Untitled - Institut Teknologi Sepuluh Nopember (ITS)

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PREFACE

International Conference on Mathematics and Sciences or abbreviated ICOMSc is a

conference which organized by Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia. ICOMSc was firstly held on 12-13 October 2011, at Majapahit Hotel, Surabaya. The conference aims to provide a forum for academics, researchers, and practitioners to exchange ideas and recent developments on mathematics and sciences. The conference was expected to foster networking, collaboration and joint effort among the conference participants to advance the theory and practice as well as to identify major trends in mathematics and sciences.

The conference was attended contributors from Indonesia, Malaysia, Thailand, Philippines, United Arab Emirates, Taiwan, Japan, Australia, Spain, and United States of America. Such a spread of participation from around the world confirms the appropriateness of the “International” label of this conference. There were 130 papers and posters presented in the conference, and there are 111 full papers are published in this proceeding.

Finally, the committee would like to thank all keynote speakers, presenters, participants, sponsors, and all those involved directly and indirectly in the first ICOMSc.

Mardi Santoso Conference Chair

Surabaya, December 2011

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Committees Honorary Committee Rector of Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesia Dean of Faculty of Mathematics and Natural Sciences of ITS Advisory Board Prof. Yoh Kohori (Chiba University, Japan) Prof. William S. Price (University of Western Sydney, Australia) Prof. Ariando (NUS, Singapore) Prof. Hadi Nur (UTM, Malaysia) Prof. Ismunandar (ITB, Indonesia) Prof. Sutawanir Darwis (ITB, Indonesia) Prof. Rustam E. Siregar (Padjadjaran University, Indonesia) Prof. Subanar (Gadjah Mada University, Indonesia) Prof. Budy Kurniawan (University of Indonesia, Indonesia) Dr. Nurul Taufiqurrahman (LIPI, Indonesia) Dr. Eniya Listiani Dewi (BPPT, Indonesia) Prof. Khairil Anwar Notodiputro (IPB, Indonesia) Prof. Darminto (ITS, Indonesia) Prof. Nur Iriawan (ITS, Indonesia) Prof. Taslim Ersam (ITS, Indonesia) Dr. Subiono (ITS, Indonesia) Dian Saptarini, MSc. (ITS, Indonesia)

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Organizing Committee Prof. Mardi Santoso Suminar Pratama, PhD. Dr. rer. pol. Heri Kuswanto Soleha, MSc. Arif Fadlan, MSc. Dr. Muhammad Mashuri Rahmah Irma Suryaningsih, MSi. Prof. Agus Rubiyanto Subchan, PhD. Dr. Didik Prasetyoko Dr. Suhartono Dr. rer. nat. Maya Shovitri Dr. Melania Suweni Muntini Dr. rer. nat. Irmina Kris Murwani Dr. Irhamah Bandung Arry Sanjoyo, MComp.Sci. Hamzah Fanzuri, PhD. M. Muryono, MSc. Prof. Basuki Widodo M. Zainul Asrori, MSc. Jerry Dwi Trijoyo Purnomo, MSc. M.Sjahid Akbar, MSc. Lukman Atmaja, Ph.D Dr. Sony Sunaryo Heny Faisal, MSc. M. Agung Pamudjo, SSos.

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Conference Secretariat Dra. Sri Hariyani Friana Ekawati, AMd. Yunita Hari listyowati Ida Srisamsuti, AMd. Dian Rachmat Saputra, SE Marsam Ahmed Usman Ali Drs. Hadi Siswanto Muzammil Drs. Ec. Suparno Cucuk Waluyo, SSos. Sumaryono

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Conference Program

Wednesday, 12 October 2011

08:00 to 09:00 Registration

09:00 to 09:30 Welcome and Opening Remarks

09:30 to 10:00 Coffee/Tea Break

10:00 to 12:00 Plenary Presentation 1 A/Prof. Gopalan Nair School of Mathematics and Statistics, The University of Western Australia, Australia Plenary Presentation 2 Prof. Hiroyuki Kitahata Department of Physics, Graduate School of Science, Chiba University, Japan

12:00 to 13:00 Lunch

13:00 to 15:00 Oral Presentation


15:30 to 16:30

19:30 to 21:00

Oral Presentation

Welcome Dinner

Thursday, 13 October 2011

08:00 to 09:00 Plenary Presentation 3 Dr. Intan Muchtadi Department of Mathematics, Institut Teknologi Bandung, Indonesia


09:30 to 11:30 Plenary Presentation 4 A/ Prof. Maria Elena de Bellard Department of Biology, The California State University Northrige, USA Plenary Presentation 5 Prof. Jyh-Chiang Jiang Theoretical & Computational Chemistry Lab, Chemical Engineering Department, National Taiwan University of Science and Technology, Taiwan

11:30 to 12:30 Lunch

12:30 to 14:45 Oral Presentation

Friday, 14 October 2011

Surabaya City Tour

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Oral Presentation Schedule

Day 1 Brantas1 Brantas 2 Bromo Adika 1 Adika 2 Adika 3 Adika 4 13:00-13:15 OM 01 OM 13 OM 25 OS 01 OS 13 OP 01 OP 13 13:15-13:30 OM 02 OM 14 OM 26 OS 02 OS 14 OP 02 OP 14 13:30-13:45 OM 03 OM 15 OM 27 OS 03 OS 15 OP 03 OP 15 13:45-14:00 OM 04 OM 16 OM 29 OS 04 OS 16 OP 04 OP 16 14:00-14:15 OM 05 OM 17 OM 30 OS 05 OS 17 OP 05 OP 17 14:15-14:30 OM 06 OM 18 OM 31 OS 06 OS 18 OP 06 OP 18

14:30-14:45 OM 07 OM 19 OM 32 OS 07 OS 19 OP 07 OP 19 14:45-15:00 OM 08 OM 20 OM 33 OS 08 OS 21 OP 08 OP 20

Coffee/Tea Break 15:30-15:45 OM 09 OM 21 OM 34 OS 09 OS 22 OP 09 OP 21 15:45-16:00 OM 10 OM 22 OM 35 OS 10 OS 23 OP 10 OP 22 16:00-16:15 OM 11 OM 23 OM 36 OS 11 OS 24 OP 11 OP 23 16:15-16:30 OM 12 OM 24 OS 12 OP 12 OP 24

