Periodic solutions of a discrete time three-species Lotka-Volterra food-chain system

The authors study the existence of the periodic solutions of the discrete time periodic three trophic level Lotka-Volterra food-chain system $$\align x_1(k+1)& = x_1(k)\exp[r_1(k)-a_{11}(k)x_1(k)-a_{12}(k)x_2(k)],\\ x_2(k+1)& = x_2(k)\exp[-r_2(k)+a_{21}(k)x_1(k)-a_{22}(k)x_2(k)-a_{23}(k)x_3(k)],\\ x_3(k+1)& = x_3(k)\exp[-r_3(k)+a_{32}(k)x_2(k)-a_{33}(k)x_3(k)]. \endalign$$ By using {\it R. E. Gaines} and {\it J. L. Mawhin}'s continuation theorem of coincidence degree theory [Coincidence degree, and nonlinear differential equations. Lect. Notes Math. 568 (1977; Zbl 0339.47031)], some sufficient conditions are derived for the existence of positive periodic solutions of the above system