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Stability and bifurcations analysis of a competition model with piecewise constant arguments
Author(s) -
Kartal S.,
Gurcan F.
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3196
Subject(s) - mathematics , piecewise , equilibrium point , constant (computer programming) , stability (learning theory) , exponential stability , differential equation , integer (computer science) , bifurcation , mathematical analysis , population model , population , pure mathematics , combinatorics , nonlinear system , physics , demography , quantum mechanics , machine learning , sociology , computer science , programming language
In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equationsd x d t = r 1 x ( t )1 − x ( t )k 1− α 1 x ( t ) y ( [ t − 1 ] ) + α 2 x ( t ) y ( [ t ] ) ,d y d t = r 2 y ( t )1 − y ( t )k 2+ α 1 y ( t ) x ( [ t − 1 ] ) − α 2 y ( t ) x ( [ t ] ) − d 1 y ( t ) ,where t ≥ 0, the parameters r 1 , k 1 , α 1 , α 2 , r 2 , k 2 , and d 1 are positive, and [t] denotes the integer part of t ∈ [0, ∞ ). x(t) and y(t) represent population density for related species. Sufficient conditions are obtained for the local and global stability of the positive equilibrium point of the corresponding difference system. We show through numerical simulations that periodic solutions arise through Neimark–Sacker bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.