Abstract
Abstract A nonlinear delay differential equation with quadratic nonlinearity, x ˙ ( t ) = r ( t ) [ ∑ k = 1 m α k x ( h k ( t ) ) − β x 2 ( t ) ] , t ≥ 0 , is considered, where α k and β are positive constants, h k : [ 0 , ∞ ) → R are continuous functions such that t − τ ≤ h k ( t ) ≤ t , τ = const , τ > 0 , for any t > 0 the inequality h k ( t ) < t holds for at least one k, and r : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function satisfying the inequality r ( t ) ≥ r 0 = const for an r 0 > 0 . It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.
Highlights
To include oscillation in population model systems, Hutchinson [, ] suggested the following delay logistic equation: dN (t) N(t – τ ) = rN(t), dtK where N(t) is the population size at time t, r > is the intrinsic growth rate of the population, τ > and K > is the carrying capacity of the population.There are many generalizations and modifications of Hutchinson’s equation [ – ]
It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation
We prove that there exists a unique positive global solution to the problem ( ), ( )
Summary
It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation. We prove that there exists a unique positive global solution to the problem ( ), ( ). Let us recall that a function x : [–τ , ∞) → R continuous on [–τ , ∞) and continuously differentiable on [ , ∞) is called a global solution to the problem ( ), ( ) if it satisfies equation ( ) on [ , ∞) and initial condition ( ).
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