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A non-autonomous competitive Lotka–Volterra system

  • Alexandre N. Carvalho
  • José A. Langa
  • James C. Robinson
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 182)

Abstract

As our first extended example we will consider a non-autonomous Lotka–Volterra model,
$$\begin{array}{rcl} \dot{u}& = u(\lambda (t) - au - bv)& \\ \dot{v}& = v(\mu - cu - dv), &\end{array}$$
(9.1)
where the parameters a, b,c,d, and μ are positive, ad>bc, and 0<λ≤λ(t)≤Λ. In line with the interpretation of this model in terms of the numbers of two competing species, we consider the dynamics only in positive quadrant \(\overline{Q} :=\{ (u,v) :\ u,v \geq 0\}\); the interesting dynamics takes place in the interior of this quadrant, {(u,v) :u,v>0}, which we denote by Q. Note that the u- and v-axes are both invariant.

Keywords

Autonomous System Order Relation Parameter Range Cell Complex Logistic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Ahmad S, Lazer AC (1995) On the nonautonomous N-competing species problems. Appl Anal 57:309–23MathSciNetMATHCrossRefGoogle Scholar
  2. Langa JA, Robinson JC, Suárez A (2003) Forwards and pullback behaviour of a non-autonomous Lotka–Volterra system. Nonlinearity 16:1277–1293MathSciNetMATHCrossRefGoogle Scholar
  3. Langa JA, Robinson JC, Rodríguez-Bernal A, Suárez A (2009) Permanence and asymptotically stable complete trajectories for non-autonomous Lotka–Volterra models with diffusion. SIAM J Math Anal 40:2179–2216MathSciNetMATHCrossRefGoogle Scholar
  4. Murray JD (1993) Mathematical biology. Springer, Berlin Heidelberg New YorkMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Alexandre N. Carvalho
    • 1
  • José A. Langa
    • 2
  • James C. Robinson
    • 3
  1. 1.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil
  2. 2.Departamento de Ecuaciones Diferenciales y Análisis NuméricoFacultad de MatemáticasSevilleSpain
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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