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Bifurcation theory of impulsive dynamical systems / / Kevin E. M. Church, Xinzhi Liu
| Bifurcation theory of impulsive dynamical systems / / Kevin E. M. Church, Xinzhi Liu |
| Autore | Church Kevin E. M. |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (xvii, 388 pages) : illustrations |
| Disciplina | 515.35 |
| Collana | IFSR international series on systems science and engineering |
| Soggetto topico |
Bifurcation theory
System analysis Teoria de la bifurcació Anàlisi de sistemes |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-64533-9 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Reading Guide -- Contents -- I Impulsive Functional Differential Equations -- I.1 Introduction -- I.1.1 Nonautonomous Dynamical Systems -- I.1.2 History Functions -- I.1.3 The Space RCR of Right-Continuous Regulated... -- I.1.4 Gelfand-Pettis Integration -- I.1.5 Integral and Summation Inequalities -- I.1.6 Comments -- I.2 General Linear Systems -- I.2.1 Existence and Uniqueness of Solutions -- I.2.2 Evolution Families -- I.2.2.1 Phase Space Decomposition -- I.2.2.2 Evolution Families are (Generally) NowhereContinuous -- I.2.2.3 Continuity under the L2 Seminorm -- I.2.3 Representation of Solutions of the ... -- I.2.3.1 Pointwise Variation-of-Constants Formula -- I.2.3.2 Variation-of-Constants Formula in theSpace RCR -- I.2.4 Stability -- I.2.5 Comments -- I.3 Linear Periodic Systems -- I.3.1 Monodromy Operator -- I.3.2 Floquet Theorem -- I.3.3 Floquet Multipliers, Floquet Exponents and... -- I.3.4 Computational Aspects in Floquet Theory -- I.3.4.1 Floquet Eigensolutions -- I.3.4.2 Characteristic Equations for Finitely ReducibleLinear Systems -- I.3.4.3 Characteristic Equations for Systems withMemoryless Continuous Part -- I.3.5 Comments -- I.4 Nonlinear Systems and Stability -- I.4.1 Mild Solutions -- I.4.2 Dependence on Initial Conditions -- I.4.3 The Linear Variational Equation and Linearized... -- I.4.4 Comments -- I.5 Existence, Regularity and Invariance of Centre Manifolds -- I.5.1 Preliminaries -- I.5.1.1 Spaces of Exponentially Weighted Functions -- I.5.1.2 η-Bounded Solutions from Inhomogeneities -- I.5.1.3 Substitution Operator and Modification ofNonlinearities -- I.5.2 Fixed-Point Equation and Existence... -- I.5.2.1 A Remark on Centre Manifold Representations:Graphs and Images -- I.5.3 Invariance and Smallness Properties -- I.5.4 Dynamics on the Centre Manifold -- I.5.4.1 Integral Equation.
I.5.4.2 Abstract Ordinary Impulsive DifferentialEquation -- I.5.4.3 A Remark on Coordinates and Terminology -- I.5.5 Reduction Principle -- I.5.5.1 Parameter Dependence -- I.5.6 Smoothness in the State Space -- I.5.6.1 Contractions on Scales of Banach Spaces -- I.5.6.2 Candidate Differentials of the SubstitutionOperators -- I.5.6.3 Smoothness of the Modified Nonlinearity -- I.5.6.4 Proof of Smoothness of the Centre Manifold -- I.5.6.5 Periodic Centre Manifold -- I.5.7 Regularity of Centre Manifolds... -- I.5.7.1 A Coordinate System and Pointwise PC1,m-Regularity -- I.5.7.2 Reformulation of the Fixed-Point Equation -- I.5.7.3 A Technical Assumption on the Projections Pc(t) and Pu(t) -- I.5.7.4 Proof of PC1,m-Regularity at Zero -- I.5.7.5 The Hyperbolic Part Is Pointwise PC1,m-Regular at Zero -- I.5.7.6 Uniqueness of the Taylor Coefficients -- I.5.7.7 A Discussion on the Regularity of the Matrices tYj(t) -- I.5.8 Comments -- I.6 Computational Aspects of Centre Manifolds -- I.6.1 Euclidean Space Representation -- I.6.1.1 Definition and Taylor Expansion -- I.6.1.2 Dynamics on the Centre Manifold in EuclideanSpace -- I.6.1.3 An Impulsive Evolution Equation and Boundary Conditions -- I.6.2 Approximation by the Taylor Expansion -- I.6.2.1 Evolution Equation and Boundary Conditions for Quadratic Terms -- I.6.2.2 Solution by the Method of Characteristics -- I.6.3 Visualization of Centre Manifolds -- I.6.3.1 An Explicit Scalar Example Without Delays -- I.6.3.2 Two-Dimensional Example with QuadraticDelayed Terms -- I.6.3.3 Detailed Calculations Associated withExample I.6.3.2 -- The u12 Coefficient -- The u22 Coefficient -- The u1u2 Coefficient -- I.6.4 The Overlap Condition -- I.6.4.1 Distributed Delays -- I.6.4.2 Transformations that Enforce the OverlapCondition for Discrete Delays -- I.6.5 Comments. I.7 Hyperbolicity and the Classical Hierarchy of InvariantManifolds -- I.7.1 Preliminaries -- I.7.2 Unstable Manifold -- I.7.3 Stable Manifold -- I.7.4 Centre-Unstable Manifold -- I.7.5 Centre-Stable Manifold -- I.7.6 Dynamics on Finite-Dimensional... -- I.7.7 Linearized Stability and Instability,Revisited -- I.7.8 Hierarchy and Inclusions -- I.8 Smooth Bifurcations -- I.8.1 Centre Manifolds Depending Smoothly on Parameters -- I.8.2 Codimension-One Bifurcations for Systems with a Single Delay: Setup -- I.8.3 Fold Bifurcation -- I.8.3.1 Example: Fold Bifurcation in a Scalar System with Delayed Impulse -- I.8.3.2 Calculation of the Function Y11(t) forExample I.8.3.1 -- I.8.4 Hopf-Type Bifurcation and Invariant Cylinders -- I.8.4.1 Example: Impulsive Perturbation from a HopfPoint -- I.8.5 Calculations Associated to Example I.8.4.1 -- I.8.5.1 The Projection Pc(t) and Matrix (t) -- I.8.5.2 Calculation of π(t) and the Matrices A(t)and B -- I.8.5.3 Calculation of n0(t): A Numerical Routine -- I.8.5.4 Calculation of h2 -- I.8.6 A Recipe for the Analysis of Smooth LocalBifurcations -- I.8.7 Comments -- II Finite-Dimensional Ordinary Impulsive Differential Equations -- II.1 Preliminaries -- II.1.1 Existence and Uniqueness of Solutions -- II.1.2 Dependence on Initial Conditions... -- II.1.3 Continuity Conventions: Right- andLeft-Continuity -- II.1.4 Comments -- II.2 Linear Systems -- II.2.1 Cauchy Matrix -- II.2.2 Variation-of-Constants Formula -- II.2.3 Stability -- II.2.4 Exponential Trichotomy -- II.2.5 Floquet Theory -- II.2.5.1 Homogeneous Systems -- II.2.5.2 Periodic Solutions of Homogeneous Systems -- II.2.5.3 Periodic solutions of Inhomogeneous Systems -- II.2.5.4 Periodic Systems Are ExponentiallyTrichotomous -- II.2.5.5 Stability -- II.2.6 Generalized Periodic Changes of Variables -- II.2.6.1 A Full State Transformation and ChainMatrices. II.2.6.2 Real Floquet Decompositions -- II.2.6.3 A Real T-Periodic Kinematic Similarity -- II.2.7 Comments -- II.3 Stability for Nonlinear Systems -- II.3.1 Stability -- II.3.2 The Linear Variational Equation... -- II.3.3 Comments -- II.4 Invariant Manifold Theory -- II.4.1 Existence and Smoothness -- II.4.2 Invariance Equation for Nonautonomous... -- II.4.3 Invariance Equation for Systems with... -- II.4.4 Dynamics on Invariant Manifolds -- II.4.5 Reduction Principle for the Centre Manifold -- II.4.6 Approximation by Taylor Expansion -- II.4.7 Parameter Dependence -- II.4.7.1 Centre Manifolds Depending on a Parameter -- II.4.8 Comments -- II.5 Bifurcations -- II.5.1 Reduction to an Iterated Map -- II.5.2 Codimension-one Bifurcations -- II.5.2.1 Fold Bifurcation -- II.5.2.2 Period-Doubling Bifurcation -- Special Case: q=1 -- II.5.2.3 Cylinder Bifurcation -- II.5.3 Comments -- III Singular and Nonsmooth Phenomena -- III.1 Continuous Approximation -- III.1.1 Introduction -- III.1.1.1 Singular Unfolding of an Impulsive Differential Equation -- III.1.1.2 Preliminaries -- III.1.1.3 Time q Map -- III.1.1.4 The Realization Problem -- III.1.1.5 A Brief Discussion on the ContinuityConvention -- III.1.2 Pointwise Convergence and the Candidate... -- III.1.3 Smoothness of the Time q Map -- III.1.4 Sensitivity and Realization -- III.1.5 An Important Comment (Or Warning)... -- III.1.6 Example: Continuous-Time Logistic... -- III.2 Non-smooth Bifurcations -- III.2.1 Overview -- III.2.1.1 Bifurcations Involving Perturbations of Impulse Times -- III.2.1.2 Bifurcations Involving Crossings of ImpulseTimes and Delays -- III.2.2 Centre Manifolds Parameterized... -- III.2.2.1 Dummy Matrix System and Robustness ofSpectral Separation -- III.2.2.2 Centre Manifold Construction -- III.2.3 Overlap Bifurcations -- III.2.3.1 Floquet Spectrum. III.2.3.2 Symmetries of Periodic Solutions -- III.2.3.3 A State Transformation that Eliminatesthe Delay -- III.2.3.4 Bifurcations of Periodic Solutions -- III.2.3.5 The Introductory Example Revisited -- III.2.4 Comments -- IV Applications -- IV.1 Bifurcations in an Impulsively Damped or Driven Pendulum -- IV.1.1 Stability Analysis: The ModelWithout Delay -- IV.1.1.1 Downward Rest Position -- Case 1: (α+1)2cos2(ρT)4α -- Case 2: (α+1)2cos2(ρT)> -- 4α -- IV.1.1.2 Upward Rest Position -- Case 1: (α+1)2cosh2(ρT)4α -- Case 2: (α+1)2cosh2(ρT)> -- 4α -- IV.1.2 Stability Analysis: The Model withDelay -- IV.1.2.1 Downward Rest Position -- IV.1.2.2 Upward Rest Position -- IV.1.3 Cylinder Bifurcation at the Downward... -- IV.1.4 Cylinder Bifurcation at the Downward... -- IV.1.4.1 Floquet Multiplier Transversality Condition -- IV.1.4.2 Computation of the First LyapunovCoefficient -- IV.2 The Hutchinson Equation with Pulse Harvesting -- IV.2.1 Dummy Matrix System: Setup for the Non-smooth Centre Manifold -- IV.2.2 Dynamics on the Centre Manifold -- IV.2.3 The Transcritical Bifurcation -- IV.3 Delayed SIR Model with Pulse Vaccination and TemporaryImmunity -- IV.3.1 Introduction -- IV.3.2 Vaccinated Component Formalism -- IV.3.3 Existence of the Disease-free Periodic Solution -- IV.3.4 Stability of the Disease-free Periodic Solution -- IV.3.5 Existence of a Bifurcation Point -- IV.3.6 Transcritical Bifurcation in Terms of Vaccine Coverage at R0=1 with One Vaccination Pulse Per Period -- Linearization -- Centre Fibre Bundle -- Projection of χ0 Onto the Centre Fibre Bundle -- Dynamics on the Centre Manifold and Bifurcation -- IV.3.7 Numerical Bifurcation Analysis -- IV.4 Stage-Structured Predator-Prey System with Pulsed Birth -- IV.4.1 Model Derivation -- IV.4.2 Stability of the Extinction Equilibrium -- IV.4.3 Analysis of Predator-Free PeriodicSolution. IV.4.3.1 Existence and Uniqueness of the Predator-Free Solution. |
| Record Nr. | UNISA-996466566303316 |
Church Kevin E. M.
