Modelling social networks in innovation diffusion processes:
the case of electricity access in rural areas
Fabio Riva’, Emanuela Colombo’, Carlo Piccardi®
“Politecnico di Milano, Department of Energy, via Lambruschini 4, Milan 20156, Italy; +39
02 2399 3866 — fabio.riva@polimi.it
‘Politecnico di Milano, Department of Energy, via Lambruschini 4, Milan 20156, Italy; +39
02 2399 3820 — emanuela.colombo@polimi.it
“Politecnico di Milano, Department of Electronics, Information and Bioengineering, via
Ponzio 34/5, Milan 20133, Italy; +39 02 2399 3566 — carlo.piccardi@polimi.it
DIGEST OF THE FULL PAPER’
1. Introduction
In rural contexts, long-term evolutions of electricity demand can be explained as a diffusion
of new electrical appliances and an increase of theit ownership and use by local people. By
relying on classical innovation diffusion models, it would be possible to simulate eventual
scenarios of electrical appliances diffusion. However, following the recommendations of
Bhattacharyya [1], reliable models should capture some of the specific socio-economic
dynamics of developing countries, especially in rural context that are affected by high
uncertainty, strong non-linear phenomena, complex diffusion mechanisms, time-
adjustments of technology perceptions.
As a first step to deal with all such complexities, with this work we start trying to introduce
an extra complexity in innovation diffusion models, ie. the modelling of social networks. We
adopt a speculative approach: we rely on an ideal case of innovation diffusion in a rural
community and we design some experiments to describe the effect of introducing social
networks in the process. Network-based diffusion scenarios are developed through discrete
agent-based modelling (ABM) approaches, and results are then compared to classical
continuous diffusion models simulated through a system-dynamics (SD) approach.
2. Material and methods
2.1. Network models
For modelling social networks in diffusion processes, we start representing an ideal rural
community through 3 types of networks: (i) Random, (ii) Barabasi & Albert, and (iii) Social.
' The full paper is available under request to the main author: fabio.riva@polimi.it
Random Erdos-Renyi (RND) network
0 5 10 15 20
Barabasi & Albert (BA) network
30
Social (SC) network
4 6
4 2 0
Rnax=11
Ruin=O
Reargei=4.000
Cu=0.0033
Runax=63
Rnin=2
Reangej=3-994
Gag=0.0153
Rmax=34
Riv=1
Farge =3-916
Cn=0.652
a 8
Figure 1. Plots of RND, BA and SC graphs for kig=4 and N=1000.
2.2. Diffusion models
The main hypothesis at the basis of the Bass model is that the social network where the
spread of an innovation takes place is assumed to be fully connected and homogenous. The
diffusion process based on this “fully connected and homogenous” hypothesis is suitable to
be formulated and simulated with the classical stock and flow diagrams of system-dynamics
(SD), as confirmed by a number of studies and books coming from SD-based literature [2]|—
11].
When investigating factors that drive the growth of energy demand, Rai and Henry [12]
suggest that «Agent-based modelling (ABM) is a powerful tool for representing the
complexities of energy demand, such as social interactions and spatial constraints». They
suggest how SD-based models may reveal some limitations in modelling the complexity of
consumer energy behaviours, referring mainly to a lack of representation of social
interactions that ensue within social networks. Many Authors focused their research on
proposing improvements and solutions to the main limitations of the continuous Bass
model. Trying to pursue the same final modelling goal and to contribute to the same effort
of other researchers, our work investigates and discusses the hypothesis of “perfect-mixing”
of adopters and non-adopters within innovation diffusion mechanisms. In particular, we try
to reject the Bass’ assumption that individuals reveal the same behaviours with respect to
their social contacts, and we report the results by modelling an ideal case of diffusion of
“electric appliances” in rural contexts of the world.
2.2.1. Modelling scenarios
In this work, we propose three cases for testing some hypothesis of diffusion mechanisms
within agent-based and SD models. Agent-based simulations have been tested for all the
three types of network (viz. RND, BA, SC), while the hypothesis at the basis of each
simulated mechanism in the three cases has been also modelled in the equivalent SD model.
