|
| Titre : |
On the Gause type prey-predator model with variable mortality rate for the predator |
| Type de document : |
texte imprimé |
| Auteurs : |
Khaoula Bektache, Auteur ; Nabil Beroual, Directeur de thèse |
| Editeur : |
Sétif:UFS |
| Année de publication : |
2024 |
| Importance : |
1 vol (81 f.) |
| Format : |
29 cm |
| Langues : |
Anglais (eng) |
| Catégories : |
Thèses & Mémoires:Mathématique
|
| Mots-clés : |
Mathématique |
| Index. décimale : |
510-Mathématique |
| Résumé : |
This thesis introduces a novel predator-prey model inspired by Gause models, incorporating a variable disappearance rate for predators. This innovation reflects the observed dependence of predator mortality on prey or predator densities in many ecological scenarios.The work establishes a graphical criterion for local asymptotic stability of positive equilibria, extending the well-known Rosenzweig-MacArthur graphical criterion designed for constant predator mortality. Furthermore, conditions for global asymptotic stability of a unique positive equilibrium and the non-existence of limit cycles are explored.Beyond stability analysis, the dissertation delves into the occurrence of a Poincaré-Andronov-Hopf bifurcation, providing an explicit formula for the first Lyapunov coefficient. The theoretical framework is then applied to various predator-prey models, with phase portraits illustrating the model's rich dynamics, including the existence of global bifurcations.
Finally, the thesis concludes with a glimpse into additional significant properties of the proposed model, paving the way for future research. For information, this work is taken from Hammoum's dissertation and Amina Hammoum, Tewfik Sari et Karim Yadi's article.
|
| Note de contenu : |
Sommaire
1 Basicconceptsofecologyanddynamicalsystems 5
1.1 Basicconceptsofecology 5
1.1.1 Somedefinitions 5
1.1.2 TypesofInteractions 7
1.2 Basicconceptsofmathematics 8
1.2.1 Somedefinitions 9
1.2.2 Conceptoflimitcycle 18
1.3 Basicconceptsintheecologicalmodel 23
1.3.1 Dynamicmodelofasinglepopulation 23
1.3.2 Two-speciesmodels 25
2 Gausetypemodelwithvariabledisappearancerate 32
2.1 GeneralmodifiedGausetypemodel 32
2.2 Aboutthegrowthratefunction 33
2.2.1 Exponentialgrowthequation 34
2.2.2 Logisticfunction 34
2.3 Aboutthefunctionresponse 35
2.3.1 LotkaVolterratypefunctionalresponse 35
2.3.2 Holling'sfunctionalresponses 35
2.3.3 Ratio-dependentfunctionalresponse 37
2.3.4 Predator-dependentfunctionalresponse 37
2.4 Classicalresults 38
2.4.1 Existence,uniqueness,positivityandboundrdness 38
2.4.2 Existenceofequilibria 39
2.4.3 Stability 39
2.4.4 Poincaré-Andronov-Hopfbifurcation 39
2.5 Specificcase(Rosenzweig-MacArthurmodel) 39
2.6 Gausetypemodelwithvariabledisappearancerate 41
2.6.1 Hsudisappearancerate 41
2.6.2 Bazykin'sdisappearancerate 42
2.6.3 CFdisappearancerate 42
2.6.4 VTdisappearancerate 43
2.7 Generalvariabledisappearancerate 44
2.8 Positivityandboundedness 45
2.9 Existenceandstabilityofequilibria 45
2.10 Globalasymptoticstability 52
2.11 Non-existenceoflimitcycles 54
3 Applicationfromtheecologicalliterature 57
3.1 Gause/RMAmodel 60
3.1.1 PAHbifurcation 60
3.1.2 Numericalsimulations 60
3.2 Hsumodel 63
3.2.1 PAHbifurcation 63
3.2.2 Numericalsimulations 63
3.3 Bazykinmodel 66
3.3.1 Numericalsimulation 66
3.4 TheCavani-Farkas(CF)model 68
3.4.1 Numericalsimulations 69
3.5 Variable-Territorry(VT)model 71
3.5.1 Numericalsimulation 71
