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2 results

  • Simple dynamics in non-monotone Kolmogorov systems

  • Lei Niu, Alfonso Ruiz-Herrera
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 2 / April 2023
    Published online by Cambridge University Press:
    02 December 2021, pp. 369-384
    Print publication:
    April 2023
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  • In this paper we analyse the global dynamical behaviour of some classical models in the plane. Informally speaking we prove that the folkloric criteria based on the relative positions of the nullclines for Lotka–Volterra systems are also valid in a wide class of discrete systems. The method of proof consists of dividing the plane into suitable positively invariant regions and applying the theory of translation arcs in a subtle manner. Our approach allows us to extend several results of the theory of monotone systems to nonmonotone systems. Applications in models with weak Allee effect, population models for pioneer-climax species, and predator–prey systems are given.

  • Multiplicity of solutions of Dirichlet problems associated with second-order equations in ℝ2

  • Part of
  • F. Dalbono, C. Rebelo
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 52 / Issue 3 / October 2009
    Published online by Cambridge University Press:
    23 September 2009, pp. 569-581
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  • We study the existence of multiple solutions for a two-point boundary-value problem associated with a planar system of second-order ordinary differential equations by using a shooting technique. We consider asymptotically linear nonlinearities satisfying suitable sign conditions. Multiplicity is ensured by assumptions involving the Morse indices of the linearizations at zero and at infinity.