Abstract. We analyse a four-dimensional compartmental system that describes the spread of ectoparasites and a disease carried by them in a population. We identify three threshold parameters that determine which of the four potential equilibria exist. These parameters completely characterize the stability properties of the equilibria and also the global behaviour of solutions. We provide a detailed description of the global attractor in each possible scenario. The key mathematical tools of the proofs are Lyapunov--LaSalle theory, persistence theory, Poincaré--Dulac criteria and unstable manifolds. In the most complicated case, the global attractor consists of four equilibria and various heteroclinic orbits connecting those equilibria, forming a two-dimensional manifold in the phase space.

DOI: 10.14232/actasm-013-004-y

AMS Subject Classification (1991): 37B25, 37C70, 92D30

Keyword(s): compartmental system, ectoparasites, global dynamics, Lyapunov function, persistence, global attractor

Received January 9, 2013. (Registered under 4/2013.)