|
| Côte titre : |
MAM/0759 |
On the Gause type prey-predator model with variable mortality rate for the predator [texte imprimé] / Khaoula Bektache, Auteur ; Nabil Beroual, Directeur de thèse . - [S.l.] : Sétif:UFS, 2024 . - 1 vol (81 f.) ; 29 cm. Langues : Anglais ( eng)
| Catégories : |
Thèses & Mémoires:Mathématique
|
| Mots-clés : |
Mathématique |
| Index. décimale : |
510-Mathématique |
| Résumé : |
This thesis introduces a novel predator-prey model inspired by Gause models, incorporating a variable disappearance rate for predators. This innovation reflects the observed dependence of predator mortality on prey or predator densities in many ecological scenarios.The work establishes a graphical criterion for local asymptotic stability of positive equilibria, extending the well-known Rosenzweig-MacArthur graphical criterion designed for constant predator mortality. Furthermore, conditions for global asymptotic stability of a unique positive equilibrium and the non-existence of limit cycles are explored.Beyond stability analysis, the dissertation delves into the occurrence of a Poincaré-Andronov-Hopf bifurcation, providing an explicit formula for the first Lyapunov coefficient. The theoretical framework is then applied to various predator-prey models, with phase portraits illustrating the model's rich dynamics, including the existence of global bifurcations.
Finally, the thesis concludes with a glimpse into additional significant properties of the proposed model, paving the way for future research. For information, this work is taken from Hammoum's dissertation and Amina Hammoum, Tewfik Sari et Karim Yadi's article.
|
| Note de contenu : |
Sommaire
1 Basicconceptsofecologyanddynamicalsystems 5
1.1 Basicconceptsofecology 5
1.1.1 Somedefinitions 5
1.1.2 TypesofInteractions 7
1.2 Basicconceptsofmathematics 8
1.2.1 Somedefinitions 9
1.2.2 Conceptoflimitcycle 18
1.3 Basicconceptsintheecologicalmodel 23
1.3.1 Dynamicmodelofasinglepopulation 23
1.3.2 Two-speciesmodels 25
2 Gausetypemodelwithvariabledisappearancerate 32
2.1 GeneralmodifiedGausetypemodel 32
2.2 Aboutthegrowthratefunction 33
2.2.1 Exponentialgrowthequation 34
2.2.2 Logisticfunction 34
2.3 Aboutthefunctionresponse 35
2.3.1 LotkaVolterratypefunctionalresponse 35
2.3.2 Holling'sfunctionalresponses 35
2.3.3 Ratio-dependentfunctionalresponse 37
2.3.4 Predator-dependentfunctionalresponse 37
2.4 Classicalresults 38
2.4.1 Existence,uniqueness,positivityandboundrdness 38
2.4.2 Existenceofequilibria 39
2.4.3 Stability 39
2.4.4 Poincaré-Andronov-Hopfbifurcation 39
2.5 Specificcase(Rosenzweig-MacArthurmodel) 39
2.6 Gausetypemodelwithvariabledisappearancerate 41
2.6.1 Hsudisappearancerate 41
2.6.2 Bazykin'sdisappearancerate 42
2.6.3 CFdisappearancerate 42
2.6.4 VTdisappearancerate 43
2.7 Generalvariabledisappearancerate 44
2.8 Positivityandboundedness 45
2.9 Existenceandstabilityofequilibria 45
2.10 Globalasymptoticstability 52
2.11 Non-existenceoflimitcycles 54
3 Applicationfromtheecologicalliterature 57
3.1 Gause/RMAmodel 60
3.1.1 PAHbifurcation 60
3.1.2 Numericalsimulations 60
3.2 Hsumodel 63
3.2.1 PAHbifurcation 63
3.2.2 Numericalsimulations 63
3.3 Bazykinmodel 66
3.3.1 Numericalsimulation 66
3.4 TheCavani-Farkas(CF)model 68
3.4.1 Numericalsimulations 69
3.5 Variable-Territorry(VT)model 71
3.5.1 Numericalsimulation 71
|
| Côte titre : |
MAM/0759 |
|
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