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Oral Presentation Schedule

Day 2 Brantas1 Brantas 2 Bromo Adika 1 Adika 2 Adika 3 Adika 4 12:30-12:45 OB 01 OB 06 OM 28 OS 20 OS 33 OS 41 OC 01 12:45-13:00 OB 02 OB 07 OM 37 OS 25 OS 34 OS 42 OC 02 13:00-13:15 OB 03 OB 08 OM 38 OS 26 OS 35 OS 43 OC 03 13:15-13:30 OB 04 OB 09 OM 39 OS 27 OS 36 OS 44 OC 04 13:30-13:45 OB 05 OB 10 OM 40 OS 28 OS 37 OS 45 OC 05 13:45-14:00 OM 41 OS 29 OS 38 OS 46 OC 06 14:00-14:15 OM 42 OS 30 OS 39 OS 47 OC 07 14:15-14:30 OS 31 OS 40 OS 48 OC 08 14:30-14:45 OS 32 OC 09

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Contents

Preface iii

Committees iv

Conference Program vii

Keynote Speakers Gopalan Nair School of Mathematics and Statistics, The University of Western Australia, Australia Analysis of Point Patternson the Planean Linear Networks Hiroyuki Kitahata Department of Physics, Graduate School of Scicence, Chiba University & PRESTO JST, Chiba 263-8522, Japan Spontaneous Motion of a Droplet Coupled with Chemical Reaction Intan Muchtadi-Alamsyah Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Algebraic Structures in Cryptography Maria Elena de Bellard Department of Biology, The california State University Northrige, USA Evolution of Glia and Neural Crest Cells Jyh-Chiang Jiang Theoretical and Computational Chemistry Laboratory Chemical Engineering Department, National Taiwan, University of Science and Technology, Taiwan The Role of Computational Chemistry in Experiments from Elucidation to Prediction

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PART I: STATISTICS AND BIOLOGY STATISTICS

OS 01 Adi Wijaya, Suhartono

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Adaptive Neuro Fuzzy Inference System For Forecasting Rice Production

OS 02 Andhie Surya Mustari, Ismaini Zain Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Spatial Tobit Regression Analysis. Case Study of Internet Usage in Java

OS 03 Adityawati N. K., Heri Kuswanto, Suhartono Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Simulation of Time Series Model in Farmers Terms of Trade (FTT) with Arima

OS 06 Yuliana Susanti, Hasih Pratiwi

Sebelas Maret University, Surakarta, Indonesia Robust Regression Model For Predicting The Availability of Soybean in Indonesia Using M-Estimation

OS 07 Alfian F. Hadi1, I Made Sumertajaya1, Rudi Iswanto2 1 Department of Statistics, Bogor Agricultural University, Bogor, Indonesia 2 Indonesian Legumes and Tuber Crops Research Institute, Malang, Indonesia Zero-Inflated Poisson Row-Column Association Models in Agricultural Trial. towards ZIP-AMMI Models

OS 08 Angsoka Dewi, Sutikno, Heri Kuswanto

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Cluster Ensemble Method for Clustering Rural Areas in Riau Province

OS 09 Ari Rusmasari, Sutikno, Setiawan

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Residual Bootstrap Approach in Spatial Regression Model

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OS 10 Brodjol Sutijo S. U., Agus Suharsono Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Tourism Data Modelling by Using Structural Vector Autoregression Approach

OS 11 Dian Andriana Research Center for Informatics, Indonesian Institute of Sciences (LIPI), Bandung, Indonesia Object Oriented Software Design of Capture Fisheries Decision Support Systems

OS 12 Margaretha Ohyver, Heruna Tanty

Department of Statistics, Faculty of Science and Technology, Universitas Bina Nusantara, Jakarta, Indonesia The Comparison of Drinking Water Filtration Using Friedman Test in Reduce Levels of Cadmium, Chromium, and Cyanide

OS 13

OS 14

Farid Ma’ruf, Bambang Widjanarko Otok Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Modelling The Underdeveloped Villages Gorontalo Province 2008 (with Multivariate Adaptive Regression Splines Approach) Hasih Pratiwi1, Subanar1, Danardono1, J. A. M. van der Weide2

1 Department of Mathematics, Gadjah Mada University, Yogyakarta, Indonesia 2 Delft University of Technology, Delft, The Netherlands Ruin Probability in Non-Life Insurance Model

OS 16 Kariyam, Edy Widodo Department of Statistics, Islamic University of Indonesia, Yogyakarta, Indonesia Clustering of Ordinal Data Based on The Weighted Ranking Pattern and Its Application

OS 17 Jerry Dwi Trijoyo Purnomo, Sindy Febri Antika

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Multilevel Regression and Its Applications

OS 18 Pudji Ismartini, Nur Iriawan, Setiawan, Brodjol Sutijo S. U. Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia

Hierarchical Structured Data Analysis Using Hierarchical Model with Bayesian Approach

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OS 19

OS20

Juwariyanto, Nur Iriawan Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Modelling Inflation in Indonesia Using Bayesian Markov Switching Arch Sukono Department of Mathematics, Faculty of Mathematics and Natural Sciences, Padjadjaran University, Bandung, Indonesia Value-At-Risk Under Asset Liability by Time Series Approach as A Framework of Portfolio Selection

OS 21 Margaretha Ari Aggorowati1, Nur Iriawan1, Suhartono1, Hasyim Gautama2 1 Department of Statistics, Faculty of Mathematics and Natural Sciences,

Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia 2 Ministry of Communication and Informatics, Indonesia Lagrange Multiplier (Lm) Test for Nonlinearity Detection in Structural Equation Modeling (Sem)

OS 22 Mohamad Samsodin, I Nyoman Budiantara

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Truncated Polynomial Spline Regression Multiresponse to Modelling Poverty Indicators in East Java

OS 23 Maslim Rajab Syafrizal, Setiawan

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia A General Spatial Two Stage Least Square Procedure For Estimating Spatial Autoregressive Model of conomic Growth In East Java 2009

OS 24 Siti Fatimah Hassan , Abdul Ghapor Hussin, Yong Zulina Zubairi

Centre for Foundation Studies in Sciences, University of Malaya, Kuala Lumpur, Malaysia Improved Efficient Approximation of The Concentration Parameter for Von Mises Distribution

OS 25 Novi Andy Dwi Setyawan, I Nyoman Budiantara

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Nonparametric Multirespon Regression Spline Approach to Modeling Determinants of Education Level in Papua Island

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OS 26 Nur Jannati Rokimah, Brodjol Sutijo S. U. Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Transfer Function Approach And Artificial Neural Networks for Modeling Inflation in East Java Province

