Configuration of planar kolmogorov cubic polynomial differential systems with the most centers

. In the paper, we study the configuration of the maximum number of centers for planar polynomial differential systems with two intersecting invariant straight lines. First, we obtain a general rule about the configuration of centers of the Hamiltonian differential systems when the number of centers reaches the maximum. Using this rule, we then give an affirmative answer to a conjecture on cubic polynomial Kolmogorov Hamiltonian systems. Last, we discuss the configuration of centers of cubic polynomial Kolmogorov differential systems with four centers, and we prove that the number of all configurations of centers for the cubic polynomial Kolmogorov differential systems is one more than that of the cubic polynomial Hamiltonian Kolmogorov differential systems in the equivalent sense.