Within each of the three cases, we simulated some different scenarios; each scenario within
each case accounts for 20 simulations per type of network, for a total of 61 simulations per
scenario: 20 for RND, 20 for BA, 20 for SC and 1 for the deterministic Bass model simulated
through a SD-based approach. In this way, our simulations statistically embrace the
stochasticity due to the process of networks creation and discrete diffusion.
For each case, the total population has been fixed equal to N = 1000; each agent represents
a household of a typical rural community in a developing country, which has received
potential access to electricity at time ¢ = 0. The simulation horizon has been set equal to T=
240 months, that is 20 years, which roughly corresponds to the lifetime of a typical off-grid
system composed by photovoltaic panels and batteries. The diffusion mechanism here refers
to the diffusion of a general type of electrical appliance, once people have received electricity
connection, or also the “decision to ask for grid connection in the house”.
Case 1. In the first case, we simulated the classical Bass model from a SD perspective and
the equivalent agent-based discrete models with the RND, BA and SC networks. We created
5 scenarios by varying the average degree (A) of the network (i.e. the “contact rate” ¢ for the
Bass model) — 4, 6, 8, 10 and 12.
Case 2. In the second case, based on the experience of the authors in the access to energy-
related research, we introduce some hypotheses that may fit with the contexts under study:
in rural areas, the effect of advertising is supposed to be minimal, especially where people
lack electricity and consequently TV, radios, mobile phones, ef. To allow the diffusion
mechanism to start and spread, and to solve the start-up problem, we consider to “seed”
some initial adopters (/.e. a portion Avof the N agents) at time ¢ = 0. We develop 6 scenarios:
for Ray equals to 4 and 8, we set an initial portion of adopters Ap equals to 1, 5 and 10% of
N. As per the previous case, we introduce the adoption fraction / equals to 0.02.
Case 3. The last case simulates the effect of splitting the entire population among two
different classes of potential adopters: the én/lventials and the imitators. From a modelling point
of view, Van den Bulte and Joshi [13] describe influentials as people who are more in touch
with new developments (i.e. affected by external influences as advertising), who in turn affect
both other influentials and the imitators.
In our work, we implement the simulations by relying on MATLAB-Simulink © computing
environment. For SD simulations, we create the stock-and-flows model by using Simulink,
while we develop specific MATLAB scripts for the agent-based diffusion models and for
generating the graphs of the networks. We firstly run the scripts for network formation, in
order to generate pools of graphs to use then within the agent-based diffusion models.
Market Simulation Word of Mouth
Adopters
Adoption Rate
Potaniial Adopters
‘Adoption Rate from Wot
Adoption +
‘fom
Aaverstsing
[fo etn
Figure 2. Example of stock and flows diagram developed in Simulink © for Case 1 and Case 2.
3. Results and discussion
In this section, we report the results of the simulations performed in the three cases. For
each simulation of each scenario, we plot the “electricity adoption curves” representing the
total number of adopters of electrical appliances A(/) at time 4 The blue curves represent the
SD model, while ted, green and yellow curves represent respectively the result of the
diffusion process on RND, BA and SC networks. For the AMB simulations with the three
types of network, the dashed lines represent the 20 simulations per scenario, and the bold
line highlights the average of the simulations. The results are then discussed by comparing
the stochastic agent-based adoption curves with the related SD model: for each scenario of
the three cases, we compare the min and max time interval needed by the agent-based
stochastic curves to reach 50% and 95% of diffusion, and we compare these values to that
of the SD model.
3.1. Case 1
For Case 1, we created 5 scenarios by varying the average degree (A...) of the network (viz.
the “contact rate” ¢ for the Bass model). Results for &...equals to 4, 8 and 12 are plotted in
Figure 3.
1000
Adopters [people]
Ss
s
Adopters [people]
Ss
s
—— SD mode!