OS 27 Mutiah Salamah, Jerry Dwi Trijoyo Purnomo, Erma Oktania Permatasari

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Weighted Spline and Its Application to Infants’ Growth

OS 28 Rina Andriani, Suhartono

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Forecasting Food Consumer Price Index by Using Intervention Analysis and Regression Splines

OS 29 Rudy Ramadani Syoer, Muhammad Mashuri

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Analysis of Group with Fuzzy C-Means and Gath-Geva Clustering Algoritm (Grouping Villages Study in Kutai Kartanegara Regency)

OS 30 Sri Subanti

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Sebelas Maret, Surakarta, Indonesia Double-Log Regression to Estimate Tourist Visits in Banaran Tourism Object Semarang Regency

OS 31 Ucik Mawarsari, Irhamah Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Missing Data Imputation Using K-Nearest Neighbor and Genetic Algorithm

OS 32 Suci Astutik, Nur Iriawan, Suhartono, Sutikno

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Spatio-temporal Rainfall Disaggregation Using Mudrain Method on Das Sampean Baru

OS 33 Wibawati, Muhammad Mashuri Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Multivariate Control Chart for Variability Based on Correlation Matrix (Wr)

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OS 35 Yulian Sarwo Edi, Heri Kuswanto, Sutikno

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Quasi Maximum Likelihood Estimator for Spatial Panel Data

OS 36

OS 38

Agung Gumilar Triyanto, Suhartono Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia A Hybrid Approach in Forecasting Indonesia Exports Yasmin Yahya1, Yasmin Yahya2, Khorn Saret3 1 Malaysian Institute of Information Technology (MIIT),

Universiti Kuala Lumpur, Kuala Lumpur, Malaysia 2 General Studies Section (Mathematics), MIIT,

Universiti Kuala Lumpur, Kuala Lumpur, Malaysia 3 Department of Forestry and Community Forestry, Phnom Penh, Cambodia Tree Species Composition. Multivariate Statistical Analysis Approach

OS 39

OS 40

Wiwiek Setya Winahju Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia The Comparison between Conway Maxwll Poison Regression and Poison in The Evaluation of Factors Influence of Infant Mortality Rate Agus Widodo, Purhadi

Department of Statistics, Faculty of Mathematics and Natural Science, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia A Comparison of Fuzzy C-Means and Fuzzy Chell Clustering Method (Case Study District/Municipalities in Java Island Based on Human Development Index Forming Variables)

OS 42

Dewi Fenty Ekasari, Sony Sunaryo

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Generalized Structured Component Analysis (GSCA) (Case Study of Poverty Characteristic in Regency of Central Java Province)

OS 43 Retno Pertiwi, Nur Iriawan

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Hierarchical Bayesian Modelling of Expenditure per Capita in Kalimantan Barat Province

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OS 44 T. Siagian 1, H. Ritonga1, Purhadi2, Suhartono2

1 BPS Statistics Indonesia, Jakarta, Indonesia 2 Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Measuring Social Vulnerability Using Minimum Message Length Clustering

OS 45 Sri Wahyuningsih, Bambang Widjanarko Otok

Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Modeling Ntp Paddy Crop with Autoregressive Integrated Moving Average (ARIMA) Method and Multivariate Adaptive Regression Spline Time Series (ITS_MARS)

OS 47 Yenita Mirawanti, Brodjol Sutijo S. U. Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Classification Using Radial Basis Functions Neural Network Case Study. Determination of Poor Households in Pasuruan in 2008

OS 48 Sony Sunaryo, Madu Ratna Department of Statistics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Wavelet Logistic Regression Analysis in Case of Binary Data

P 05 Alfian F. Hadi Department of Statistics, Bogor Agricultural University, Bogor, Indonesia Some Numerical Examples of Row-Column Association Models

BIOLOGY

OB 01 Alfonds Andrew Maramis1, Aloysius Duran Corebima2 1 Department of Biology, Faculty of Mathematic and Natural Sciences,

State University of Manado, Manado, Indonesia 2 Biology Education Doctoral Program, Postgraduate Program, State

University of Malang, Malang, Indonesia Chlorophyllin Supplementation Able to Normalized The Ratio of SGOT/SGPT in Serum of Mice That Have Increased due to Repeated Exposure of Formalin-Contaminated Fish

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OB 02 Bong Yii Bonn1, Asma Ahmad Shariff2, Abdul Majid Mohamed2, Amir Feisal Merican3 1 Institute of Graduate Studies, University of Malaya, Kuala Lumpur,

Malaysia 2 Center for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia 3 Institute of Biological Sciences, Faculty of Science,

University of Malaya, Kuala Lumpur, Malaysia Body Mass Index of Malaysian School Children and Adolescents

OB 04

OB 05

Ismail Forest Management Program Studies, Faculty of Agriculture, 17th August 1945 University of Samarinda, Samarinda, Indonesia The Effect of Shape of Plots on Accuration of Stand Basal Area Data in Inventarisation, in Bahau River Forest at Bulungan Regency, East Kalimantan Province Utami Sri Hastuti1, Permata Ika Hidayati2, Henny Nurul Khasanah1 1 Biology Department, State University of Malang, Malang, Indonesia 2 Educational Biology Program, Post Graduate Department,

State University of Malang, Malang, Indonesia Isolation of Proteolytic and Lipolytic Bacterias in Preserved Man Mackarel Fish (Rastrelliger canagurta) with Cabbage Lettuce (Lactucca sativa) Ensiling Fermentation and Kepayang Seed (Pangium edule Reinw) and to Confirm The Protein and Lipid Hydrolysis Index

OB 06 Solikin Purwodadi Botanic Garden, Indonesian Institute of Sciences, Purwodadi, Indonesia Generative Phenology of Fruits Plant in Purwodadi Botanic Garden As Effect of Climate Change

OB 07 Tutik Nurhidayati1, Nurul Jadid1, Safita Merediana2, Listy Anggraeni2, Ni Ketut Dewi Indrayati2 1 Department of Biology, Faculty of Mathematics and Natural Sciences,

Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia 2 Agricultural Department, Bangkalan, Indonesia Indegeneus Vesicular Arbuscular Mycorrhizae (VAM) Isolated at Pangpong and Petong Village at Bangkalan Madura

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OB 08 Utami Sri Hastuti1, Sundari2 1 Biology Department, State University of Malang, Malang, Indonesia 2 Biology Department, Khairun University, Ternate, Indonesia Biodiversity of Soil Rhizosphere Mycoflora on Potato Farm at Tosari and Batu and Antagonis Ability Some Species of Antagonistic Mold Toward Fusarium Spp.

