- 212.2 The classical competition model without individuals’ strategy. - 473.3 Generating graph of the individual-based pred a t or - p re y model. - 523.3.3 Analysis of the generating graph. - APPLICATION: MODELING OF SOME REFERENCE ECOSYS-TEMS 574.1 Modeling of the thiof-octopus system. - Nguyen Ngoc Doanh Nguyen Phuong Thuyiii ACKNOWL EDG EME NT SFirst of all, I would l i ke to express my sincere gra t itu d e t o my supervisor, Dr.Nguyen Ngoc Doanh for his patient guidance, encouragement and valuable advicesthroughout my PhD research. - Dr.habil. - Le Van Hien for their suggestionsand input t h at led to the imp rovement of the thesis. - Sidy Ly for their collaborationin research.It would have been much more difficult for me to complete this work without thesupport and friendship of the members of the “Discrete Mathematics” Semin ar atthe Institute of Mathemati cs, Vietnam Academy of Science and Technology (VAST),the “Applied Mathematical Models in Control and Ecosystems” Seminar at HanoiUniversity of Science and Technology and the “Modeling and Si mulation of Comple xSystem” Seminar of WARM Team at MSLab, Faculty of Computer Science andEngineering, Thuyloi University. - Nguyen The Vinh.Thank you so mu ch.I would like to thank all the members of the Applied Mathematics Depar t m ent,School of Applied Mathemati c s and Informatics, Hanoi University of Science andTechnolog y for their encouragement and help in my work.I would like t o express my gratefulness to my beloved family, to my parents whoalways encourage and help me at every stages of my personal and academic life andhave been longing to see this achievement come true. - For my lovely children,Tra and Khang, their accompany definitely give me a strong motivation to reach tothis point.Hanoi, December 2018Nguyen Phuong Thuyv LIST OF ABBREVIATIONSEBM : Equation-Based ModelIBM : Individual-Based ModelGBM : Graph-Based M odelLSE : Local Superior resource ExploiterLIE : Local Inferior resource ExploiterBPH : Brown Plant Hopper1 LIST OF TABLESTable 2.1 Equilibria of aggregated model (2.13) and local stability analysis . - 40Table 3.1 The statistics for several complex systems. - 51Table 3.2 The statistics for several s teps of the simulation of the predator-prey competition system. - 54Table 3.3 Statistics about the cliques of the graphs a t step 1 of the simulationof the predator-prey competition system. - 5 5Table 3.4 Statistics about the cliques of the graphs at step 530 of the simu-lation of the predator-prey competition system. - F is the mathematical function which representsgeneral laws applied to all members of the compartments [83. - 15Figure 2.1 Comparison of solutions of system (2.3) with their approxima-tions through the aggregated system (2.10) for the both biotic andabiotic resource cases. - This figure shows the evolutions in timeof each of the four state variables of system (2.3) (R, C1, C1Cand C1R) and their approximations obtained from the aggregatedsystem (2.10) (R , C1, kC2/H(C1) and (αC1+ α0)C2/H(C1) ,respectively), for the same parameter values (r = 3. - 31Figure 2.2 Comparison of solutions of system (2.11) wi th their approxima-tions through the aggregated system (2.13) for the both biotic andabiotic resource cases. - This figure shows the evolutions in timeof each of the four state var iabl es of system (2.11) (R, C1C,C1Rand C2) and their approximations obtained from the aggre-gated system (2.13) (R, mC1/L(C2), (βC2+ β0)C1/L(C2) andC2, respectively), for the same parameter values (r = 5. - 34Figure 2.3 The outcomes of model (2.11) with the biotic resource. - 413 Figure 2.4 The outcomes of model (2.11) with the abiotic resource. - In eachcorresponding simulation, parametershave the same values as inthe case of biotic resource and the values of S and the values ofK are exactly the same. - 42Figure 2.5 The left panel is about dom ai ns of the space (l, d1N, β) for thedifferent outcomes of model 2.13 of the abiotic resource case. - 43Figure 2.6 The left panel is about domains of the space (l, d1N, β0) for thedifferent outcomes of model 2.13 of the biotic resource case. - a) at step 10, b) at step 100, c) at step 200,d) at step 300. - 49Figure 3.3 Evolution of the number of individuals of each species. - 50Figure 3.4 Individual B as ed Model (on the left) and the corresponding DiskGraph Based Model (on the right. - 54Figure 3.5 Distribution of degree in several simulation steps: a) at step 1, b)at step 530, c) at step 1000, d) at step 2500. - 56Figure 4.1 Example of t he case where the inferior competitor wins globallyin model 1. - Parameters are chosen as follows r1= 0.7. - 6 1Figure 4.2 Example of t he case where the inferior competitor wins globallyin model 2. - Parameters are chosen as follows r1= 0.9. - 64Figure 4.3 Example of the case where the inferior competitor wins globally inmodel 2: A comparis on between the aggregated model (blue dots)and the complete model (red curve). - 654 Figure 4.4 Example of t he case where the inferior competitor wins globallyin model 3. - 68Figure 4.5 Example of the case where the inferior competitor wins globally inmodel 3. - The case: rice wins globally in competition.Parameters values are chosen as follows: r1= 0.7. - 79Figure 4.10 Equi l ibr i a and local stability a nalys i s of the reduced model. - 87Figure 