A simple parasite model with complicated dynamics

 During their first year of life sheep acquire parasites through grazing, and simultaneously build up an immunity to infection. At the beginning of each year non-immune lambs are introduced onto contaminated pasture. We represent this process by differential equations describing the within-year dynamics, and defining a difference equation that describes the between-year dynamics. An example with two system parameters is analysed in detail. It is shown that regions exist in parameter space where periodic (between-year) or aperiodic solutions occur. Parasite control schemes could change the system dynamics from a stable equilibrium to complicated long-term fluctuations.