Discrete-time analogues of predator–prey models with monotonic or nonmonotonic functional responses

Discrete-time analogues of predator–prey models with monotonic and nonmonotonic functional responses are introduced, respectively. The discrete-time analogues are considered to be numerical discretizations of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models with monotonic functional responses. Moreover, it is the first time that multiplicity of periodic solutions are studied when modeled with nonmonotonic functional responses  . Unlike other types of functional responses, nonmonotone functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation Lx=λNxLx=λNx are inapplicable to our case and some new arguments are employed for the first time.