~RND NETWORK
~~ BA NETWORK
~~ - SC NETWORK
100 120 140
Time [months]
160
180 200 220 240
~~ BA NETWORK
~~ - SC NETWORK
100 120 140
Time [months]
160 180 200 220 240
1000
Adopters [people]
8
Ss
300 fi
Ui
200 i
qi
100 MY
My ~~~ SC NETWORK
0
50 100 150 200
Time [months]
Figure 3. Diffusion curves for Case 1: results for Aig=4, 8 and 12
3.2. Case 2
For Case 2, we created 6 scenarios by varying the values of initial adopters for all the diffusion
processes with &,. of the network (vz. the “contact rate” ¢) equals to 4 and 8. Results for Ray
equals to 4 and Ap= 1%, 5%, 10% of N are plotted in Figure 4.
14000
900
‘Adopters [people]
8
8
180 200 220 240
60 80 100 120 140 160
Time [months]
6
‘Adopters [people]
8 8 8
8 8
20 40 60 80 100 120 140 160 180 200 220 240
Time [months]
600
Adopters [people]
eg
8g
és
g
s
Ss
SC NETWORK
20 40 60 80 100 120 140 160 180 200 220 240
Time [months]
Figure 4. Diffusion curves for Case 2: results for kig=4 and Ag = 1%, 5%, 10%,
3.3. Case 3
For Case 3, we created 6 scenarios by varying the value of »— i. the relative importance that
imitators attach to influentials’ versus other imitators’ behaviour (Ow S1) — for all the
diffusion processes with Aa, of the network (vz, the “contact rate” ¢) equals to 4 and 8.
Results for a= 4 and w= 0.03, 0.15, 0.75 are plotted in Figure 5, left side, while results for
Ray= 8 are on the tight side.
‘Adopters [people]
Adopters [people]
Adopters [people]
2 40 60 80 100 120 140 160 180 200 220 240
Time [months]
20 40 60 80 100 120 140 160 180 200 220 240
Time [months]
20 40 80 80 100 120 140 160 180 200 220 240
Time [months]
Adopters [people]
Adopters [people]
‘Adopters [people]
8
2
8
100 120 140 160 180 200 220 24
Time [months]
20 40 60 80
100 120 140 160 180 200 220 2aC
Time [months]
20 40 60 8
100 120 140 160 180 200 220 24C
Time [months]
Figure 5, Diffusion curves for Case 3: results for ay=4 (left) and 8 (right). From top to bottom, »=0.03, 0.15, 0.75.
The next Figure 6 represents the fraction of the population adopting over time / for the 4
diffusion processes when Aa. is equals to 4, to highlight some particular patterns due to the
subdivision of the population among influential and imitator households.
0035} |
ar a er ae
Tien {months}
Time {months}
120 160
Tio {months}
mo 20
zo 240
i ais
0.04) |
i\
aos/ |
) a a Se ee oe ee
te rene ne et
i
_
isi :
; = /
te ent
sas is
0.02 | [Total } a [Total
ose! 4
os) |
i |
ozs |
I. |
oot |
Ls
fa ae ee ss
rie rol ‘i cin
= a)
m0 240 10 40 mo 240
om feo
Tio [months
Figure 6. Adoption fraction over time / for the diffusion process of Case 3 when kix=4. From top to bottom: SD model,
ABM on RND, BA, SC networks, and »=0.03 (left), »=0.15 (centre), »=0.75 (right).
i080
Time {months}
In many simulations, the agent-based processes do not reach 100% of adoption for Ra, = 4,
and the relative portion of population is numerically relevant in the cases resumed in Table
1, While RND-networks present some isolated nodes that prevent complete adoption, the
lacking adoption by some agents in case of BA- and SC-network processes is due to the too
short simulation horizon.
