A non-autonomous competitive Lotka–Volterra system

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


As our first extended example we will consider a non-autonomous Lotka–Volterra model,
$$\begin{array}{rcl} \dot{u}& = u(\lambda (t) - au - bv)& \\ \dot{v}& = v(\mu - cu - dv), &\end{array}$$
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.


Autonomous System Order Relation Parameter Range Cell Complex Logistic Equation 
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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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