OB 09 I. Trisnawati D.T., M. Muryono Ecologycal Laboratory, Department of Biology, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Agrobiodiversity Study on Local Arthropods Community in Paddy Fields, Sampang, Madura, Indonesia. Preliminary Study on Biological Test Estimated Potential Impact of Sea Level Rise

OB 10

P 04

P 07

M. Muryono, I. Trisnawati D.T. Ecologycal Laboratory, Department of Biology, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Bio-Assessment The Use of Organic and Inorganic Fertilizer's on Paddy Field Using SRI (System of Rice Intensification) Method. The Effect on Growth and Yield of Rice Ciherang Cultivar Fauziah, Solikin UPT Balai Konservasi Tumbuhan Kebun Raya Purwodadi – LIPI, Jl. Raya Surabaya-Malang Km 65 Purwodadi, Pasuruan – Jawa Timur. Scarcity and Diversity of Dioscorea spp. in Pasuruan, East Java Agung Sri Darmayanti and Lia Hapsari Purwodadi Botanical Garden, Indonesian Institute of Sciences, Jl. Raya Surabaya – Malang Km. 65, Pasuruan, East Java, Indonesia The Hydrological Aspects on Some Selected Local Plants: Flamboyan (Delonix regia), Sawo (Manilkara achras), Dau (Dracontomelon dao), and Bintaro (Cerbera manghas) and Its Potential Role for Land and Water Conservation

P 08

Solikin Purwodadi Botanical Garden, Indonesian Institute of Sciences, Jl. Raya Surabaya- Malang km 65, Purwodadi, Pasuruan, Indonesia Conservation of Plants for Cancer in Purwodadi Botanical Garden

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P 09

P 10

Mariana Rengkuan1, Aloysius Duran Corebima2, Sutiman Bambang Sumitro3, Mohamad Amin2

1 Department of Biology, Manado State University, North Celebes, Indonesia

2 Department of Biology, Malang State University, East Java, Indonesia 3 Department of Biology, Brawijaya University, East Java, Indonesia Sequence Variations Identification at Growth Hormone Gene of Composite Breed Cattle from BBIB Singosari and UPA Pasuruan Tarzan Purnomo Department of Biology, Faculty of Mathematics and Natural Sciences, State University of Surabaya, Surabaya, Indonesia Biodiversity of Plankton and Lead Content in Waters Contaminated Lapindo Mud

P 11 Tarzan Purnomo1, Yenny Risjani2, Sukoso2, and Marsoedi2 1 Department of Biology, Faculty of Mathematics and Natural Sciences,

State University of Surabaya, Surabaya, Indonesia 2 Faculty of Agriculture, Brawijaya University, Malang, Jl.Veteran,

Malang, Indonesia Potential of Plants in situ As Fitoremediator Lead of Land Contaminated Lapindo Mud

P14

P 15

Siti Nurfadilah, Janis Damaiyani, Ridesti Rindyastuti, Tarmudji, Pa’i Purwodadi Botanic Gardens, LIPI, Jl. Surabaya-Malang Km. 65, Purwodadi, Pasuruan, East Java 67163, Indonesia The Acclimatization of Paraphalaenopsis labukensis Shim, A. Lamb & C.L. Chan (Orchidaceae) on Different Types of Potting Media Nengah Dwianita Kuswytasari, Maya Shovitri, Rendra Dwi Andriyadi Biology Departement, Faculty of Mathematic and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Soil Mold Diversity Along The Coastal Wonorejo, Surabaya

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PART II: MATHEMATICS, CHEMISTRY, PHYSICS MATHEMATICS

OM 01 Abraham P. Racca, Emmanuel A. Cabral Ateneo de Manila University, Loyola Heights, Quezon City, Philippines Primitives of Henstock-Kurzweil Integrable Functions

OM 04 Tigor Nauli Research Center for Informatics, Indonesian Institute of Sciences (LIPI), Indonesia Spliced Alignment Approach to Solve Exon Chaining Problem in Gene Prediction

OM 09 Alvida Mustika Rukmi Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Uncertain Data Clustering by Geometry Computation

OM 11 Basuki Widodo, M. Siing, Febrian Eka Priyangga Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Numerical Simulation of The Model Stream Flow and Sedimentation in The Main River in The Form of Sinusoidal Derived from Two Affluent Streams in The Form of Sinusoidal

OM 13 Rohan de Silva CQUniversity Sydney, Sydney, Australia A Method to Calculate The Target Score in Rain Interrupted Limited over Cricket Matches

OM 16 Nova Hadi Lestriandoko, Ekasari Nugraheni Research Center for Informatics, Indonesian Institute of Sciences (LIPI), Bandung, Indonesia A Block-Based Wavelet Watermarking Using The Contour of Blocks

OM 17 Erna Apriliani, Subchan, Santi Hartini Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Implementation of Ensemble Kalman Filter (EnKF) Method to Estimate The Position of Mobile Robot

OM 21 Jesús Trello1, Alfredo Serret2 1 Universidad Autónoma de Madrid, Spain 2 Grupo EUROVAL, Madrid, Spain Social Value of Investments. A Fuzzy Indicator

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OM 22 Valeriana Lukitosari, Suparno, I Nyoman Pujawan, Basuki Widodo Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Maintenance Spare Parts Planning with Uncertainty Failure Rates

OM 23 Jonald P. Fenecios, Emmanuel A. Cabral Ateneo de Manila University, Loyola Heights, Quezon City, Philippines On The Properties of Baire Class One Functions Using Epsilon-Delta

OM 25 G. Naoum-Adil, J. Mohammed-Lemease Abu Dhabi University, Abu Dhabi, UAE P-Compactly Packed and Coprimarily Packed Modules

OM 28 Isnaini Rosyida, Kintafani Tessa Olivia Department of Mathematics, Faculty of Mathematics and Natural Sciences, Semarang State University, Semarang, Indonesia Modeling a Traffic Regulation at Intersection Street Using Fuzzy Graph

OM 30 Pipark Chansuriya, Nongkhran Sasom, Sarawut Saenkarun Department of Mathematics, Statistics and Computer Science, Faculty of Sciences, Ubon Ratchathani University, Warin Chamrap, Ubon Ratchathani 34190, Thailand Finite Dimensional Simple Poisson Modules Over Certain Poisson Algebras

