically model systems with time-varying parameters. The LVC equations easily generalize to models with more than two competitors.

This paper analyzes the behavior of the LVC system for the case of a dominant competitor at equilibrium being replaced by an invading competitor. Given a system at equilibrium, where the dominant competitor is at market saturation, a small population of an invading competitor, possessing a competitive advantage over the dominant competitor, is introduced into the system. The analysis describes the behavior of the sigmoidal curve of growth of the invading competitor and lists the different classes of behavior as a function of the model parameters.

Several researchers have used the LVC equations to model competing technologies. Bhargava [5] examines the LVC equations as a substitution model, illustrating the variation in substitution curve shape as a function of the model parameters and showing that in certain cases the model reverts to logistic substitution. Bhargava uses time-varying parameters to show that the LVC model can approximate the time response of the Sharif–Kabir [6] and modified “Non-Symmetrical Responding Logistic” (NSRL) [7] substitution models.

Farrell [8] describes the LVC equations as one of several tools for modeling the “daily struggle for existence” among technologies, as opposed to modeling the long-term evolutionary growth of technology. He uses the LVC equations to model lead-free cans replacing soldered cans, tufted carpets replacing woolen carpets, ball point pens replacing fountain pens and nylon tire cord replacing rayon tire cord. Farrell also outlines a method to derive LVC model parameters from experimental data.

Modis [9] discusses using the LVC equations in generalized form to model six types of interaction between technologies: competition, predator–prey, mutualism, commensalism, amensalism and neutralism. Modis gives examples of dynamic shifting of the relation between two technologies from one type of interaction to another and proposes methods to manipulate the interaction to gain advantage in the market.

Pistorius and Utterback [10] discuss the use of the LVC equations to model interaction of technologies in three modes: pure competition, symbiosis and predator–prey. Mathematically, this is done by changing the algebraic signs of the competition coefficients in the two competitor LVC equations [11], yielding pure competition when the competition coefficients are both positive, symbiosis when they are both negative and predator–prey when they are of opposite sign. In an earlier paper, Pistorius and Utterback [12] discuss modeling oscillatory behavior of the interaction of technologies and investigate possible chaotic behavior in economic systems modeled by the LVC equations.

Porter et al. [13] examine the LVC equations in detail, discussing the physical meaning of the model parameters and showing that many types of behavior such as linear, exponential, confined exponential, logistic and Gompertz can be related to the LVC equations at different limiting values of model parameters. Porter's work adds comments on the analogy between biological niches and technological niches, discussing the hierarchical structure of technological niches and stressing the importance of modeling competition between technologies at equivalent levels on the technological hierarchy.

Section 2 of this paper briefly discusses the general behavior of the LVC system using the analysis of nullclines in the phase plane. This analysis establishes the initial conditions and the value of model parameters for which a dominant competitor is replaced by an invader with a competitive advantage. Given these conditions, Section 3 analyses LVC behavior to find the asymptotic time responses that occur during the initial and final stages of substitution. Section 4 discusses the four classes of LVC substitution responses: (1) left asymmetric, (2) logistic, (3) right asymmetric with ε₂−1 asymptote and (4) right asymmetric with γ asymptote. A response diagram is developed, showing the occurrence of each of the response classes as a function of the LVC parameters. Section 4 further discusses conditions under which LVC substitution reverts to pure logistic behavior and shows that logistic behavior results if the sum of the two competitor populations is fixed, a reasonable assumption in many technological substitution situations. Section 4 also shows that LVC substitution is a “coupled logistic” process and graphically demonstrates this concept. Section 5 introduces a graphical technique for comparing substitution curves and uses this technique to compare LVC substitution to the Gompertz, Bass, NSRL and Sharif–Kabir substitution models. Section 6 demonstrates several concepts from this paper using the example of lead-free cans replacing soldered cans.

This paper presents several concepts, techniques and results that are useful for modeling competitive systems and technological substitution. Normalization of the LVC equations to a three-parameter model allows detailed characterization of LVC systems. Prediction of the asymptotic behavior of LVC substitution enables research into techniques to estimate LVC model parameters from experimental data. Prediction of response classes based on LVC parameters adds significant insight into the mechanism of competition. The conditions under which LVC substitution reverts to logistic substitution are particularly important. Reversion of LVC substitution to logistic substitution when the sum of competitor populations is fixed is a result that, with further investigation, may explain the abundance of examples of logistic substitution in competitive situations. Graphical analysis of substitution curves whose asymptotic behavior approaches a confined exponential at saturation allows comparison of substitution curve shapes in a meaningful way. The normalized Bass model in one-parameter normalized form provides insight into the range of curve shapes generated by the model. Comparison of LVC substitution to other substitution models shows that LVC curves have their own unique asymptotic behavior and cannot be fit to other substitution curves with arbitrarily small error. Yet, this analysis demonstrates that the LVC model can mimic the behavior of several substitution curve families over the full range of curve shapes exhibited by those families.