‘Table 1. Percentage of maximum adoption at t=241 months for RND, BA and SC processes at &ay=4
max adoption
w=0.03 w=0.15 w=0.75
RND
BA
sc
93.2-98.0 95.7-97.9 71.6-97.8
79.0-99,9 - -
91.3-99.5 97.2-99.6 75.2-100,
The decline of fisheries in Japan described by a simple
dynamic model
Ilaria Perissi(1), Alessandro Lavacchi (2), Toufic el Asmar(3), Ugo Bardi (1, 4)
1. Consorzio Interuniversitario Nazionale per la scienza e la tecnologia dei Materiali
(INSTM); at Universita di Firenze, Chemistry Department, Via della Lastruccia 3, Sesto
Fiorentino, Italy, tel. +390554573119; email: ilariaperissi@gmail.com
2. Consiglio Nazionale delle Ricerche, CNR-ICCOM; Via Madonna del Piano 10, 50019
Sesto Fiorentino, Italy; tel. +390555225250; email: alessandro.lavacchi@cnr.it
3. Food and Agriculture Organization, FAO; Roma, Italy, Tel. +390657055739, Fax.
+390657053057, email: toufic.elasmar@fao.org
4. Dipartimento di Scienze della Terra, Universita di Firenze; at Chemistry Department, Via
della Lastruccia 3, 50019 sesto Fiorentino, Italy; tel +390554573118, fax:+390554573120;
email: ugo.bardi@unifi,it
Keywords: overfishing, exploitation, Lotka-Volterra, system dynamics
Abstract
Fisheries have been a historical playground for dynamic models involving depletion and
resource overexploitation, inspiring Vito Volterra in the development of what was probably
the first system dynamic model of resource depletion. The model is known today as the
Lotka-Volterra (LV) model. In the present paper, we examine the specific case of the
Japanese fisheries by means of a simple dynamic system based on the original LV model.
We assume that the prey is the fish stock and the predator is an aggregated parameter
that takes into account the capital stock of the fishing industry, introducing in the model
innovative elements beyond the populations of predators and prey. The results confirm
those of earlier work (Ugo Bardi & Lavacchi, 2009) on the behavior of the 19" century
whale fishery and show that the LV model can be used for the quantitative description of a
real-world model.
Introduction
The overexploitation of the world's fisheries is a much debated problem in view of the
depletion of many fish stocks (Myers & Worm, 2003), (Worm et al., 2009), (Lotze &
Worm, 2009), (Pauly, 2009) (Watson et al., 2013). The question is obviously complex and
the yields of fisheries depend on a variety of physical and economic factors which may be
difficult to disentangle from each other. In the present paper we wish to contribute to the
understanding of the problem by presenting evidence that, at least in some cases, the
behavior of fisheries can be quantitatively understood by means of a simple dynamic
In case of Ang = 8, the maximum adoption fraction is less than 100% in 16 simulation of SC
model, and the final adopters range from 909 to 1000.
4. Conclusion
Since electricity use in remote contexts reveal many complex dynamics to deal with, we start
investigating the effect of social network by relying on an ABM approach. We introduced 3
types of networks — (i) Random (RND), (ii) Barabasi & Albert (BA), and (iii) Social (SC) —
and we simulated 3 different Cases of diffusion processes, designing many experiments and
different scenarios for each one, and finally comparing the results with the corresponding
diffusion model simulated through a system-dynamics (SD) approach in continuous time.
We adopt a speculative approach, being the final aim of the paper to investigate how the
modelling of social network may impact of diffusion process.
In Case 1, the diffusion curves generated with both the AMB and SD approaches show all
the same trend, that is the typical S-curve trend of classical continuous diffusion models.
However, the agent-based diffusion processes take longer to complete, especially the SC-
based processes, with some processes reaching 95% of diffusion from 6 to 77 months later
the SD one.
In Case 2, the agent-based processes nose up in the first months, sometimes reaching 50%
of adoption till 24 months before the Bass model — in the case of BA networks —, while they
slowly approach the plateau at the end, sometimes reaching 95% of adoption till 136 months
later the SD model, or even without reaching it — in the case of SC networks.
In Case 3, the ABM processes presents high variability and stochastic uncertainty, and the
final diffusion curves highly depend on the initial assignment of roles of influential and the
relative importance that imitators attach to influentials’ versus other imitators’ behaviour.