OM 31 Alongkot Suvarnamani Department of Mathematics, Faculty of Science and Technology, Rajamangala Univerity of Technology Thanyaburi (RMUTT), Thanyaburi, Pathum Thani, Thailand Solutions of The Diophatine Equation 2x + py = z2

OM 32 Subiono, Zumrotus Sya’diyah

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Lyapunov-Max-Plus-Algebra Stability in Predator-Prey Systems Modeled by Timed Petri Net with The Entire Holding Times Are Considered

OM 33 Putu Wira Angriyasa1, Zuherman Rustam1, Jacub Pandelaki2 1 Department of Mathematics, Faculty of Mathematics and Natural

Sciences, University of Indonesia, Jakarta, Indonesia 2 Department of Radiology, Faculty of Medicine

University of Indonesia, Jakarta, Indonesia Brain Cancer (Astrocytoma) Classification Using Kernel Perceptron

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OM 34 Sandra Yuwana, Devi Munandar Research Center for Informatics, Indonesian Institute of Sciences (LIPI), Indonesia Interoperability Ontologies Using Prompt Algorithm for The Integrated Social Sciences

OM 35 M. Tauviqirrahman1,2, R.Ismail1,2, J. Jamari2, D.J. Schipper2 1 Laboratory for Surface Technology and Tribology,

University of Twente, Enschede, The Netherlands 2 Department of Mechanical Engineering, University of Diponegoro,

Semarang, Indonesia Numerical Method in The Performance Analysis of Journal Bearing

OM 39 M. Isa Irawan1, Maya Shovitri2, Alfi Yusrotis Zakkiyah1, Siti Fauziyah1

1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia

2 Department of Biology, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia

Mathematics and Computation Approach for Biological Sequences Analysis

P 01 Suhud Wahyudi, Sumarno, Suharmadi Sanjaya Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Metric Dimension of Graph and Its Aplication on The Minimum Placement of Fire Censorship in The Building (Case Study of Mathematics Department Building of ITS

P 02 M. Isa Irawan, Asri Bektipratiwi, Muis Kamaruddin, Oni Soesanto

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia The New Fast Learning Methods of Neural Networks for Pattern Recognition

P 03 Sadjidon, Sunarsini Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Normed-2 on Sequence Space

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia The Application of The Speech Identification System on Developing Control System of The Moving Robot

OM 42 Nurul Hidayat, Basuki Widodo

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CHEMISTRY

OC 01 Anita Alni1, Loh Teck Peng2 1 Universitas Surabaya, Surabaya, Indonesia 2 Nanyang Technological University, Singapore InCl3 Catalyzed Mukaiyama-Michael Reaction and Its Application towards Synthesis of Polirachitide A

OC 02 Saharman Gea Department of Chemisty, Faculty of Mathematics and Natural Sciences, University of Sumatera Utara, Medan, Indonesia Feasibility Study into The Use of Bacterial Cellulose as Scaffold Tissue Engineering

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LYAPUNOV-MAX-PLUS-ALGEBRA STABILITY IN

PREDATOR-PREY SYSTEMS MODELED BY

TIMED PETRI NET WITH THE ENTIRE HOLDING

TIMES ARE CONSIDERED

Subiono

Faculty of Mathematics and Natural Science-ITS

[email protected]

Zumrotus Sya’diyah

Graduated Mathematics Master Student-ITS

[email protected]

Abstract. In this paper, we construct a model of predator-prey systems withtimed Petri net and analyze the stabilization of the systems. We discuss a timedPetri net model with the entire holding times are considered. Furthermore, weanalyze the periodic behavior of the systems. Using the Lyapunov-max-plus-algebra stability theory, we will obtain the sufficient condition for the stabiliza-tion problem. This sufficient condition lead us to the stable timed Petri netmodel. The periodic duration of the oscillation in these systems will be alsodetermined. The analysis will also use the interval matrix in max plus algebra.In this case, every holding time in timed Petri net is viewed as an interval value.

Keywords: Predator-prey Systems, Timed Petri Net, Max-Plus.

1 Introduction

Generally, the state of systems changes as time changes. The state spaces are ex-pected to be changed at every tick of the clock. These kinds of systems are calledtime driven systems. There are some of them which evolve in time by the occur-rence of events at possible irregular time intervals, i.e. not necessarily coincidingwith clock ticks. In this case, the state transition is a result of the other harmonicevents. This kind of systems is called event driven systems [3]. Event driven systemswith discrete states are called by discrete event systems.

Max plus algebra is the useful approach to represent the discrete event systems.This approaching makes us possible to determine and analyze various kinds of sys-tems properties. The model of them will be linear over max plus algebra. In this

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Proceeding of the International ConferenceISBN 978-602-19142-0-5 on Mathematics and Sciences (ICOMSc) 2011

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kind of systems, event is more decisive than time [9]. We can analyze the systemsin max plus algebra easier and simpler than the conventional one because of thislinearity [10].

Petri net is a mathematical modeling tool which can be applied to represent thestate evolution of the discrete event systems. Petri net is called autonomous if everyits transition has at least an input place, i.e. does not have a transition which isalways be enabled [2].

In the previous research, the predator-prey systems have been modeled withtimed Petri net which is consistent with the real predator-prey behavior in real life[6]. We will modify this model with adding some holding times, condition and event.This model is inspired by the timed Petri net model of queueing systems with oneserver that discussed in [10] and the timed Petri net of the predator-prey systemdiscussed in [13]. Furthermore we analysis the periodic behaviour of the system byconsidering the interval matrix in max plus algebra. In this case, every holding timein timed Petri net is viewed as an interval value.

For this discussion, we need the theory of conventional and interval max plusalgebra, timed Petri net, and Lyapunov stability in systems modeled by Petri net.These theories will be discussed in the next section.

1.1 The Max Plus Algebra and Interval Max Plus Algebra

In this subsection we will give an introduction of max plus and interval max plusalgebra that will be used in the next discussion.

