In this paper, we analyze the Babesiosis transmission dynamics on bovine and tick populations. Ticks play a role of infectious agents and vector of the protozoan Babesia hemo‐parasite. In this sense, we set out a mathematical model with constant size population for the evolution of the infected bovines with Babesiosis and analyze its qualitative dynamics. Statistical data are used to estimate some of the parameters of the model. Numerical simulations of the model varying the parameters show different scenarios about the spread of the disease. Copyright © 2011 John Wiley & Sons, Ltd.

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In this work, we provide conditions to obtain fixed point theorems for parallel dynamical systems over graphs with (Boolean) maxterms and minterms as global evolution operators. In order to do that, we previously prove that periodic orbits of different periods cannot coexist, which implies that Sharkovsky’s order is not valid for this kind of dynamical systems.