Advances in Discrete Dynamical Systems, Difference Equations and Applications: 26th ICDEA, Sarajevo, Bosnia and Herzegovina, July 26-30, 2021

Contents
A Discrete-Time Predator-Prey Model with Selection and Mutation
	1 Introduction
	2 The Model
	3 The Pure-Selection Model Without Mutation
		3.1 Equilibria and Their Stabilities
		3.2 Persistence
	4 The Full Model with Mutation
		4.1 Coexistence and Persistence
	5 Numerical Simulations
		5.1 Numerical Simulations for the Pure Selection Case
		5.2 Numerical Simulations for a Selection-Mutation Case
	6 Concluding Remarks
	References
On the Dynamics and Asymptotic Behaviour of the Mean Square of Scalar Linear Stochastic Difference Equations
	1 Introduction
	2 Standing Assumptions and Precise Formulation of the Problem
	3 Volterra Summation Equations for the Mean Square
	4 Asymptotic Behaviour, p=1
		4.1 The Simple Case q=1
		4.2 Preparatory Results
		4.3 Asymptotic Behaviour
		4.4 Covariance Dynamics
	5 The Case When α=0
	References
Border Collision and Heteroclinic Bifurcations in a 2D Piecewise Smooth Map
	1 Introduction
	2 Investigated Map
	3 Phase Portrait Transformations
		3.1 Fold Border Collision Bifurcation
		3.2 Heteroclinic Tangles: Destruction and Creation of a Closed Invariant Curve
	4 Conclusion
	References
Using Homotopy Link Function with Lipschitz Threshold in Studying Synchronized Fluctuations in Hierarchical Models
	1 Introduction
	2 Analytical Derivation of Drive-Response System
		2.1 Complete Synchronization Using Contraction Mapping Theorem
		2.2 Local Dynamics, Attractors and Attracting Set of Drive-Response System
	3 Numerical Results
	4 Conclusion
	References
Solving Third-Order Linear Recurrence Relations with Applications to Number Theory and Combinatorics
	1 Introduction
	2 Previous Works
	3 Preliminaries
	4 Main Results
		4.1 General Solution
		4.2 Summatory Formula
	5 Applications
		5.1 Tribonacci Numbers
		5.2 Tribonacci Polynomials
	6 Future Work
	References
A Survey on Max-Type Difference Equations
	1 Introduction
	2 Preliminaries
	3 Relation with Piecewise Linear Difference Equations
	4 A Version of Lyness Difference Equation with Maximum
		4.1 Case k=1,  l=0: xn+1 = max{xn,A }xn-1
		4.2 Case k=1,  l=1: xn+1 = max{xn,A}xnxn-1
		4.3 Case k=2,  l=1: xn+1 = max{xn2,A}xnxn-1
		4.4 A Generalization of the Lyness' Max-Type Difference Equation
	5 Reciprocal Difference Equation with Maximum
		5.1 Constant Coefficients
		5.2 Periodic Coefficients
		5.3 Powers in the Denominator
	6 Other Classes of Max-Type Difference Equations
	7 Rank-Type Difference Equations
	8 Applications
	9 Open Problems
	10 Conclusions
	References
Catalan Numbers Recurrence as a Stationary State Equation of the Probabilistic Cellular Automaton
	1 Introduction
	2 The Automaton
	3 Equations for Expected Values
		3.1 Stationary State
		3.2 Balance Equation for the Density
		3.3 Balance Equation for the Total Number of Clusters
		3.4 Balance Equation for the Number of M-Clusters
	4 Catalan Numbers Recurrence
	5 Comments
	References
Oscillation of Second Order Impulsive Neutral Difference Equations of Non-canonical Type
	1 Introduction
	2 Oscillation Criteria
	References
On the Robustness Property of Nonuniform Exponential Dichotomies
	1 Introduction
	2 Robustness Property
	References
Implicit Linear First Order Difference Equations Over Commutative Rings
	1 Introduction
	2 Implicit Linear Difference Equation bxn+1=axn+f
	3 Periodic Solutions of Implicit Linear Difference Equations
	4 Quasi-polynomial Solution of Implicit Linear Difference Equation
	5 The Case of a Local Ring
	References
Global Attraction and Repulsion of a Heteroclinic Limit Cycle in Three Dimensional Kolmogorov Maps
	1 Introduction
	2 Carrying Simplex of Competitive Kolmogorov Maps
	3 Global Attraction and Repulsion of a Heteroclinic Limit Cycle
	4 An Example
	5 Conclusion
	References
Bifurcation and Stability of a Ricker Host-Parasitoid Model with a Host Constant Refuge and General Escape Function
	1 Introduction
	2 Equilibrium Points
	3 Linearized Stability Analysis
		3.1 Local Stability of the Exclusion Equilibrium Point
		3.2 Linearized Stability of the Coexistence Equilibrium
	4 Local Bifurcations of Equilibrium Points
		4.1 Period-Doubling Bifurcation of Exclusion Equilibrium Point
		4.2 Transcritical Bifurcation of Exclusion Equilibrium Point
		4.3 Local Bifurcations of Positive Equilibrium Point
	5 Boundedness of the Solutions
	6 Global Attractivity
	7 Uniform Persistence
	8 Examples
	9 Conclusion and Further Discussion
	References
SageMath Tools for Stability and Bifurcation for Discrete Dynamical Systems
