The collaborative evaluation of enterprise innovation populations is a hot issue. The Lotka–Volterra model is a mature method used to evaluate the interaction mechanism of populations and is widely used in innovation ecology research studies. The Lotka–Volterra model mainly focuses on the quantitative characteristics of the interactive populations. The growth mechanisms cannot explain all the synergy mechanisms of the innovative populations. The collaborative evaluation between enterprise innovation populations is a typical multiobjective evaluation problem. The multichoice goal programming model is a mature method to solve multiobjective optimization problems. This paper combines the Lotka–Volterra model and multichoice goal programming method to construct a three-stage multiobjective collaboration evaluation method based on Lotka–Volterra equilibrium. An evaluation example is used to illustrate the application process of this method. The method proposed in this paper has excellent performance in computing, parameter sensitivity analysis, and model stability analysis.
Innovation activity is a systems engineering issue. It is difficult for a single organization to have all the resources necessary for innovation. It has dynamic evolution, symbiosis, and concurrence formed by the connection and transmission of material flow and talent flow among various innovation populations (IPs) under a specific innovation environment in a certain region. The innovation resources in a certain region are limited. The basic problem addressed by theoretical research and social practice in the fields of economics and management is how to allocate resources optimally [
The population scale in a region determines the population density. Existing studies generally use the population density index to analyze the relationship between population scale and innovation performance. Some studies found that industrial clusters have a positive impact on enterprise innovation [
The Lotka–Volterra model is often used in population ecology to analyze the cooperative or competitive relationship of populations. Studies have shown that the introduction of Lotka–Volterra, a population competition model in biology, into market competition and diffusion has produced better analysis results [
Three-stage multiobjective collaboration evaluation method.
As shown in Figure
In this study, the interaction model of IPs is constructed based on the perspectives of ecological theory and innovation theory, which form the theoretical basis of this study. A multichoice model is used to determine the appropriate scale of the IP under resource constraints, population synergy, and maximum output targets. To resolve the multichoice optimization problem mentioned above while taking into account the collaborative development of the IP, a comprehensive method is required. Based on the perspective of resource constraints, this paper constructs a dynamic model of the growth for the IP. This paper estimates the suitability of population size by using the multichoice goal programming method. Two proposed models are constructed to obtain the appropriate population scale. This research constructs the theoretical model from two aspects, namely, input constraint and output maximization, which is theoretically innovative. Our research also has practical significance because it provides an appropriate analysis method for various innovation subjects to analyze and plan the development of IPs.
Over the past 10 years, the concept of innovation ecosystem has become popular among the rapidly growing literature [
Related literature uses the theory of organizational ecology to study the coevolution process of corporate populations. Baum, Korn, and Kotha empirically studied the competitive advantages and survival rate of incumbents and new entrants in the telecommunication service industry after the technical standards have been established [
Adomavicius, Bockstedt, Gupta, and Kauffman qualitatively studied the interdependence and mechanism of the three types of technologies: components, basic common technologies, and products and applications [
Almost all of the above-mentioned studies are based on a certain industry, starting from the enterprise level, using the number of enterprises entering and exiting the market as the research data source, using ecological related models to verify the evolution of a single population or the coevolution relationship between two subpopulations. In reality, especially in high-tech industries, it is more common to form an interinfluenced and interdependent technological innovation ecosystem around the industrial chain. There are few relevant literatures focusing on technological populations in the innovation ecosystem, and qualitative research is the main focus.
In this section, the growth model of the innovation populations (IPs) is to be constructed. Based on the Lotka–Volterra model, an innovation population relationship model is proposed, and the equilibrium point is analyzed.
