Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model

Abstract

Sufficient conditions are obtained for the existence of positive periodic solution of the following discrete Lotka-Volterra commensal symbiosis model

$$\begin{array}{rcl}

x_1(k+1)&=& x_1(k)\exp\big\{ a_1(k)-b_1(k)x_1(k)+c_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\},

\end{array}

$$

where $ \{b_{i}(k)\}, i=1, 2, \{c_1(k)\} $ are all positive $\omega$-periodic sequences, $\omega $ is a fixed positive integer, $\{a_{i}(k)\}$ are $\omega$-periodic sequences, which satisfies $\overline{a}_i=\frac{1}{\omega}\sum\limits_{k=0}^{\omega-1}a_i(k)>0, i=1,2$.