Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients
Date
2003-12-01
Authors
Palamides, Panos K.
Galanis, George N.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
In this paper we study the generalized logistic equation
du/ dt = α(t)un - b(t)un+(2k + 1), n, k ∈ ℕ,
which governs the population growth of a self-limiting specie, with α(t), b(t) being continuous bounded functions. We obtain a unique global, positive and bounded solution which, further, plays the role of a frontier which clarifies the asymptotic behavior or extensibility backwards and further it is an attractor forward of all positive solutions. We prove also that the function
∅(t) = 2k+1√α(t)/ b(t)
plays a fundamental role in the study of logistic equations since if it is monotone, then it is an attractor of positive solutions forward in time. Furthermore, we may relax the boundedness assumption on α(t) and b(t) to a boundedness of it. An existence result of a positive periodic solution is also given for the case where α(t) and b(t) are also periodic (actually we derive a necessary and sufficient condition for that). Our technique is a topological one of Knesser's type (connecteness and compactness of the solutions funnel).
Description
Keywords
Generalized logistic equation, Asymptotic behavior of solutions, Periodic solutions, Knesser's property, Consequent mapping, Continuum sets
Citation
Palamides, P. K., & Galanis, G. N. (2003). Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients. Electronic Journal of Differential Equations, 2003(119), pp. 1-13.
Rights
Attribution 4.0 International