Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics
Abstract
1. Introduction and Preliminaries
2. Main Results
2.1. Global Attractivity of Some Cooperative Discrete Dynamical Systems via Difference Inequalities
- (a)
- is a cooperative map.
- (b)
- ,, are sequences of the real components in such that and
- It remains to prove the global attractivity of the equilibrium point , when , . If for several , but not for all i, and , for all remaining , then we have
2.2. Global Stability of Some Additive Cooperative Discrete Dynamical Systems
- (a)
- T maps the positive quadrant into the invariant set .
- (b)
- For all values of the parameters, the system has the equilibrium point .
- (c)
- There is at least one and at most two equilibrium points.
- (d)
- The point is the unique equilibrium if and only ifIn this case, is globally asymptotically stable.
- (e)
- A positive interior fixed point exists if and only if condition (26) is not satisfied, that is whenIn this case, is globally asymptotically stable on .
3. Examples of Cooperative Evolutionary Models
- (i)
- if , then there exists only the zero equilibrium point ,
- (ii)
- if , then there exist two equilibrium points: and ,
- (iii)
- if , then there exist three equilibrium points: and two positive equilibrium points .
- (i)
- if , then ,
- (ii)
- if , then and ,
- (iii)
- if , then and .
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MDPI | Multidisciplinary Digital Publishing Institute |
DOAJ | Directory of open access journals |
TLA | Three letter acronym |
LD | Linear dichroism |
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Kulenović, M.R.S.; Nurkanović, M.; Nurkanović, Z.; Trolle, S. Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics. Axioms 2024, 13, 730. https://doi.org/10.3390/axioms13110730
Kulenović MRS, Nurkanović M, Nurkanović Z, Trolle S. Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics. Axioms. 2024; 13(11):730. https://doi.org/10.3390/axioms13110730
Chicago/Turabian StyleKulenović, Mustafa R. S., Mehmed Nurkanović, Zehra Nurkanović, and Susan Trolle. 2024. "Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics" Axioms 13, no. 11: 730. https://doi.org/10.3390/axioms13110730
APA StyleKulenović, M. R. S., Nurkanović, M., Nurkanović, Z., & Trolle, S. (2024). Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics. Axioms, 13(11), 730. https://doi.org/10.3390/axioms13110730