Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

Daniel Franco, Christopher Guiver, Hartmut Logemann, Juan Perán

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)


Boundedness, persistence and stability properties are considered for a class of nonlinear, possibly infinite-dimensional, forced difference equations which arise in a number of ecological and biological contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes), disturbances induced by seasonal or environmental variation, or migration. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We present sufficient conditions for the non-zero equilibrium to be stable in a sense which is strongly inspired by the input-to-state stability concept well-known in mathematical control theory. In particular, our stability concept incorporates the impact of potentially persistent forcing. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). The theory is applied to a number of examples from population dynamics.
Original languageEnglish
Pages (from-to)1029-1076
Number of pages48
JournalJournal of Mathematical Biology
Issue number3
Early online date6 Jun 2019
Publication statusPublished - 1 Aug 2019


  • Absolute stability
  • Density-dependent population models
  • Environmental forcing
  • Forced systems
  • Global asymptotic stability
  • Infinite-dimensional systems
  • Input-to-state stability
  • Integral projection models
  • Lur’e systems
  • Population persistence

ASJC Scopus subject areas

  • Modelling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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