1.1.1 Max Plus Algebra

The domain of max plus algebra is the set Rǫ = R ∪ {ε = −∞}, where R the set ofreal number. The basic operations in max plus algebra are maximization (denotedby ⊕) and addition (denoted by ⊗). For x, y ∈ Rǫ, we get x⊕ y = max{x, y} andx⊗ y = x+ y. The set Rǫ with operation maximization and addition will be writtenas Rmax. For two matrices A and B over Rmax, the addition of the matrices givenby:

[A⊕B]i,j = ai,j ⊕ bi,j = max{ai,j , bi,j},

where i = 1, 2, · · · ,m and j = 1, 2, · · · , n, and the multiplication of two matricesA ∈ R

m×pmax and B ∈ R

p×nmax is given by:

[A⊗B]i,j = ai,j ⊗ bi,j =

p⊕

k=1

(ai,j ⊗ bi,j) ,

with i = 1, 2, · · · ,m and j = 1, 2, · · · , n. Let be given A ∈ Rm×nmax , a directed graph of

matrix A is dinoted by G(A) = (E,V ). More detail about this graph can be foundin [12]. A sequence of arc p = (i1, i2), (i2, i3), · · · , (il−1, il) of a graph is called path .A path is called elementer if its nodes has only one incoming and one outgoing

LYAPUNOV-MAX-PLUS-ALGEBRA STABILITY IN PREDATOR-PREY ....

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arc. A circuit is a close elementer path. The average weight of a path p is the weightof path p divided by the lenght of path p, i.e.

|p|w|p|l

=

(

ai2,i1 + ai3,i2 + · · ·+ ail,il−1

)

l − 1.

A circuit mean is the average weight of a circuit which also acts as the eigen valueof matrix A. The detail explanation about the eigen value and eigen vector in maxplus algebra can be seen in [9].

An algorithm to compute an eigen value and a corresponding eigenvector of asquare matrix A can be found in [8]. This algorithm is an iterative algorithm of alinear equation bellow:

x(k + 1) = A⊗ x(k), k ≥ 0. (1)

For non negative number M ≥ 0, let be a matrix Am ∈ Rn×nmax for 0 ≤ m ≤ M

and x(m) ∈ Rnmax for −M ≤ m ≤ −1, then the difference equation of of Mth-order

will be written as follow:

x(k) =

M⊕

m=0

(Am ⊗ x(k −m)) , k ≥ 0. (2)

A difference equation of Mth-order with A0 6= ε can be transformed into adifference equation of 1st-order that is given by equation (1), as follow:

x(k + 1) = A⊗ x(k) (3)

with

A =

A∗0 ⊗A1 A∗

0 ⊗A2 · · · · · · A∗0 ⊗AM

E εεε · · · · · · εεε

εεε E. . . · · · εεε

......

. . .. . .

...εεε εεε · · · E εεε

(4)

and

A∗

0 =

n−1⊕

i=0

A⊗i0 , (5)

the matrix E is identity matrix and εεε is the zero matrix. The information aboutthis equation can be found in [10]. The the equation (3) that satisfies (4) and (5)will be called as the autonomous equation.

1.1.2 Interval max Plus Algebra

In this section we will discuss about the interval max plus algebra. The domain ofinterval max plus algebra is all of the closed interval in Rmax . Let be the closedinterval in Rmax defined as follows:

xdef= { [x, x | x ∈ Rmax, x ≤m x �m x} .


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The, the set of this interval is defined by:

I(R)ξdef= { [x, x] | x, x ∈ R, ε �m x �m x} ∪ {ξ}

with ξ = [ε, ε]. There is also two kinds of operators in interval max plus algebra.They are ⊕ and ⊗ which are defined below:

x⊕y = [x⊕ y, x⊕ y] and x⊗y = [x⊗ y, x⊗ y]

with x, y ∈ I(R)ξ more explanation of interval max plus can be found in [5]. Theset of matrices of size m× n in interval max plus algebra are denoted by I(R)m×n

max ,where

I(R)m×nmax =

{

A = [A,A]∣

∣ A,A ∈ Rm×nmax

}

.

If A = [A,A] ∈ I(R)m×nmax , then there are lower bound and upper bound matrix of A,

these ones areA = (Ai,j) ∈ R

m×nmax and A = (Ai,j) ∈ R

m×nmax .

Furthermore, the operation between two matrices in interval max plus algebra aredefined as follows:

(A⊕B)i,j = Ai,j⊕Bi,j, (α⊗A)i,j = α⊗Ai,j and (A⊗B)i,j = Ai,j⊗Bi,j

with α ∈ R. The eigen vector and eigen value of a matrices in interval max plusalgebra is an interval vector and eigen value is an interval value. This description iswritten in the definition below.

Definition 1 Let A is a matrix in I(R)n×nmax , λ = [λ, λ] ∈ I(R)max, with λ, λ ∈ R

and v = [v, v] is a vector in I(R)nmax. If all ones satisfy the following equation

A⊗v = λ⊗v,

then the interval scalar λ = [λ, λ] and the vector v = [v, v] are respectively calledthe eigenvalue and eigenvector of matrix A.

There are also possible eigenvalue, possible eigenvector and universal eigenvalueand universal eigenvector of a matrix in the interval max plus algebra which areexplained as follows.

Definition 2 Let A is a matrix in I(R)n×nmax , with A = [A,A] and λ = [λ, λ] ∈

I(R)max, with λ, λ ∈ R. If λ is an eigenvalue of a matrix in the interval matrixA, then λ is called a possible eigenvalue of A. And if λ is an eigenvalue of thewhole matrices in the interval matrix A, then λ is called an universal eigenvalue

of A. Furthermore, the corresponding eigenvectors are respectively called possible

eigenvector and universal eigenvector.

The algorithm to find the possible eigenvalue, eigenvector and universal eigen-value, eigenvector can be found in [4].


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1.2 Petri Net and Timed Petri net

In this section we discuss about Petri net, timed Petri net and their theory relatedto the problem discussed in this paper. Petri net is a 4-tuple (P, T,A,w), where P

is a finite set of places, i.e. P = {p1, p2, · · · , pm}, T is a finite set of transitions, i.e.T = {t1, t2, · · · , tn}, A is a set of arcs, i.e. A ⊆ (P × T ) ∪ (T × P ) and w is weightfunction, i.e. w : A → {1, 2, · · · }.

Petri net graph consist of two shapes. These ones are rectangles or lines and theothers are circles. The rectangles represent the transition and the circles representthe place. Mostly, a transition and a place can be respectively interpreted as anevent and a condition such as an event can be occurred.

