Positive periodic solution of a discrete commensal symbiosis model with Holling II functional response

Abstract

Sufficient conditions are obtained for the existence of positive periodic solution of the following discrete commensal symbiosis model with Holling II functional response

$$

x_1(k+1)&=& x_1(k)\exp\Big\{ a_1(k)-b_1(k)x_1(k)+\di\frac{c_1(k)x_2(k)}{e_1(k)+f_1(k)x_2(k)}\Big\},

$$

$$

x_2(k+1)&=& x_2(k)\exp\big\{ a_2(k)-b_2(k)x_2(k)\big\},

$$

where $ \{b_{i}(k)\}, i=1, 2, \{c_1(k)\} \{e_1(k)\}, \{f_1(k)\} $ are all positive $\omega$-periodic sequences, $\omega $ is a fixed positive integer, $\{a_{i}(k)\}$ are $\omega$-periodic sequences, which satisfies $\overline{a}_i=\frac{1}{\omega}\sum\limits_{k=0}^{\omega-1}a_i(k)>0, i=1,2$.The results obtained in this paper generalized the main results of Xiangdong Xie, Zhansshuai Miao, Yalong Xue (Commun. Math. Biol. Neurosci. 2015 (2015), Article ID 2).