Dynamical properties of discrete Lotka-Volterra equations
The authors consider the Lotka-Volterra system of differential equations in the plane of the form \[ \frac{dx}{dt}=\lambda x(1-ax-by),\quad \frac{dy}{dt}=\mu y(1-cx-d y) \] where \(\lambda>0\), \(\mu>0\), \(a\geq 0\), \(b\geq 0\), \(c\geq 0\), \(d\geq 0\), \(a+b+c+d>0\). Define the Lotka-Volterra map \(F:\mathbb{R}^2\to\mathbb{R}^2\) by \[ F(x,y)=(\lambda x(1-ax-by),\;\mu y(1-cx-d y)),\quad (x,y)\in\mathbb{R}^2. \] Denote by \(F^n\) the iterates of \(F\) under composition of functions. Consider the smooth analytic semigroup action \(\mathbb{Z}_+\times \mathbb{R}^2\to\mathbb{R}^2\) defined by \((n,X)\mapsto F^n(X)\) constituting a smooth semidynamical system on \(\mathbb{R}^n\), where \(\mathbb{Z}_+\) is the additive semigroup of nonnegative integers and \(X=(x,y)\in\mathbb{R}^2\). The authors consider the special uncoupled case of Lotka-Volterra map \[ F(x,y)=(\lambda x(1-x),\mu y(1-y)),\quad (x,y)\in\mathbb{R}^2 \] and point out a few of the features of the iterates of this map. Then the dynamics of the simple quadratic coupled special case of the Lotka-Volterra map \[ F(x,y)=(\lambda x(1-y),\mu y(1-x)),\quad (x,y)\in\mathbb{R}^2 \] is investigated. In this case for \(0<\lambda\), \(\mu <1\) the unit square is invariant with respect to the map \(F\), for \(\lambda=\mu\) we have \(F(L_\sigma)\subset L_{\lambda\sigma}\) where \[ L_\sigma= \{(x,y)\in\mathbb{R}^2;\;y=x+\sigma,\quad \sigma\in\mathbb{R}\}, \] for \(\lambda=\mu>1\) the authors give some properties of the dynamical system of iterates of \(F\) and also if \(\lambda>1\), \(\mu>1\) and \((\lambda-1)(\mu-1)<4\). Finally the authors consider \[ F(x,y)=(\lambda x(1-x-\alpha y),\mu y(1-\beta x-y)),\quad (x,y)\in\mathbb{R}^2 \] with \(\alpha, \beta, \lambda, \mu\) positive numbers, summarize some basic features of this map and its dynamic behaviour by proving that there is a chaotic regime generated by an orientation-reversing horseshoe map that the authors call a twisted horseshoe. Moreover the authors show that the compact invariant set corresponding to the twisted horseshoe is essentially locally repelling -- indicating that this population state is highly unstable.
scientific article; zbMATH DE number 1755351
scientific article