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
Bifurcation theory of impulsive dynamical systems / / Kevin E. M. Church, Xinzhi Liu
| Bifurcation theory of impulsive dynamical systems / / Kevin E. M. Church, Xinzhi Liu |
| Autore | Church Kevin E. M. |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (xvii, 388 pages) : illustrations |
| Disciplina | 515.35 |
| Collana | IFSR international series on systems science and engineering |
| Soggetto topico |
Bifurcation theory
System analysis Teoria de la bifurcació Anàlisi de sistemes |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-64533-9 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Reading Guide -- Contents -- I Impulsive Functional Differential Equations -- I.1 Introduction -- I.1.1 Nonautonomous Dynamical Systems -- I.1.2 History Functions -- I.1.3 The Space RCR of Right-Continuous Regulated... -- I.1.4 Gelfand-Pettis Integration -- I.1.5 Integral and Summation Inequalities -- I.1.6 Comments -- I.2 General Linear Systems -- I.2.1 Existence and Uniqueness of Solutions -- I.2.2 Evolution Families -- I.2.2.1 Phase Space Decomposition -- I.2.2.2 Evolution Families are (Generally) NowhereContinuous -- I.2.2.3 Continuity under the L2 Seminorm -- I.2.3 Representation of Solutions of the ... -- I.2.3.1 Pointwise Variation-of-Constants Formula -- I.2.3.2 Variation-of-Constants Formula in theSpace RCR -- I.2.4 Stability -- I.2.5 Comments -- I.3 Linear Periodic Systems -- I.3.1 Monodromy Operator -- I.3.2 Floquet Theorem -- I.3.3 Floquet Multipliers, Floquet Exponents and... -- I.3.4 Computational Aspects in Floquet Theory -- I.3.4.1 Floquet Eigensolutions -- I.3.4.2 Characteristic Equations for Finitely ReducibleLinear Systems -- I.3.4.3 Characteristic Equations for Systems withMemoryless Continuous Part -- I.3.5 Comments -- I.4 Nonlinear Systems and Stability -- I.4.1 Mild Solutions -- I.4.2 Dependence on Initial Conditions -- I.4.3 The Linear Variational Equation and Linearized... -- I.4.4 Comments -- I.5 Existence, Regularity and Invariance of Centre Manifolds -- I.5.1 Preliminaries -- I.5.1.1 Spaces of Exponentially Weighted Functions -- I.5.1.2 η-Bounded Solutions from Inhomogeneities -- I.5.1.3 Substitution Operator and Modification ofNonlinearities -- I.5.2 Fixed-Point Equation and Existence... -- I.5.2.1 A Remark on Centre Manifold Representations:Graphs and Images -- I.5.3 Invariance and Smallness Properties -- I.5.4 Dynamics on the Centre Manifold -- I.5.4.1 Integral Equation.
I.5.4.2 Abstract Ordinary Impulsive DifferentialEquation -- I.5.4.3 A Remark on Coordinates and Terminology -- I.5.5 Reduction Principle -- I.5.5.1 Parameter Dependence -- I.5.6 Smoothness in the State Space -- I.5.6.1 Contractions on Scales of Banach Spaces -- I.5.6.2 Candidate Differentials of the SubstitutionOperators -- I.5.6.3 Smoothness of the Modified Nonlinearity -- I.5.6.4 Proof of Smoothness of the Centre Manifold -- I.5.6.5 Periodic Centre Manifold -- I.5.7 Regularity of Centre Manifolds... -- I.5.7.1 A Coordinate System and Pointwise PC1,m-Regularity -- I.5.7.2 Reformulation of the Fixed-Point Equation -- I.5.7.3 A Technical Assumption on the Projections Pc(t) and Pu(t) -- I.5.7.4 Proof of PC1,m-Regularity at Zero -- I.5.7.5 The Hyperbolic Part Is Pointwise PC1,m-Regular at Zero -- I.5.7.6 Uniqueness of the Taylor Coefficients -- I.5.7.7 A