The agent-based diffusion processes take longer to complete, specifically the SC- and BA-
based processes when influentials have low importance — with some processes reaching 95%
of diffusion till 136 months later the SD model, or even without reaching it —, and the RND-
based processes when influentials have high importance — with some processes reaching 95%
of diffusion till 183 months later the SD model, or even without reaching it.
The results obtained in this paper confirm how the ABM of social networks in diffusion
ptocesses may considerably impact on diffusion mechanisms, leading to unexpected agents’
rates of adoption and timing to complete the process. Such understanding may be pivotal
for local electricity utilities, which manage off-grid systems, especially when they make their
investment plans, and define the electricity tariffs for guaranteeing a positive return of
investment. Indeed, unreliable prediction may have an unpredictable ruinous impact on
utilities’ financial resources and stability, since they may overestimate or underestimate the
amount of energy that people are forecasted to consume at time /
References
{1] S.C. Bhattacharyya and G. R. Timilsina, “Modelling energy demand of developing
countries: Are the specific features adequately captured?,” Exergy Policy, vol. 38, no. 4,
pp. 1979-1990, 2010.
[2] J. D. Sterman, Business dynamics: systems thinking and modeling for a complex world. London,
United States, 2000.
BI)
4]
5]
[8]
PI
[10]
[14]
[12]
(13)
H. Rahmandad and J. D. Sterman, “Reporting guidelines for simulation-based
research in social sciences,” Syst. Dyn. Rev., vol. 28, no. 4, pp. 396-411, 2012.
P. Milling, “Decision support for marketing new products,” in The 1986 Conference of
the System Dynamics Society, 1986, pp. 787-793.
R. M. Mooy, D. J. Langley, and J. Klok, “The ACMI adoption model: predicting the
diffusion of innovation,” in 2004 International System Dynamics Conference, 2004.
S. Y. Park, J. W. Kim, and D. H. Lee, “Development of a market penetration
forecasting model for Hydrogen Fuel Cell Vehicles considering infrastructure and
cost reduction effects,” Exergy Policy, vol. 39, no. 6, pp. 3307-3315, 2011.
L. Santa-Eulalia, D. Neumann, and J. Klasen, “A Simulation-Based Innovation
Forecasting Approach Combining the Bass Diffusion Model, the Discrete Choice
Model and System Dynamics: An Application in the German Market for Electric
Cars,” in SIMUL 2011: The Third International Conference on Advances in System Simulation,
2011, no. c, pp. 81-87.
W.-S. Chen and K.-F, Chen, “Modeling Product Diffusion By System Dynamics
Approach,” J. Chinese Inst. Ind. Eng., vol. 24, no. 5, pp. 397-413, 2007.
P. M. Milling, “Modeling innovation processes for decision support and management
simulation,” Sys. Dyn. Rev., vol. 12, no. 3, pp. 211-234, 1996.
F. H. Maier, “New product diffusion models in innovation management—a system
dynamics perspective,” Syst. Dyn. Rev., vol. 14, no. 4, pp. 285-308, 1998.
M. Kune, “System Dynamics and Innovation: A complex problem with multiple levels
of analysis,” in 30th International Conference of the System Dynamics Society, 2012.
V. Rai and A. D. Henry, “Agent-based modelling of consumer energy choices,” Nat.
Clim. Chang,, vol. 6, no. 6, pp. 556-562, 2016.
C. Van den Bulte and Y. V. Joshi, “New Product Diffusion with Influentials and
Imitators,” Mark. Sci., vol. 26, no. 2, pp. 400-421, 2007.
model that assumes that overfishing is the main parameter involved in the decline of the
fish stock and, as a consequence, of the fishery yield.
The model we are using is derived from the well known “Lotka-Volterra” (LV) (Alfred J.