Section snippets

Formulation of the LVC equations

For two-competitor systems, the LVC formulation consists of two autonomous nonlinear differential equations, which, using the notation of Bazykin [4], are $d x_{1} d t = a_{1} x_{1} K_{1} (K_{1} −x_{1} −α_{2} x_{2})$ $d x_{2} d t = a_{2} x_{2} K_{2} (K_{2} −x_{2} −α_{1} x_{1})$ where

•
x₁ and x₂ are the populations of technologies 1 and 2. These can be expressed several ways, e.g., rate of units sold, volume of market share, etc.
•
a₁ and a₂ are the rate constants of technologies 1 and 2. In the absence of any competition and in an infinite market, the growth rate of

Formulation of the LVC equations as a substitution model

Assume u₁ is the dominant competitor and u₂ is the invading competitor. The curve of interest is the sigmoidal curve described by u₂ as it grows from near zero to final saturation, at unity.

General behavior of the LVC system during substitution

The behavior of the LVC system during substitution can be divided into four classes. Fig. 6 shows a response diagram of the LVC system as a function of ε₁, ε₂ and γ. Note that, in accordance with the limitations of , , ε₁ ranges from zero to unity, while ε₂ ranges from unity to infinity. The four classes of behavior, as noted in Fig. 6, are (1) left asymmetric, (2) logistic, (3) right asymmetric with (ε₂−1) asymptote and (4) right asymmetric with γ asymptote.

In the following explanations, note

Graphical method of comparing substitution curves

This section presents a graphical method for comparing substitution curve shapes for curves whose asymptotic behavior near unity approaches that of a confined exponential, 1−exp(−τ). Such models include the logistic model, Bass model, NSRL model, Gompertz model and others. In this technique, all curves are normalized in time to τ=At where A is the rate constant of the asymptotic confined exponential that controls the curve behavior near unity. Using a confined exponential curve as a reference,

Example: lead-free cans replacing soldered cans

This section presents an example of technological substitution taken from Farrell [8], in which lead-free cans replace soldered cans. Fig. 21 contains a plot of the number of lead-free cans and soldered cans manufactured from 1979 to 1988, data that were digitized from a figure in Farrell's paper [8]. A plot of the sum of these two curves is also shown. Using data acquired from the Can Manufacturer's Institute, the number of cans manufactured from 1979 to 1999 for the purpose of vegetable

Conclusions

This paper describes the behavior of the LVC equations during substitution of a dominant competitor by an invader possessing a competitive advantage. The substitution behavior falls into four classes depending on the LVC parameters: left asymmetric, logistic, right asymmetric with 1−ε₂ asymptote and right asymmetric with γ asymptote. LVC substitution reverts to pure logistic substitution under a range of conditions and under the constraint that the sum of competitor populations is fixed. LVC

Steven Morris is a PhD student in Electrical Engineering at Oklahoma State University, Stillwater, OK, USA. Previously, he was a Senior Research Scientist at the Amoco Production Research Center in Tulsa, OK, USA.

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Citation Excerpt :
For instance, the authors in [6] used the competition system for the mobile company data in Korea. The authors in [7] use the competition system as a technological substitution model. Moreover, this competition system is used by the authors for Korean stock market, modeling and policy implications, dynamics of the markets, and banking system [8–11].
The main focus of this paper is to investigate the dynamics of the competition between rural and commercial bank with in the framework of fractal–fractional Atangana–Baleanu derivative sense. Initially, we take a model from literature and then utilize the real data since 2004–2014. We estimate effectively the parameters for the competition fractal–fractional model. Then, we provide a novel numerical technique for fractal–fractional model with many graphical results. We show graphical results for data fitting with the suggested operators and show that these two operators give the same results for the data fitting but varying both the values of the fractal and fractional order we have reasonable results . The results of the data fitting with these two operators are reliable and good are shown. Moreover, we present by comparing these two operators for many values of the fractal and fractional order parameters $θ$ and $α$ .

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Dr. David Pratt is an associate professor at the School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK, USA.

¹: Tel.: +1-918-744-6055.

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Analysis of the Lotka–Volterra competition equations as a technological substitution model

Abstract

Introduction

Section snippets

Formulation of the LVC equations

Formulation of the LVC equations as a substitution model

General behavior of the LVC system during substitution

Graphical method of comparing substitution curves

Example: lead-free cans replacing soldered cans

Conclusions

Technological Forecasting and Social Change

Technological Forecasting and Social Change

Technological Forecasting and Social Change

Technological Forecasting and Social Change

Technological Forecasting and Social Change

Research Policy

Technological Forecasting and Social Change

Technological Forecasting and Social Change

Technological innovation diffusion: The proliferation of substitution models and easing the user's dilemma

IEEE Transactions on Engineering Management

Nonlinear dynamics of interacting populations