	1 Introduction
	2 Parameter Curves
		2.1 One-Dimensional Maps: Parameter-Derivative Plane
		2.2 Two-Dimensional Maps: tr-det Plane
	3 Basins of Attraction
		3.1 A Closeness Function
		3.2 An Algorithm
		3.3 One-Dimensional Maps
		3.4 Two-Dimensional Maps
	4 Stability Region in Parameter-Parameter Plane
		4.1 One-Dimensional Maps
		4.2 Two-Dimensional Maps
	5 Conclusions
	References
Pullback Attractors of Nonautonomous Lattice Difference Equations
	1 Introduction
	2 Set Up
		2.1 Some Basic Estimates
	3 Nonautonomous Discrete Time Lattice Dynamical System
		3.1 Existence of a Pullback Attractor
		3.2 Existence of a Forward ω-Limit Sets
	4 Discrete Time Skew Product Lattice Systems
		4.1 Some Basic Estimates
		4.2 Existence of a Random Attractor
	References
Global Dynamics of Modified Discrete Lotka-Volterra Model
	1 Introduction and Preliminaries
	2 Global Dynamic Results
	3 Global Dynamics of System (1)
		3.1 Local Stability Results
		3.2 Global Stability Results
	References
Nonwandering Sets and Special α-limit Sets of Monotone Maps on Regular Curves
	1 Introduction
	2 Preliminaries
	3 Nonwandering Sets of Monotone Maps on Regular Curves
	4 The Space of Minimal Sets with Respect to the Hausdorff Metric
	5 On the Continuity of Limit Maps ωf and αf
	6 On Special α-limit Sets
		6.1 Relation Between Nonwandering Sets, α-limit Sets and Special α-limit sets
		6.2 Further Results on Special α-limit Sets
	References
Asymptotic Stability, Bifurcation Analysis and Chaos Control in a Discrete Evolutionary Ricker Population Model with Immigration
	1 Introduction
	2 An Evolutionary Discrete Model
	3 Local Dynamics
	4 Bifurcations Analysis
		4.1 Existence of Bifurcations About the Positive Fixed Point of the Model
		4.2 Neimark–Sacker Bifurcation About (x(b0), u(b0))
		4.3 Period-Doubling Bifurcation About (x(b0), u(b0))
	5 Chaos Control
		5.1 State Feedback Control
		5.2 Pole Placement Method
		5.3 Chaos Control Using Hybrid Method
	6 Numerical Simulations
	7 Concluding Remakes
	References
Weighted Norms In Advanced Volterra Difference Equations
	1 Introduction
	2 Boundedness of Solutions
	3 Weighted Norm
	4 Applications
	5 Concluding Remarks
	References
Comparison of Tests for Oscillations in Delay/Advanced Difference Equations with Continuous Time
	1 Introduction
	2 The Main Test
	3 The Comparison to the Higher-Order Functional Equations
		3.1 Nowakowska-Werbowski Conditions
		3.2 Zhang-Choi Conditions
		3.3 The Comparison to the Second-Order Functional Equations
	4 The Comparison to the Delay Difference Equations
		4.1 Ladas-Pakula-Wang Conditions
		4.2 Zhang-Yan-Zhao Conditions
		4.3 Zhang-Yan-Choi Conditions
		4.4 Zhang-Yan Conditions
	5 Summary
	References
Krause Mean Processes Generated by Cubic Stochastic Matrices IV: Off-Diagonally Uniformly Positive Nonautonomous Cubic Stochastic Matrices
	1 Introduction
	2 Krause Mean Processes
	3 Quadratic Stochastic Processes
	4 The Krause Mean Process Generated by the Quadratic Stochastic Operator
	5 The Main Result
	6 Discussions
	7 Conclusion
	References
Linearization for Difference Equations with Infinite Delay
	1 Introduction
	2 Preliminaries
	3 Main Result
		3.1 Uniform Exponential Dichotomy Case
	References
A Method to Derive Discrete Population Models
	1 Introduction
	2 Discrete Model Derivation
	3 Analysis of the Discrete Model
		3.1 black Extinction and Competitive Exclusion
		3.2 Relationship between the Discrete Model and black its Continuous black Counterpart
		3.3 Importance of the Composition of the Model Parameters
	4 Examples of Models Obtained Using the Derivation Method
		4.1 Single Species Models
		4.2 Multi-Species Models
	5 Conclusion
	References
Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models
	1 Introduction
	2 Preview: Turnover/Reproduction Trichotomy
		2.1 Iteroparous Populations with Mating
		2.2 Individual Development Modeled by Feller Kernels
	3 More About the Spectral Radius
		3.1 An Example by Bonsall [Sect. 2(iv)]ch23Bon58
		3.2 Lower and Upper Bounds for the Spectral Radius
		3.3 Commutation Rules
		3.4 Monotonicity of the Spectral Radius
		3.5 Cartesian Products
		3.6 Existence of (Lower) Eigenvectors
		3.7 Left Resolvents
	4 Turnover Versus Reproduction Number
		4.1 Starting Point: Next Generation Operator
		4.2 Starting Point: Basic Turnover Operator
		4.3 Synopsis
		4.4 Cones with Uniform Order Units
	5 Additive Perturbations of Rank-One Operators
	6 Iteroparous Populations with Mating
		6.1 Iteroparous Populations More Concrete
		6.2 A Rank-Structured Population
	7 Discussion
	References