What next?

A discrete-time prey-predator model

In the population models we have seen, differential equation models imply a continuous overlap of generations. But many species have no overlap whatsoever between successive generations and so population growth is in discrete steps. We get difference equations defined by a mapping. As we have seen, one needs at least three variables (i.e., a three-dimensional phase space) to get deterministic chaos with differential equations. Remarkably, in discrete time, chaos can show up with a single variable evolving according to simple quadratic mapping! Here we present a prey-predator model given by the mapping

$$ \begin{cases} x_{t+1}=r x_t \,(1-x_t -\alpha y_t)\\ y_{t+1}=y_t\,(\beta x_t -d) \end{cases} $$

where $t=0,1,2,\ldots$ and $r,\alpha,\beta,d$ are positive parameters. In the digital experiment, a bunch of initial conditions $(x_0,y_0)$ are launched. For the default parameters, you can observe a strange attractor.