4.11 The case: rice wins globally in competition. - a1= 0.2;a2= 0.2. - 92Figure 4.12 The case: rice disappears on patch 2. - a2= 0.7;e1= 0.6. - 93Figure 4.13 The case: the existence of rice and BPH on both patches. - Motivat io nThe growth and degradation of populations in the nature and the struggl e of onespecies to dominate other species have been an interesting topic for a long time. - Parameters riand Kiarethe growth rate and the carrying capacity of the species i, i ∈ {1, 2}, respectively.Param et er aijis interspecific compet i t i ve coefficient representing the negative effectof species j on the growth of species i, i 6= j, i, j ∈ {1 , 2}.The ecological meaning of th i s model is that two species coexist only if th e effectsof their co m petition are small. - When the competing effects of two specie s are large,one of the two species will be extinct. - Competition is noted when one of the two speciesis absent, r esu l t i n g in an increase t h e remaining species. - This result is contrary to theexclusion principle of the classical competition model. - This result is alsocontr a r y to t h e p r i n ci p l e of com petitive exclusion.The main reason for the l i m i t a t i on of Lotka-Volt er r a’ s classic competition modelis that there ar e too many assumptions in the model, such as the assumption thatthe environment is homogeneous and stable (expressed by the carrying capacitiesKifor the specie i, i ∈ {1, 2. - the behavior of the individual species is the sameand the competi t i o n is expressed only by interspecific competitive coefficient aij.Meanwh i l e, these factors appear frequently and play a very important role. - Individuals of the same species or of different speciesmay have different behaviors. - Inaddition, individ u a l s may also change their behaviors frequently according to thechan ge of th e environment as stu d i ed i n Therefore, the development of new models that take into account the complexenvi r on m ents and the behaviors of individuals has been interested by many mathe-maticians. - Following are some recent approaches.• The complex environment and individual migration behavior in competitiveecosystems. - The competition process and the migration process have the sametime scale or different time scal es.• Aggressive behavior of individuals in competitive system.• Age structure (ad u l t gr o u p an d i m m a t u r e gr o u p ) i n th e competitive system.2. - To reach this goal, we divide this thesis into 4 main work packages:- Developin g models analyzing the effects of complex environments and aggressivebehavior of the two competing ecosystems.- Developing models analyzing the effect of age structure (adul t and juven i l e)the studied competing ecosystems.- Building disk-graph based models to study competing ecosystems.7 - Implementation and simulation experiments.3. - Research methodsTo reach the goal of the thesis, the following methods will be possibly considered:• Equation-ba sed and individual-based modeling methods are undertaken tomodel the reference systems at different time- sca l es an d l evels of complexity.• Methods of d y n am i ca l systems and ordinary differe ntial equa ti o n s are ded-icated to the study of the obtained mathematical models. - In particularly,method of aggregation of variables will be used, if it is necessary, to reducethe complexity of the models.• Methods relating to graph theory are considered t o investigate some generatedgraph models from the individual-based ones.4. - Results and applicationsThe thesis presents different models and simulations which can be applied in the-oretical as well as empirical study in competitive ecosystems. - From the theoreti ca lpoint o f view, the author h as successfully developed sever a l model s (some continuesmodels for the case where two consumer species exploit a common resource withdifferent competitive strategies) and simulations (some discret e models for prey-predator systems: from the individual-based model to th e generating graph of theindividual-based model). - In the ap p l i c at i o n point of view, the author has presentedsome models which are very useful for different case studies such as (Case 1) Thiofand Octopus Competition in Senegal Coast and (Case 2) Brown Hopper-Plant andRice.5. - The structure and results of the thesisThe main part of this thesis is divided into four chapters:• Chapter 1 pr ese nts the concept of competition in ecology systems as well as theapproaches to study competing ecosystems including continuous models anddiscrete models. - Chapter 2 presents some continu o u s models for the case where two consumerspecies exploit a common resource with different competiti ve strategies.• Chapter 3 presents some discrete models for prey-predator systems: from theindividual-based model to the generating grap h of the individual-ba sed model.8 • Chapter 4 presents the modeling of two ecology systems: the brown pla nthopper system and the thiof-octopus system.The main content of the thesis is b a sed on the articles listed in “List of pub-lished works of thesis”.These results have been presented in- International Workshop on Selected Problems in Optimiza ti o n and ControlTheory at Institute for Advanced Study in Mathematics (VI-ASM
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