Analysis of the interaction among rice, weeds, inorganic fertilizer, and a herbivore in a composite farming paddy ecosystem.

As one of the Globally Important Agricultural Heritage Systems (GIAHS), rice field composite farming is an ecological measure in rice production, which can reduce the amount of chemical fertilizers, pesticides and herbicides. This research studies the interaction among rice, weed, inorganic fertilizer and herbivore in a composite farming paddy ecosystem. We develop a differential equation model to analyze the relations and interactions among those components. Results show the existence of an equilibrium for paddy and weed extinction, one or two equilibria for rice extinction, an equilibrium for weed extinction, and an equilibrium for rice and weed coexistence. Based on the obtained stability conditions of these equilibria, measures are proposed to avoid the existence or the stability of equilibria for rice extinction. Other measures are proposed to lead to a stable equilibrium for weed extinction, which is the most desirable result in rice production. Conditions for maximizing the yield of rice are also obtained by taking the relative mortality of rice as variable. In addition, we discover the existence of Hopf bifurcation phenomenon in the system, and develop the critical value of Hopf bifurcation by taking the artificial fertilizer rate as the bifurcation parameter. Our findings provide effective guidance and insights for rice production in a composite farming paddy ecosystem.