Lotka, 1925) (Volterra, 1926) model (also know as the “Foxes and Rabbits” model). It is
based on two coupled partial differential equations that describe the behavior of the
simplest possible trophic chain in biological systems: a predator and a prey. The structure
of this simple model can be seen as the ancestor of modern “system dynamics” (Forrester,
1989) a method of simulation of complex systems also based on coupled differential
equations. The LV model was found to perform poorly for real biological systems (Hall,
1988) but we can show here that it can be used to describe the historical data for at least
some cases of a well known economic system: fisheries. We assume that the prey is the
fish stock and the predator is an aggregated parameter that takes into account the capital
stock of the fishing industry. We report here an example of this approach for the case of
the Japanese fishing industry, where we observed that the model can provide a good fit
with the historical data. We do not claim that this model is of general validity for the
world's fishing industry. However, the fact that it can provide a good insight for at least
some cases show that overexploitation is an important parameter that needs to be taken
into account when studying the economics of fisheries .
Model outline
The two equations that form the Lotka-Volterra model are well known and can be written
as follows:
R’ = kokR-kiCR
C’= k,CR — k3C
In this model, R is an exploitable 'resource stock’ (the prey) and Cis the ‘capital stock'
(the predator). We define as R’and C’as the flow (the variation as a function of time) of
the stocks of resources and capital. Further parameters of the model are the initial stocks
of resource (R,) and of capital (C,). The model depends on four constants, namely: how
fast the resource renews itself (ko), how efficiently the resource is extracted or produced
(k:), how efficiently the resource is transformed into capital (kz) and how rapidly capital
depreciates (ks). The dimensions of these constants depend on the units used for the
capital and resource stocks.
It is well known that this version of the LV model generates infinite oscillations in the
amount of both stocks. In the examination of the historical behavior of fisheries, there are
examples of multiple cycles of oscillations that may be related to this behavior. However,
in this study we have examined systems where a single cycle is observed in the historical
record and where, therefore, the reproduction coefficient (ko) can be considered as close
to zero. In other words, in these systems, the reproduction of the stock is so slow that the
system behaves as if the stock were not renewable.
In this form, the model generates a declining sigmoid curve for the R stock, that behaves
as if it were a non renewable resource. As a consequence, the flow of the stock, R' (the
first derivative of the stock amount) appears as a sigmoid curve. Also the capital
parameter follows a bell shaped curve, but shifted forward in time with respect to the flow
of the resource stock. These two latter parameters, R' and C, are especially important
when using the model to describe historical cases, since they can be estimated by means
of suitable proxies, whereas the actual stock, R, is usually not an easily available datum.
In the model, R' is normally taken as the “landings”, that is the fishery yield, whereas the
capital, it could be represented by the labor force (in number of person or in salary)
employed in the fishery, the number of fishing vessels, the tonnage of the fishing fleet;
and, in recent statistics, also data of investment in currency or expressed by economic
indexes are available.
This is the model that was used in the present study, details about the fitting procedure
and the calculation of the goodness of fit are reported in the appendix.
Model testing and discussion
The LV model described in the previous section was tested in several cases of historical
fisheries. Here, we report the case of the fishery sector of Japan. This is an important
fishery that ranks as the 6th in the world in terms of productivity, harvesting more than
3.6 million metric tons of fish in 2012, according to the Food and Agriculture Organization
(FAO).
Here we report the evolution of the Japanese total fish catch and the national
Disbursement of Fishery, from 1962 to 2000 (Source: Statistics Bureau, Ministry of
Internal Affairs and Communications, website http://www.stat.go.jp/).
The catch data are expressed as the quantity, in weight, of fish. The disbursement is
expressed in currency and it includes expenses for the fisheries’ in terms of wages, fuel,
fishing boats capital and replacement and equipment, thus, it is a direct measurement of
the capital investment in the sector. As shown in the figure, the historical data show a
decline in the fishery yield starting with approximately 1980. The capital expenditure
peaks approximately with the peak yield, but it declines less rapidly afterward.
1,2
oo? e Japan Fish Catch (weight)
- Capital Fishery investment (Yen)
>
0,0 T T T
1970 1980 1990 2000
year
Fig. 1. Lotka Volterra modeling of Japanese total Catch (production-prey) and the Disbursement of Fishery
(capital-predator) from 1962 to 2000. Normalizing factors: catch 1.26 10’ Tons, disbursement 1.35 10° Yen.
