Almost periodic solutions for Fox production harvesting model with delay

Abstract

Abstract. By utilizing the continuation theorem of coincidence degree theory, we shall prove that a Foxproduction harvesting model with delay has at least one positive almost periodic solution. Some pre-liminary assertions are provided prior to proving our main theorem. We construct a numerical examplealong with graphical representations to illustrate feasibility of the theoretical result.Keywords. Almost periodic solution; Fox production harvesting model; Coincidence degree theory. AMS subject classification: 34K14. 1 Introduction Consider the following equation of population dynamics [1, 2]x ′ (t) = −xF(t,x) +xG(t,x), x ′ (t) =dxdt, (1)where x = x(t) is the size of the population, F(t,x) is the per–capita harvesting rateand G(t,x) is the per–capita fecundity rate. Let G(t,x) and F(t,x) be defined in theformF(t,x) = α(t) and G(t,x) = β(t)ln γ K(t)x(t), γ > 0then equation (1) becomesx ′ (t) = −α(t)x(t) +β(t)x(t)ln γ K(t)x(t), (2)where α(t) is a variable harvesting rate, β(t) is an intrinsic factor and K(t) is a varyingenvironmental carrying capacity. The positive parameter γ is referred to as an interac-tion parameter [1, 3, 4]. Indeed, if γ > 1 then intra–specific competition is high whereasif 0 < γ < 1 then the competition is low. For γ = 1, equation (2) reduces to a classical