Bifurcations and chaos in a time-discrete integral model of population dynamics
The purpose of this paper is to study the dynamic properties of integral models, mainly one-species models. The study is made using the model \[ u_{n+1}(x)=a\int_ M\alpha(x,y)u_ n(x,y)dy\;\exp\left[-\int_ M\beta(x,y)u_ n(x,y)dy\right], \] where \(M\) is a one-dimensional uniform space and the kernels \(\alpha\) and \(\beta\) are of the form \[ \alpha(x,y)=\begin{cases} a_ 0 & \| x-y\|\leq r_ \alpha\\ 0 & \| x-y\|>r_ \alpha\end{cases},\qquad\beta(x,y)=\begin{cases} b_ 0 & \| x- y\|\leq r_ \beta\\ 0 & \| x-y\|>r_ \beta\end{cases}. \] The following normalization condition is assumed to be satisfied: \[ \int_ M\alpha(x,y)dy=\int_ M\beta(x,y)dy=1, \] that is \(2 r_ \alpha a_ 0=2 r_ \beta b_ 0=1\). \(L_ 2(M)\) will be used as a phase space. The presence of dissipator caused by spatial interaction is detected. Several bifurcations of codimensions 1 and 2 are investigated. The Lyapunov exponents and dimension of attractors is shown to be a common feature of chaotic behaviour.
scientific article; zbMATH DE number 64799