Data Source: Statistics Bureau, Ministry of Internal Affairs and Communications (http://www.stat.go.jp/).
The goodness of fit (GOF) is obtained by calculating the normalized means square errors (NMSE) function
(see supplemental data). For the fitted data: NMSE Catch fit: 0.81; NMSE Disbursement fit: 0.92.
The LV model can describe these trends reasonably well; indicating that the fish stock and
the capital stock relate to each other in a prey/predator relationship where the yield of the
fishery depends on the product of the remaining resource stock times the available capital
stock. As the fish stock declines, the profits of the industry decline, too, and less resources
are available for replacing the obsolescence of the fishing capital, which therefore declines
as well. This is a typical behavior of these predator/prey systems.
Recent data from the Statistical Handbook of Japan 2015 (OECD, 2015) show that, since
2000, the Japanese fish production trend is still declining. The value of the catch is
decreasing with a rate of 25% from 2000 to 2014. For the same period, the Statistical
Handbook of Japan 2015 also reports the number of Enterprises and the number of
workers engaged in the Fishery sector. The values of such entities, even thought they are
not expressed in currency, can be reasonable assumed proportional to the capital effort
invested in the sector. The data show that the trend, for both, is declining: in particular,
from 2000 to 2014, the number of enterprises is reduced by 39%, while the number of
workers is reduced by 33%.
Conclusion
The history of fisheries tells us of how the yield of the system is determined as the result
of an intensive ‘fish extraction’ effort. Fishes are a renewable resources, and this is
undoubtedly true, as long as the velocity of the resource reproduction is faster respect to
the velocity of depletion. But in fact, in particular after the II world war, the rapid growth
of the economy ,the development of the open access market economy, the ‘velocity of
fishing’ (power fishing) in several different kind of fish supply chain, experienced a fast
speed up respect to the ‘velocity of the fish stock rebuilding’ that remained almost
invariant or even lowered, because more and more younger spawns were caught or
trapped to satisfy the growing fish demand. Thus, in such a situation, a net resource
outflow depletes the fish stock and this flow can not be balanced by the inflow due to the
biological renewability of the resource itself.
Therefore, it is clear that fish can behave as a non-renewable resource, as shown by
another historical case, the collapse of the US whaling industry collapse (Starbuck, 1989).
(U Bardi, 2007), which was also found to fit the Lotka_Volterra model (Ugo Bardi &
Lavacchi, 2009b). There are several other cases of collapsing fish stocks also indicating
that these stocks may behave as non-renewable resources, for instance the Californian
pacific sardina in the 1950s (Wolf, 1992),
With this example, we show that the prey-predator dynamic can describe the behavior of
an economic system: the whole fishing sector of a country operating in a global open
market framework. The innovative approach of this model is that it is a quantitative
application of system dynamics that emphasizes the role of depletion and the feedback
relationships of the various parameters of the models. The reasonably good fitting of the
model with the historical data does not mean that depletion is the only forcing that affects
the system, but it invites to consider depletion as an important parameter even in systems
that are normally defined as “renewable”.
Acknowledgement
This work was partially supported by the MEDEAS project, funded by the European Union’s
Horizon 2020 research and innovation program under grant agreement No 691287. The
opinion expressed in the present work are those of the author's only and are not to be
attributed to any organ of the European Union
References
Alfred J. Lotka. (1925). Elements of Physical Biology. Williams and Wilkins Company.
http://doi.org/10.2105/AJPH.15.9.812-b
Bailey, J. (2016). Adventures in cross-disciplinary studies: Grand strategy and fisheries
management. Marine Policy, 63, 18-27. http://doi.org/10.1016/j.marpol.2015.09.013
Bardi, U. (2013). Mind Sized World Models. Sustainability, 5(3), 896-911.
http://doi.org/10.3390/su5030896
Bardi, U., & Lavacchi, A. (2009). A simple interpretation of Hubbert’s model of resource
exploitation. Energies, 2(3), 646-661. http://doi.org/10.3390/en20300646
Forrester, J. (1989). The beginning of system dynamics Banquet Talk at the international meeting of
the System Dynamics Society. McKinsey Quarterly. Stuttgart, Germany.