A Petri net marking vector is a vector of size m × 1 , where m is the numberof place in the Petri net. This vector is denoted by x = [x(p1), x(p2), · · · , x(pm)]T ,where i-th element of this vector represents the number of token in i-th place pi and

T

denotes transposition. There are backward incidence matrices and forward incidencematrices in Petri net which are respectively denoted by Ab with Ab(i, j) = w(pi, tj)and Af with Af (i, j) = w(tj , pi), where i = 1, 2, · · · ,m and j = 1, 2, · · · , n. In termof Ab and Af an incidence matrix A is given by

A = Ab −Af . (6)

Timed Petri net is a 6-tuple (P, T,A,w,M0, V ) whereM0 is an initial marking vector,i.e. M0 : P → {0, 1, 2, 3, · · · } and V is time structure related to the entire place inPetri net. This structure of time called holding time which means as the time atoken have to spend in a place before contributing to the enabling of a transition.More discussion about Petri net is given in [1] and for the more complete explanationof timed Petri net can be found in [6] and [7].

1.3 Lyapunov Stability in Systems Modeled by Petri net

The stability theory of Lyapunov is described in the following proposition.

Proposition 1 A Petri net is stable if there is m × 1 strictly positive vector Φ ,such that

∆v = eTATΦ ≤ 0,

where e is a transition firing vector n × 1 and A an incidence matrix m × n of agraph.

�

Moreover, the Petri net which does not satisfy the proposition above will beobserved whether this Petri net is stabilizable. The criteria of a Petri net is calledas a stabilizable which one is defined in Proposition 2 as following.


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Proposition 2 A Petri net is stabilizable if there is a transition firing vector e, suchthat

∆v = Ae ≤ 0,where e =d−1∑

k=0

ek.

�

The detail information of The Propositions can be found in [7].

2 Temed Petri net Model and Its analysis

In this section, we discuss about the analysis of the timed Petri net model ofpredator-prey systems.

The Timed Petri Net Model

One constructed of the model is described in Figure 1. Where the conditions (places

f

D

s

C5

d

Ib

E

b

C4

C3

C1

R

A

t

C2

Figure 1: The timed Petri net model of predator-prey systems

in the systems) are R: Preys are resting, A:Preys are in activity, I:Predators areidle, D:Preys in danger, and E:Preys are being eaten. The transition that drive thesystem are f:Preys have finished their rest, t: Preys are at threat, s: Predator isstarting to attack the prey, and d:Predator departs (leave the preys). The holdingtimes corresponding to place R, A, I, D and E, respectively are C1, C2, C3, C4 andC5 with Ci ∈ R

+, i = 1, 2, 3, 4, 5 where C1 and C3 satisfies

C1 > C3 (7)


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This condition (7) means that the predator have enough time to get the prey. Wewill use (7) in determining the value of the holding times. Using Proposition 1 weget the vector Φ = [r1, 0, r2, r2, r2]

T with r1, r2 ∈ R+. Because of the vector Φ is not

strictly positive, then we conclude that the system is not stable. Now we analyzewhether this system is stabilizable or not. From Proposition 2 we get the firingvector

e = [y, y, y]T , y ∈ N. (8)

So, we can also say that the system is stabilizable. From Figure 1 we can also obtainthe equation below:

x1(k) = C1 ⊗ x1(k − 1)x2(k) = C2 ⊗ x1(k) ⊕C3 ⊗ x4(k − 1)x3(k) = C4 ⊗ x2(k)x4(k) = C5 ⊗ x3(k)

wherex1(k) : time of prey finishing rest at k-thx2(k) : time of prey threating at k-thx3(k) : time of predator attacking prey at k-thx4(k) : time of predator leaving prey at k-th.

These equations can be formed into a matrices equation as follows:

x1(k)x2(k)x3(k)x4(k)

=

ε ε ε ε

C2 ε ε ε

ε c4 ε ε

ε ε C5 ε

⊗


⊕

C1 ε ε ε

ε ε ε C3

ε ε ε ε

ε ε ε ε

⊗

x1(k − 1)x2(k − 1)x3(k − 1)x4(k − 1)

From this matrices equation we get the standard autonomous equation below:

x(k + 1) = A⊗ x(k), with

A =

C1 ε ε ε

C1 + C2 ε ε C3

C1 + C2 + C4 ε ε C3 + C4

C1 + C2 + c4 + C5 ε ε C3 +C4 + C5

,


=


(9)

The graph representation of matrix A is given in Figure 2. So, we obtain the eigenvalue of the system that is given by:

λ = max{C1, C3 + C4 + C5}.

Based on (7), let the value of each holding times are C1 = 6, C2 = 2, C3 = 4, C4 =C5 = 1. Then we have matrix A and eigenvalue λ

A =

6 ε ε ε

8 ε ε 49 ε ε 510 ε ε 6

and λ = max{6, 6} = 6.


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7

b

b

b

b1

2

3

4C1

C1+ C2 C

3

C1 + C2 + C4 + C5

C1 +

C2 +

C4 C3

+C4

C3 + C4 + C5

Figure 2: The graph representation of A

The period of the system is equal to λ = 6. Now, according to (7) we consider thatthe holding times are interval values, i.e., C1 = [4, 8], C2 = [1, 3], C3 = [2, 6], C4 =[1, 2], C5 = [1, 2]. We get the matrix

A =

[4, 8] [ε, ε] [ε, ε] [ε, ε][5, 11] [ε, ε] [ε, ε] [2, 6][6, 13] [ε, ε] [ε, ε] [3, 8][7, 15] [ε, ε] [ε, ε] [4, 10]

So, we get the two kinds of eigenvalues here. They are lower and upper eigenvalues,which are λ = 4 and λ = 10 . So, the period of the system is the value in [4, 10]. Furthermore, let be choosen a vector v = [8, 6, 9, 9]T . Using the algorithm ofpossible eigenvector in [4], we can show taht that vector v is a possible eigenvectorof matrix interval A with its corresponding possible eigenvalue is 7 and its suitablematrix is

A♯ =

7 ε ε ε

5 ε ε 48 ε ε 78 ε ε 7

.

Indeed, it is easy to see that

A♯ ⊗ v =

15131616

= 7⊗

8699

= λ⊗ v.

