Dynamics of almost periodic solutions for a discrete Fox harvesting model with feedback control
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RESEARCH
Open Access
Dynamics of almost periodic solutions for a discrete Fox harvesting model with feedback control Jehad Alzabut* *
Correspondence: [email protected] Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
Abstract We consider the following discrete Fox harvesting model with feedback control of the form
) – α (n) – γ (n)u(n)}, x(n + 1) = x(n) exp{β (n) lnr ( K(n) x(n) u(n) = –μ(n)u(n) + ν (n)x(n).
Under the assumptions of almost periodicity of the coefficients, sufficient conditions are established for the existence and uniformly asymptotical stability of almost periodic solutions of this model. The persistence as well as the boundedness of solutions of the above system are discussed prior to presenting the main result. Examples are provided to illustrate the effectiveness of the proposed results. MSC: 39A11; 34K14 Keywords: discrete Fox harvesting model; almost periodic; persistence and boundedness; feedback control; uniformly asymptotical stability
1 Introduction Consider the following equation of population dynamics [, ]: x (t) = –xF(t, x) + xG(t, x),
x (t) =
dx , dt
()
where x = x(t) is the size of the population, F(t, x) is the per-capita harvesting rate and G(t, x) is the per-capita fecundity rate. Let G(t, x) and F(t, x) be defined in the form F(t, x) = α(t) and G(t, x) = β(t) ln
r
K(t) , x(t)
r > ,
then equation () becomes K(t) x (t) = x(t) β(t) lnr – α(t) , x(t)
()
where α(t) is a variable harvesting rate, β(t) is an intrinsic factor and K(t) is a varying environmental carrying capacity. The positive parameter r is referred to as an interaction parameter [, , ]. Indeed, if r > then intra-specific competition is high, whereas © 2012 Alzabut; 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.
Alzabut Advances in Difference Equations 2012, 2012:157 http://www.advancesindifferenceequations.com/content/2012/1/157
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if < r < , then the competition is low. For r = , equation () reduces to the classical Gompertzian model with harvesting [, ]. Equation () is called a Fox surplus production model that has been used to build up certain prediction models such as microbial growth model, demographic model and fisheries model. This equation is considered to be an efficient alternative to the well known r-logistic model. Specifically, the Fox model is more appropriate upon describing lower population density; we refer the reader to [, , , –] and, in particular, to the recent paper [] for more information. Ecosystems in the real world are continuously disturbed by unpredictable forces which can result in changing some biological parameters such as survival rates. In ecology, a question of practical interest is whether or not an ecosystem can w
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