7. Conclusions

In this chapter we have analysed the dynamics of a predator-prey dynamical system in discrete time (see also [12]). We have provided conditions for the global stability of the fixed points corresponding the co-extinctions of the predator-prey as well as for the extinction of predators and survival of preys. For some parameter regions, we have identified hyperchaos (i.e. more than one positive Lyapunov exponent; see [17]). A deep analysis of the existence and properties of the invariant set has been provided for a wide region of the parameter space containing the most biologically relevant dynamics. We have identified the presence of escaping zones in the phase space at which species populations go out of the domain (e.g. they overcome the carrying capacity) and then the iterates become negative, meaning that populations go to extinction. By means of iteration, we have characterised a very complicated shape of the escaping regions, presumably with a highly entangled, fractal topology. These escaping regions could be responsible for species extinctions evolving in discrete time. Although early experimental research allowed to identify deterministic chaos in insect populations [3], as far as we know, no empirical proofs about this phenomenon have been described.