When there are abundant resources, populations can grow at geometric or exponential rates. As resources are depleted, population growth rate slows and eventually stops. This is known as logistic population growth. The environment limits population growth by changing birth and death rates. On average, small organisms experience increases per capita at higher rates and more variable populations, while large organisms have lower increase rates per capita and less variable populations. In view of resource constraints and the need for specialization, it is difficult for any single firm to develop and commercialize a technology-based offering from start to finish [
The study of enterprise IP dynamics should consider the influence of regional constraints. The ecological sense of a population is a collection of certain organisms within a given time and space. The region where the population grows is a relatively homogeneous nonlinear region that is different from the surrounding environment. There are universal temporal and spatial constraints in natural hierarchy systems.
The spatial distribution characteristics of different ecological regions such as size, shape, boundary, nature, and distance make up different ecological zones, forming the differences of ecosystem and regulating population growth. The model of enterprise IP dynamics focuses on the quantity change in the IP. Its changing rule is based on the nonlinear growth principle of biological population quantity. The growth model of most species is nonlinear in nature. The number of innovative enterprises may change rapidly with the influence of incentive-based policies and innovation resources in a given period and in a given area.
Competition and synergy within populations are also important factors, based on the intraspecific competition principle of biological populations. Competition exists within the biological population. The larger the population scale is, the more intense the competition will be. Competition among populations has the function of population size adjustment. There is also a certain competition mechanism in the IP, and this competition mechanism will suppress the excessive expansion of the IP, to some extent.
Therefore, intraspecific competition is also one of the processes for the survival of the fittest. Thus, this mechanism should be an important component of the growth model for entrepreneurial population. There is also a competition or synergy relationship between the different IPs. Based on the points raised above, this study uses the growth dynamics model of biological population theory to investigate the development characteristics of IPs.
According to the logistic model, we construct an internal relationship model of innovation population 1 (IP1) as follows:
If
If
According to the logistic model, this paper constructs an internal relationship model of innovation population 2 (IP2) as follows:
Among them,
Equation (
The Lotka–Volterra model of dual-population or multipopulation growth is a differential dynamic system to simulate the dynamic relationship between populations in the innovation ecosystem. Based on the numerical value of ① When ② When ③ When ④ When ⑤ When
In innovation activities, competition can occur between populations that use common resources. Symbiosis in the innovation ecosystem does not exclude competition. Innovative populations in completely or part of the same living space need to conduct technology, talent, and market interaction in the factor market. The competition for capital and then separating, expanding, and alliance niche occupy a more favorable living position, enhance its core competitiveness, and form new offspring to adapt to the environment of the innovation ecosystem through mass reproduction. However, when one party in the innovation ecosystem relies on another core or dominant population to obtain resources and living space, a parasitic relationship is formed. Under the parasitic relationship, the symbiotic subject has a one-way exchange of interests. Because of the one-way asymmetric exchange, this state is not extensive. Therefore, the system will gradually develop in the direction of symbiosis that is conducive to mutual dependence and mutual benefit.
The equilibrium point of the evolution of the innovation populations means that the output of both parties has reached the maximum and remained stable. The following uses the stability analysis of the equilibrium point to discuss the symbiosis stability of the innovative populations 1 and 2. When the two populations reach a symbiotic stable state, the differential equations can be expressed as
Solving the equations can get the equilibrium point of the symbiotic relationship between the two IPs:
The Jacobian matrix is used to solve the equilibrium point of the symbiotic evolution model of the innovation population in the innovation ecosystem. Obtain the determinant Det (J) and trace Tr (J) of the Jacobian matrix. When Det (J) > 0 and Tr (J) < 0, the local equilibrium point is in a stable state; otherwise it is not a stable equilibrium point. The following is a comparative analysis of the determinants and traces of the Jacobian matrix (Table
Equilibrium point and stability conditions of innovation populations.
Equilibrium point | Det (J) | Tr (J) | Stability conditions |
---|---|---|---|
Unstable | |||
As shown in Table
Evolution mode among innovation populations.