Discussion on the Regularity of the Matrices tYj(t) -- I.5.8 Comments -- I.6 Computational Aspects of Centre Manifolds -- I.6.1 Euclidean Space Representation -- I.6.1.1 Definition and Taylor Expansion -- I.6.1.2 Dynamics on the Centre Manifold in EuclideanSpace -- I.6.1.3 An Impulsive Evolution Equation and Boundary Conditions -- I.6.2 Approximation by the Taylor Expansion -- I.6.2.1 Evolution Equation and Boundary Conditions for Quadratic Terms -- I.6.2.2 Solution by the Method of Characteristics -- I.6.3 Visualization of Centre Manifolds -- I.6.3.1 An Explicit Scalar Example Without Delays -- I.6.3.2 Two-Dimensional Example with QuadraticDelayed Terms -- I.6.3.3 Detailed Calculations Associated withExample I.6.3.2 -- The u12 Coefficient -- The u22 Coefficient -- The u1u2 Coefficient -- I.6.4 The Overlap Condition -- I.6.4.1 Distributed Delays -- I.6.4.2 Transformations that Enforce the OverlapCondition for Discrete Delays -- I.6.5 Comments. I.7 Hyperbolicity and the Classical Hierarchy of InvariantManifolds -- I.7.1 Preliminaries -- I.7.2 Unstable Manifold -- I.7.3 Stable Manifold -- I.7.4 Centre-Unstable Manifold -- I.7.5 Centre-Stable Manifold -- I.7.6 Dynamics on Finite-Dimensional... -- I.7.7 Linearized Stability and Instability,Revisited -- I.7.8 Hierarchy and Inclusions -- I.8 Smooth Bifurcations -- I.8.1 Centre Manifolds Depending Smoothly on Parameters -- I.8.2 Codimension-One Bifurcations for Systems with a Single Delay: Setup -- I.8.3 Fold Bifurcation -- I.8.3.1 Example: Fold Bifurcation in a Scalar System with Delayed Impulse -- I.8.3.2 Calculation of the Function Y11(t) forExample I.8.3.1 -- I.8.4 Hopf-Type Bifurcation and Invariant Cylinders -- I.8.4.1 Example: Impulsive Perturbation from a HopfPoint -- I.8.5 Calculations Associated to Example I.8.4.1 -- I.8.5.1 The Projection Pc(t) and Matrix (t) -- I.8.5.2 Calculation of π(t) and the Matrices A(t)and B -- I.8.5.3 Calculation of n0(t): A Numerical Routine -- I.8.5.4 Calculation of h2 -- I.8.6 A Recipe for the Analysis of Smooth LocalBifurcations -- I.8.7 Comments -- II Finite-Dimensional Ordinary Impulsive Differential Equations -- II.1 Preliminaries -- II.1.1 Existence and Uniqueness of Solutions -- II.1.2 Dependence on Initial Conditions... -- II.1.3 Continuity Conventions: Right- andLeft-Continuity -- II.1.4 Comments -- II.2 Linear Systems -- II.2.1 Cauchy Matrix -- II.2.2 Variation-of-Constants Formula -- II.2.3 Stability -- II.2.4 Exponential Trichotomy -- II.2.5 Floquet Theory -- II.2.5.1 Homogeneous Systems -- II.2.5.2 Periodic Solutions of Homogeneous Systems -- II.2.5.3 Periodic solutions of Inhomogeneous Systems -- II.2.5.4 Periodic Systems Are ExponentiallyTrichotomous -- II.2.5.5 Stability -- II.2.6 Generalized Periodic Changes of Variables -- II.2.6.1 A Full State Transformation and ChainMatrices. II.2.6.2 Real Floquet Decompositions -- II.2.6.3 A Real T-Periodic Kinematic Similarity -- II.2.7 Comments -- II.3 Stability for Nonlinear Systems -- II.3.1 Stability -- II.3.2 The Linear Variational Equation... -- II.3.3 Comments -- II.4 Invariant Manifold Theory -- II.4.1 Existence and Smoothness -- II.4.2 Invariance Equation for Nonautonomous... -- II.4.3 Invariance Equation for Systems with... -- II.4.4 Dynamics on Invariant Manifolds -- II.4.5 Reduction Principle for the Centre Manifold -- II.4.6 Approximation by Taylor