This paper is concerned with the l1-gain performance analysis and positive filter design for a positive discrete-time Markov jump linear system (MJLS) by using a linear programming (LP) approach. First, by constructing a linear stochastic Lyapunov function and introducing an “equivalent” deterministic positive discrete-time linear system, necessary and sufficient conditions in the form of LP are derived for stochastic stability and l1-gain performance of the positive discrete-time MJLS. Then a sufficient condition on the existence of the desired positive l1-gain filter is provided. The desired positive l1-gain filter can be designed by solving a standard LP problem. The pest’s structured population dynamic model is employed to illustrate the effectiveness of the proposed method.