Influencing factor value | Symbiotic relationship | Stable equilibrium point |
---|---|---|
Parasitic relationship | ||
Favor symbiosis | ||
Asymmetric symbiosis | ||
Symmetry symbiosis |
As shown in Table
Because of the interdependence between the two populations, the population size cannot be zero, so the points
The only nonnegative solution can be obtained by solving the equations above. That is, the equilibrium point is
Objective programming is an effective method for solving the multiobjective programming problem. Its basic idea is to determine a desired value (objective value or ideal value) for each objective function of the multiobjective programming problem. However, due to the limitations of various conditions, these objective values are often impossible to achieve. Therefore, positive or negative deviation variables are introduced into each objective function to represent the situation where the objective value is either exceeded or not reached. To distinguish the importance of each objective, the priority and weighting coefficient of the objective are introduced. Then, constraint equations are established for all objective functions. From this new set of constraints, the scheme to minimize the combination deviation is obtained. The foundations of the objective programming model are simple and easy to understand, and the model and its hypothesis are in line with reality. Compared with other methods, the objective programming method has more flexibility, effectiveness, and convenience in use and implementation when dealing with multiobjective problems.
In recent years, multichoice goal programming (MCGP) has been widely used to resolve many practical decision-making problems. Chang et al. [
Here,
MCGP is a linear form of objective programming, which can be solved by some common linear programming software. This is because, in minimizing the objective function, the objective function can be infinitely close to the value of the objective. In the same way, in minimizing the objective function, the objective value can also approach the upper bound of the objective infinitely.
Embed the Lotka–Volterra model and the symbiotic population equilibrium point as constraints into the MCGP model to obtain the Lotka–Volterra-MCGP model:
Multichoice goal programming embedded with Lotka–Volterra equilibrium is a linear form of objective programming, which can be solved by some common linear programming software.
Based on the classic Cobb Douglas production function, this paper chooses the accumulated assets of R and D investment as the main investment indicator and adds R and D human capital to the model robustness test part to measure enterprise innovation investment. For the research on enterprise innovation output, many studies use the number of patents to measure [
The two related innovation populations (IP1 and IP2) have a synergistic effect between them. Variable interpretation and data selection are as follows (shown in Table
Relevant data for the example.
Area | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
19186 | 1610 | 157887 | 16448 | 1190 | 417 | 22214 | 1262 | 180101 | 2027 | |
18872 | 1456 | 166445 | 14874 | 1113 | 345 | 21586 | 1044 | 188031 | 1801 | |
14150 | 1328 | 131966 | 13567 | 998 | 325 | 14843 | 983 | 146809 | 1653 | |
12283 | 1239 | 172787 | 12658 | 944 | 248 | 13938 | 750 | 186725 | 1487 | |
11133 | 1080 | 186220 | 11033 | 909 | 208 | 12362 | 629 | 198582 | 1288 | |
7712 | 964 | 118919 | 9848 | 795 | 108 | 9406 | 327 | 128325 | 1072 | |
2257 | 525 | 71781 | 5363 | 714 | 333 | 6726 | 1008 | 78507 | 858 | |
2159 | 417 | 46976 | 4260 | 713 | 300 | 3618 | 908 | 50594 | 717 | |
2508 | 684 | 23925 | 6988 | 693 | 113 | 2429 | 342 | 26354 | 797 | |
2236 | 571 | 14616 | 5833 | 632 | 115 | 1823 | 348 | 16439 | 686 | |
1831 | 401 | 7835 | 4097 | 658 | 290 | 1600 | 878 | 9435 | 691 | |
1695 | 324 | 5436 | 3310 | 700 | 269 | 1078 | 814 | 6514 | 593 |
We can get the IP1 and IP2 interpopulation relationship model (Figure
Figure
Interpopulation model of IP1 and IP2.
In the innovation ecosystem, multiple related populations affect each other. There are also interactive behaviors within the IP of enterprises. This kind of mutual influence behavior can be expressed as coordination or competition. Due to the constraints of the total resources in the innovation environment, the collaboration between populations or within populations may not necessarily promote innovation. The total resource variable is set in the model, which fully reflects the resource constraint mechanism.