Expansion -- II.4.7 Parameter Dependence -- II.4.7.1 Centre Manifolds Depending on a Parameter -- II.4.8 Comments -- II.5 Bifurcations -- II.5.1 Reduction to an Iterated Map -- II.5.2 Codimension-one Bifurcations -- II.5.2.1 Fold Bifurcation -- II.5.2.2 Period-Doubling Bifurcation -- Special Case: q=1 -- II.5.2.3 Cylinder Bifurcation -- II.5.3 Comments -- III Singular and Nonsmooth Phenomena -- III.1 Continuous Approximation -- III.1.1 Introduction -- III.1.1.1 Singular Unfolding of an Impulsive Differential Equation -- III.1.1.2 Preliminaries -- III.1.1.3 Time q Map -- III.1.1.4 The Realization Problem -- III.1.1.5 A Brief Discussion on the ContinuityConvention -- III.1.2 Pointwise Convergence and the Candidate... -- III.1.3 Smoothness of the Time q Map -- III.1.4 Sensitivity and Realization -- III.1.5 An Important Comment (Or Warning)... -- III.1.6 Example: Continuous-Time Logistic... -- III.2 Non-smooth Bifurcations -- III.2.1 Overview -- III.2.1.1 Bifurcations Involving Perturbations of Impulse Times -- III.2.1.2 Bifurcations Involving Crossings of ImpulseTimes and Delays -- III.2.2 Centre Manifolds Parameterized... -- III.2.2.1 Dummy Matrix System and Robustness ofSpectral Separation -- III.2.2.2 Centre Manifold Construction -- III.2.3 Overlap Bifurcations -- III.2.3.1 Floquet Spectrum. III.2.3.2 Symmetries of Periodic Solutions -- III.2.3.3 A State Transformation that Eliminatesthe Delay -- III.2.3.4 Bifurcations of Periodic Solutions -- III.2.3.5 The Introductory Example Revisited -- III.2.4 Comments -- IV Applications -- IV.1 Bifurcations in an Impulsively Damped or Driven Pendulum -- IV.1.1 Stability Analysis: The ModelWithout Delay -- IV.1.1.1 Downward Rest Position -- Case 1: (α+1)2cos2(ρT)4α -- Case 2: (α+1)2cos2(ρT)> -- 4α -- IV.1.1.2 Upward Rest Position -- Case 1: (α+1)2cosh2(ρT)4α -- Case 2: (α+1)2cosh2(ρT)> -- 4α -- IV.1.2 Stability Analysis: The Model withDelay -- IV.1.2.1 Downward Rest Position -- IV.1.2.2 Upward Rest Position -- IV.1.3 Cylinder Bifurcation at the Downward... -- IV.1.4 Cylinder Bifurcation at the Downward... -- IV.1.4.1 Floquet Multiplier Transversality Condition -- IV.1.4.2 Computation of the First LyapunovCoefficient -- IV.2 The Hutchinson Equation with Pulse Harvesting -- IV.2.1 Dummy Matrix System: Setup for the Non-smooth Centre Manifold -- IV.2.2 Dynamics on the Centre Manifold -- IV.2.3 The Transcritical Bifurcation -- IV.3 Delayed SIR Model with Pulse Vaccination and TemporaryImmunity -- IV.3.1 Introduction -- IV.3.2 Vaccinated Component Formalism -- IV.3.3 Existence of the Disease-free Periodic Solution -- IV.3.4 Stability of the Disease-free Periodic Solution -- IV.3.5 Existence of a Bifurcation Point -- IV.3.6 Transcritical Bifurcation in Terms of Vaccine Coverage at R0=1 with One Vaccination Pulse Per Period -- Linearization -- Centre Fibre Bundle -- Projection of χ0 Onto the Centre Fibre Bundle -- Dynamics on the Centre Manifold and Bifurcation -- IV.3.7 Numerical Bifurcation Analysis -- IV.4 Stage-Structured Predator-Prey System with Pulsed Birth -- IV.4.1 Model Derivation -- IV.4.2 Stability of the Extinction Equilibrium -- IV.4.3 Analysis of Predator-Free PeriodicSolution. IV.4.3.1 Existence and Uniqueness of the Predator-Free Solution. |
| Record Nr. | UNINA-9910483135703321 |
|
Church Kevin E. M.
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||