Lotze, H. K., & Worm, B. (2009). Historical baselines for large marine animals. Trends in Ecology
& Evolution, 24(5), 254-62. http://doi.org/10.1016/j.tree.2008.12.004
OECD. (2015). Japan 2015. OECD Economic Surveys. Retrieved from
http://dx.doi.org/10.1787/eco_surveys-jpn-2015-en
Papert, S. (1980). Mindstorms: Computers, children, and powerful ideas. NY: Basic Books. NY:
Basic books.
Radovich, J. (1982). The Collapse of the California Sardine Fishery What Have We learned ?
CalCOFI Reports, XXIII, 56-78.
Starbuck, A. (1989). History of the American Whale Fishery. Castle. Retrieved from
https://books.google.it/books?id=zGYRAAAAYAAJ
The World Bank. (2013). FISH TO 2030 Prospects for Fisheries and Aquaculture, (83177), 102.
http://doi.org/83177-GLB
Thurstan, R. H., Brockington, S., & Roberts, C. M. (2010). The effects of 118 years of industrial
fishing on UK bottom trawl fisheries. Nature Communications, 1(2), 1-6.
http://doi.org/10.1038/ncomms1013
Tower, W. S. (1907). A History of the American Whale Fishery. Pub. for the University. Retrieved
from https://books.google.it/books?id=du4JAAAAIAAJ
Volterra, V. (1928). Variations and fluctuations of the number of individuals in animal species living
together. Journal Du Conseil Permanent International Pour L’ Exploration de La Mer, 3(1), 3—
51. http://doi.org/10.1093/icesjms/3.1.3
Watson, R. A., Cheung, W. W. L., Anticamara, J. A., Sumaila, R. U., Zeller, D., & Pauly, D. (2013).
Global marine yield halved as fishing intensity redoubles. Fish and Fisheries, 14(4), 493-503.
http://doi.org/10.1111/j.1467-2979.2012.00483.x
Wolf, P. (1992). Recovery of the pacific sardine and the california sardine fishery. CalCOFI Rep,
33, 76-86. Retrieved from
http://www.calcofi.org/publications/calcofireports/v33/Vol_33_Wolf.pdf
Supplemental information on the model
The model was implemented using MATLAB&Simulink as shown in figure 2.
GI
Experimental Capital comparing ALL
NexpDx
simDx
Experimental Production
Simulated Production
[=|
‘Simulated Capital
Fig. 2. Simulink blocks’ model to obtain the graphical solution of the Lotka Volterra equations adapted for
the study of the fishery dynamics. NexpDx and NexpY are respectively the production and the capital
available data. SimDx and SimY are the simulated production and capital: they correspond to the fitting
curves.
In the implementation of the model, because data of R’ and data of C can be very
different in order of magnitude, we normalized each data series. The fitting procedure is
very sensitive on the initial value of the k1, k2 and k3. Thanks to the graphic solution of
LV with Simulink (fig. 2), we can obtain a set of initial guesses for such constants that are
successively optimized with Matlab employing a nonlinear least squares routine, using as
objective function the sum of the square of residuals (SSE, sum of squared errors of
prediction) here represented by the deviations of the LV predicted data from actual
empirical values of data. All the fitting are provided using the unconstrained nonlinear
optimization method based the Nelder-Mead algorithm. The goodness of fit is generated
by calculating the normalized mean square errors (NMSE) function. NMSE measure the
discrepance between the historical data and the model estimated value. The NMSE value
is calculated by the Matlab toolbox facilities 'goodnessoffit': NMSE equal to 1 represents
the perfect fit, NMSE equal to zero means that real data are no better than a straight line
at matching the model.
