Stable positive periodic solutions of the time-dependent logistic equation under possible hereditary influences

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25 Citations (Scopus)

Abstract

The logistic equation, generalized to include time-dependent but periodic coefficients and a functional, hereditary interaction term, is shown to have a positive periodic solution provided the time-dependent net birth rate has a positive average. Under more restrictive conditions on the interaction term and the net birth rate, this solution is shown to be uniformly asymptotically stable. The approach is to treat the problem as one of the bifurcation of nontrivial positive solutions from the identically zero solution using, roughly speaking, the average of the net birth rate as a nonlinear eigenvalue.

Original languageEnglish (US)
Pages (from-to)747-754
Number of pages8
JournalJournal of Mathematical Analysis and Applications
Volume60
Issue number3
DOIs
StatePublished - 1977

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Logistic Equation
Positive Periodic Solution
Logistics
Nonlinear Eigenvalue
Periodic Coefficients
Term
Asymptotically Stable
Interaction
Positive Solution
Bifurcation
Zero
Influence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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title = "Stable positive periodic solutions of the time-dependent logistic equation under possible hereditary influences",
abstract = "The logistic equation, generalized to include time-dependent but periodic coefficients and a functional, hereditary interaction term, is shown to have a positive periodic solution provided the time-dependent net birth rate has a positive average. Under more restrictive conditions on the interaction term and the net birth rate, this solution is shown to be uniformly asymptotically stable. The approach is to treat the problem as one of the bifurcation of nontrivial positive solutions from the identically zero solution using, roughly speaking, the average of the net birth rate as a nonlinear eigenvalue.",
author = "Cushing, {Jim M}",
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AB - The logistic equation, generalized to include time-dependent but periodic coefficients and a functional, hereditary interaction term, is shown to have a positive periodic solution provided the time-dependent net birth rate has a positive average. Under more restrictive conditions on the interaction term and the net birth rate, this solution is shown to be uniformly asymptotically stable. The approach is to treat the problem as one of the bifurcation of nontrivial positive solutions from the identically zero solution using, roughly speaking, the average of the net birth rate as a nonlinear eigenvalue.

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