In this case, the number of authorized patents is taken as a measure of innovation output. We use the objective solution of MCGP and the solution of equilibrium value to construct the suitability of population.
The related functions and parameters are listed below:
In basis of MCGP-achievement, this problem can be formulated as follows:
Considering the symbiotic relationship [
The problem is solved using the LINGO [
Solution of the MCGP model.
Area | Sample observations | MCGP model | L-MCGP model | |||||
---|---|---|---|---|---|---|---|---|
IP1 | IP2 | MIP1 | MIP2 | LMIP1 | LMIP2 | |||
19186 | 1190 | 15069 | 2041 | 18275 | 1402 | 0.10 | 0.10 | |
18872 | 1113 | 17546 | 506 | 16815 | 1162 | 0.10 | 0.10 | |
14150 | 998 | 12279 | 1667 | 15074 | 1092 | 0.10 | 0.10 | |
12283 | 944 | 17987 | 0 | 17037 | 850 | 0.30 | 0.10 | |
11133 | 909 | 19129 | 0 | 18310 | 733 | 0.56 | 0.10 | |
7712 | 795 | 12361 | 0 | 11952 | 366 | 0.19 | 0.10 | |
2257 | 714 | 6716 | 758 | 6304 | 1126 | 0.10 | 0.10 | |
2159 | 713 | 3384 | 1334 | 4733 | 1008 | 0.10 | 0.10 | |
2508 | 693 | 0 | 2273 | 7764 | 380 | 0.10 | 0.10 | |
2236 | 632 | 0 | 1418 | 6481 | 386 | 0.10 | 0.10 | |
1831 | 658 | 0 | 813 | 4552 | 975 | 0.10 | 0.10 | |
1695 | 700 | 0 | 561 | 3677 | 904 | 0.10 | 0.10 |
As seen in Table
This paper evaluates the level of synergy among innovative populations in different regions based on the similarity between sample values and optimized values. The higher the similarity, the better the synergy. The evaluation matrix is
This study uses the entropy method to determine the weight of the two populations in the collaborative evaluation. Use the technique for order preference by similarity to an ideal solution (TOPSIS) method to evaluate the similarity, and sort the regions in the sample at the level of population coordination. The ideal scale of the two populations can be regarded as two criteria for evaluating the synergy of the innovation system.
Entropy weight is an objective weight method. The advantage of entropy weight method reduces the subjective impact of decision makers and increases objectivity [
For example, Mohsen [
Assume Step 1: normalize the evaluation matrix: Step 2: compute entropy: Step 3: the weights of each criterion are calculated:
Calculated based on the data in the example:
Technique for order preference by similarity to an ideal solution (TOPSIS) is a popular method proposed by Hwang and Yoon [ Step 1: construct the normalized decision matrix Step 2: construct weighted normalized decision matrix Step 3: determine the positive-ideal solution (PIS) and negative-ideal solution (NIS), denoted, respectively, as where Step 4: calculate the distances of each alternative from positive-ideal solution (PIS) and negative-ideal solution (NIS): Step 5: calculate the closeness coefficient and rank the order of alternatives:
As shown in Table
Result of TOPSIS.
Area | Rank | |||||
---|---|---|---|---|---|---|
0.011 | 0.035 | 0.165 | 0.083 | 0.334 | 4 | |
0.024 | 0.008 | 0.014 | 0.088 | 0.866 | 1 | |
0.011 | 0.016 | 0.087 | 0.093 | 0.517 | 2 | |
0.056 | 0.016 | 0.098 | 0.063 | 0.391 | 3 | |
0.085 | 0.029 | 0.163 | 0.042 | 0.205 | 8 | |
0.050 | 0.071 | 0.255 | 0.035 | 0.120 | 12 | |
0.048 | 0.069 | 0.249 | 0.037 | 0.130 | 10 | |
0.031 | 0.049 | 0.203 | 0.059 | 0.225 | 6 | |
0.062 | 0.052 | 0.216 | 0.030 | 0.122 | 11 | |
0.050 | 0.041 | 0.185 | 0.046 | 0.200 | 9 | |
0.032 | 0.053 | 0.212 | 0.056 | 0.209 | 7 | |
0.024 | 0.034 | 0.161 | 0.072 | 0.309 | 5 |
The superiority of the model can be tested by the model operation time. Sample data of each year is formulated as MCGP and Lotka–Volterra-MCGP and then solved by LINGO on a PC with CPU time of 3.2 GHz. The average relative performance of MCGP and Lotka–Volterra-MCGP is measured by CPU time. The CPU times of MCGP and Lotka–Volterra-MCGP are both the same of 00 : 00 : 00 (hh : mm : ss). The two models have no significant difference in computing time, and both can be solved quickly.