We found that the systems represented by timed Petri net in Figure 1 said tobe unstable but stabilizable. This stabilization by firing vector in (8). This firingvector means that the prey should not be in activity before they are in threat. Fromthis interpretation, we can construct the stable timed Petri net model pictured inFigure 3. The places and transitions of this model have the same representationwith the previous one with C1, C2, C3 and C4 respectively are the holding times ofplaces R, I,D and E, where now

C1 > C2. (10)


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b

b

R

C1

I C2t

C3D

E C4

s

d

Figure 3: The stable timed Petri net model of predator-prey systems in Figure 1

We will use (10) in determining the value of holding times. Now we will investigatethe stabilization property of this system. Using Proposition 1, we get the vector Φ =[r1, r2, r2, r2]

T with r1, r2 ∈ R+. This vector shows us that the system represented

by Figure 3 is stable, because the vector Φ is strictly positive. And from Figure 3we get the following equations:

x1(k) = C1 ⊗ x1(k − 1)⊕ C2 ⊗ x3(k − 1)x2(k) = C3 ⊗ x1(k)x3(k) = C4 ⊗ x2(k).

These equations can be formed into matrix equation below:

x1(k)x2(k)x3(k)

=

ε ε ε

C3 ε ε

ε C4 ε

⊗

x1(k)x2(k)x3(k)

⊕

C1 ε C2

ε ε ε

ε ε ε

⊗

x1(k − 1)x2(k − 1)x3(k − 1)

.

From this equation, we can obtain the standard autonomous equation:

x(k + 1) = A⊗ x(k)

with

A =

C1 ε C2

C1 + C3 ε C2 + C3

C1 + C3 +C4 ε C2 + C3 + C4

and

x1(k)x2(k)x3(k)

=

x1(k)x2(k)x3(k)

. (11)

From Figure 4, we get the eigen value of matrix is given by:

λ = max

{

C1,C1 + C2 + C3 + C4

2, C2 + C3 + C4

}

.


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b

b

b1

2

3C1

C1 + C2 C2 + C

3

C2

C1 + C3 + C4C2 + C3 + C4

Figure 4: The graph representation of matrix A in (11)

Based on (10), we consider the holding times values are C1 = 6, C2 = 4 and C3 =C4 = 1, then we get the eigen value of matrix is

λ = max

{

C1,C1 + C2 + C3 + C4

2, C2 + C3 + C4

}

= max{, 6, 6, 6} = 6.

It means that the period of oscillation system is equal to 6.Now, according to (10) let the holding times values are the interval C1 =

[5, 9], C2 = [2, 5], C3 = [1, 3] and C4 = [1, 2]. Then we obtain the eigen value ofinterval matrix A is [5, 10] . Furthermore, let be choosen a vector v = [7, 8, 10]T .Using the algorithm about the possible eigenvalue and eigen vector in [4], we obtainthat vector v is a possible eigenvector of interval matrix A with the possible eigen-value is equal to 6 and its suitable matrix which is included in interval matrix A

is

A♯ =

6 ε 37 ε 49 ε 6

.

Indeed, it is easy to see that

A♯ ⊗ v =

131416

= 6⊗

7810

= λ⊗ v.

3 Concluding Remark

In this section we give some notes of discussing especially in predator-prey systemmodeled by timed Petri net. There is one model of timed Petri net that can beconstructed here. This model is the modification of the previous timed Petri netmodel by adding some places and transitions. The modification is also inspired bythe Petri net model of the queueing system with one server.

The model is unstable but stabilizable. This stabilization is done by firing thetransitions based on firing vector e = [y, y, y]T , where y ∈ N, then we identify thatthis firing vector is the sufficient condition for each model to be stabilizable. By the


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interpretation of this sufficient condition, we can construct the new timed Petri netwhich is stable. The last note of this discussion is that the period of oscillation sys-tem can be determined by calculating the eigenvalue of the matrix in the standardautonomous equation.

For the future work, the discussing can be continued in constructing more com-plex the modification model and describing the period of the system with the holdingtimes are fuzzy number in max plus algebra.

References

[1] Adziya, D., Membangun Model Petri Net Lampu Lalu-lintas dan Simulasinya,Thesis of Mathematics Department FMIPA, Institut Teknologi Sepuluh Nopem-ber (ITS), Surabaya, 2008.

[2] Bacelli. F, G. Cohen, G.J. et.al, Synchronization and Linearity: An Al-

gebra for Discrete Event System, Web-edition, 2001.

[3] Necora, Ion., Model Predictive Control for Max-Plus-Linear and Piecewise

Affine Systems, Technise Universiteit Delft. Netherland, 2006.

[4] Novitasari, Ratna., Analisis Masalah Generator Dari Possible Dan Univer-

sal Eigenvector Pada Matriks Interval Dalam Aljabar Max-Plus, Thesis of Math-ematics Department FMIPA, Institut Teknologi Sepuluh Nopember, Surabaya,2008.

[5] Rejeki, Sri., Analisis Sistem Jaringan Antrean Dengan Elemen-Elemen Ma-

triks Adjasen Berupa Interval Dalam Aljabar Max-Plus, Thesis of MathematicsDepartment FMIPA, Institut Teknologi Sepuluh Nopember, Surabaya, 2010.

[6] Retchkiman, Zvi., Mixed Lyapunov-Max-Plus Algebra Approach to the Sta-bility Problem for a two Species Ecosystem Modeled with Timed Petri Nets,International Mathematical Forum Vol.5 No.28, 1393-1408, 2010.

[7] Retchkiman, Zvi.,The Stability Problem for Discrete Event Dynamical Sys-tems Modeled with timed Petri Nets Using a Lyapunov-Max-Plus Algebra Ap-proach , International Mathematical Forum Vol.6 No.11, 541-566, 2011.

[8] Subiono and J vander Woude.,Power Algorithms for (Max,+)-and Bipar-tite (Min,Max,+)-systems, Int. Journal Discrete Event Dynamic Systems: The-

ory and Applications 10(4), 369-389, 2000.

[9] Subiono., The existence of eigenvalues for reducible matrices in Max-Plus

Algebra, Mathematics Department FMIPA- I T S, Surabaya, 2008.

[10] Subiono., Aljabar Max Plus dan Aplikasinya: Model Sistem Antrian, Mathe-matics Department FMIPA- I T S, Surabaya, 2008.


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[11] Subiono and Nur Sofiyana.,Using Max-Plus Algebra In The Flow ShopScheduling, IPTEK, The Journal for Technology and Science, Vol.20, No.3,83-87, August 2009.

[12] Subiono., Aljabar Max Plus dan Terapannya, Mathematics DepartmentFMIPA, Institut Teknologi Sepuluh Nopember (ITS), Surabaya, 2010.

[13] Subiono and Zumrotus. S., Lyapunov-Max-Plus-Algebra Stability In

Predator-Prey Systems Modeled With Timed Petri Net, Mathematics Depart-ment FMIPA- I T S, Surabaya, 2011.


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