On the control of certain interacting populations

Felix Albrecht*, Harry Gatzke, Abraham Haddad, Nelson Wax

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

We study the time optimal control of the system x ̇1 = x1f{hook}1(x1, x2) + u1(t) g1(x1), x ̇2 = x2f{hook}2(x1, x2) + u2(t)g2(x2), where x1 is the size of the population of one species, x2 is the population size of the second species, f{hook}1 and f{hook}2 are the fractional growth rates of the respective species, g1 and g2 are nowhere vanishing functions of class C1(0, + ∞), and the control u(t) = (u1(t), u2(t)) takes on values in a closed rectangle. The functions f{hook}1 and f{hook}2 are chosen to represent prey-predator, competitive, and symbiotic interactions. We show, for the various interactions, that a time optimal control, if it exists, must be "bang-bang," and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.

Original languageEnglish (US)
Pages (from-to)578-603
Number of pages26
JournalJournal of Mathematical Analysis and Applications
Volume53
Issue number3
DOIs
StatePublished - Mar 1976

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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