In order to test the robustness of the Lotka–Volterra-MCGP model, this paper changes the relevant constraints and model parameters to obtain the following new model:
Model optimization results are shown in Table
Result of the robustness model.
Area | LMP′1 | LMP′2 | |||||||
---|---|---|---|---|---|---|---|---|---|
19186 | 1190 | 30892 | 2027 | 560464 | 23023 | 1547 | 18134 | 1218 | |
18872 | 1113 | 26909 | 1801 | 547288 | 22646 | 1447 | 17847 | 1140 | |
14150 | 998 | 25894 | 1653 | 415350 | 16980 | 1297 | 13153 | 1004 | |
12283 | 944 | 21685 | 1487 | 386207 | 14740 | 1227 | 12040 | 1002 | |
11133 | 909 | 19630 | 1288 | 356207 | 13360 | 1182 | 10974 | 970 | |
7712 | 795 | 16510 | 1072 | 269328 | 9254 | 1034 | 7885 | 881 | |
2257 | 714 | 10326 | 858 | 109308 | 2708 | 928 | 2142 | 734 | |
2159 | 713 | 8025 | 717 | 94996 | 2591 | 927 | 1821 | 652 | |
2508 | 693 | 7248 | 797 | 110352 | 3010 | 901 | 2306 | 690 | |
2236 | 632 | 5495 | 686 | 98384 | 2683 | 822 | 2034 | 623 | |
1831 | 658 | 3597 | 691 | 80564 | 2197 | 855 | 1731 | 673 | |
1695 | 700 | 2947 | 593 | 93990 | 2034 | 910 | 1602 | 716 |
As shown in Table
Enterprise population interaction is a common topic [
This paper combines the Lotka–Volterra model and the multiobjective decision model to build an analysis path that takes both input and output perspectives into account. At the same time, the paper used optimization values to evaluate the population size suitability, which makes the evaluation more operable. The results show that the combination of Lotka–Volterra model and multichoice goal programming model can better evaluate the scale suitability of IPs within a region.
In the enterprise innovation ecosystem, multiple related populations affect each other. There are also interactive behaviors within the IP of enterprises. This kind of mutually influential behavior can be expressed as coordination or competition. Due to the constraints of the total resources in the innovation environment, the collaboration between populations or within populations may not necessarily promote innovation. The total resource variable is set in the model, which fully reflects the resource constraint mechanism. The innovation ecosystem can carry out self-organized evolution, and cooperation among populations and among enterprises within populations can promote a win-win situation among multiple populations. However, the fact that there are negative cases in reality challenges the self-organized evolution of the innovation ecosystem. The reason lies in the lack of detailed and in-depth analysis of the mechanisms at play in the innovation ecosystem. The innovation ecosystem is a complex system, and its operation is inevitably affected by the interaction between the subsystems.
In this study, the different life cycles of IP development are not considered, and the demand for resources of IPs in different life cycle are different. Future studies could consider the characteristics of population life cycle development and add life cycle factors into the special analysis model.
The experimental data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This work was supported by the National Social Science Foundation of China (no. 20BGL203). The authors thank Professor Ching-Ter Chang of Chang Gung University for his suggestions on this paper.