Abstract
In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov’s second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Highlights
Introduction and preliminariesTheory of ordinary differential equations plays an important role for solving fundamental problems in population dynamics
To the best of our knowledge, it is the first time in the literature that a differential equation with piecewise constant argument of generalized type is reduced into a nonautonomous difference equation
If we have a differential equation with generalized piecewise constant argument whose switching moments are ordered arbitrarily, it generates a nonautonomous difference equation
Summary
We discuss the existence of bounded solutions for the reduced nonautonomous difference equation. A continuous function w : Sρ → R is said to be positive definite if (i) w( ) = and (ii) w(u) > for all u = , u ∈ Sρ. If there exists a positive definite scalar function V (k, u) ∈ C[N × Sρ, R+] such that V( )(k, u(k)) ≤ , the trivial solution u = of the difference equation ( ) is stable. If there exists a positive definite scalar function V (k, u) ∈ C[N × Sρ, R+] such that V( )(k, u(k)) ≤ –α( u(k) ), where α ∈ K, the trivial solution u = of the difference equation ( ) is asymptotically stable.
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