On a delay population model with quadratic nonlinearity
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RESEARCH
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On a delay population model with quadratic nonlinearity Leonid Berezansky1 , Jaromír Baštinec2 , Josef Diblík2,3* and Zdenˇek Šmarda2 *
Correspondence: [email protected] 3 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering and Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic Full list of author information is available at the end of the article
Abstract A nonlinear delay differential equation with quadratic nonlinearity, x˙ (t) = r(t)
m
αk x(hk (t)) – β x (t) , 2
t ≥ 0,
k=1
is considered, where αk and β are positive constants, hk : [0, ∞) → R are continuous functions such that t – τ ≤ hk (t) ≤ t, τ = const, τ > 0, for any t > 0 the inequality hk (t) < t holds for at least one k, and r : [0, ∞) → (0, ∞) is a continuous function satisfying the inequality r(t) ≥ r0 = const for an r0 > 0. It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.
Introduction To include oscillation in population model systems, Hutchinson [, ] suggested the following delay logistic equation: N(t – τ ) dN(t) = rN(t) – , dt K 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 [–]. In particular, a delay logistic equation with several delays, x˙ (t) = x(t) α –
m
βk x(t – τk ) ,
()
k=
where α, βk and τk > are positive constants, is considered in [, p.]. Equation () can be viewed as one with quadratic nonlinearities of the unknown function x. For more work on the stability and boundedness of equations and systems related to (), one can refer to [–]. © 2012 Berezansky et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Berezansky et al. Advances in Difference Equations 2012, 2012:230 http://www.advancesindifferenceequations.com/content/2012/1/230
Page 2 of 9
In the monograph [, p.], the author considers the following population model with quadratic nonlinearity: x˙ (t) =
m
αk x(t – τk ) – βx (t),
t ≥ ,
()
k=
where αk > , β > , τk > , and with the initial condition x(t) = ϕ(t),
t ∈ –τ * , ,
()
where ϕ : [–τ * , ] → R is a continuous function, τ * = maxk=,...,m τk and ϕ(t) > if t ∈ [–τ * , ]. As can simply be verified, equation () has a unique positive equilibrium x(t) = K , t ∈ [–τ * , ∞), where K=
α , β
α :=
m
αk .
()
k=
Theorem [, Corollary .., p.] The positive equilibrium K is a global attractor of problem (), (). This result is different from almost all known results on the stability for nonli
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