Stability of a Paddy Ecosystem with Time Delay

Abstract

Abstract-A delay differential equation model of paddy ecosystem was put forward to reveal the interaction among rice, weeds and inorganic fertilizer on the system. The results show that, the system exists a rice and weed extinction equilibrium, and it also exists a rice extinction or weed extinction equilibrium. Their stable and unstable conditions are obtained. Moreover, Hopf bifurcations occur at the rice extinction or weed extinction equilibrium as the delay crosses some critical values. According to the conditions, some measures to increase rice yield were recommended.

Hopf bifurcation controlling for a fractional order delayed paddy ecosystem in the fallow season

“…As everyone knows, there are a lot of interacting components in a paddy ecosystem, such as weeds, inorganic salts, light intensity, herbivore, especially human activities, which make it become a complex nonlinear system. Recently, some mathematical models of paddy ecosystem have been proposed and some valuable results have been achieved [1][2][3][4][5].…”

“…In addition to studying the stability of equilibria for fractional order systems, many scholars have also discussed the existence of Hopf bifurcations [13][14][15][16][17]. By extending the integer order paddy ecosystem discussed in [3] to fractional order case, Zhou et al established a fractional order paddy ecosystem with time delay [4]. They obtained the sufficient conditions of the stability of equilibria and the existence of Hopf bifurcation, which are the generalization of the corresponding conditions of integer order systems in [3].…”

Stability and Hopf bifurcation analysis in a fractional-order delayed paddy ecosystem

“…It is a common research method to analyze the interaction of all factors in a population system by using mathematical models [1][2][3][4][5][6][7][8][9][10][11]. As far as we know, there are only a few mathematical models that have been established for paddy ecosystems [12][13][14].…”

“…They also found that the system can exhibit Hopf bifurcation phenomenon and gave the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter [13]. Wang et al [12] further studied the interaction of rice, weeds, and inorganic fertilizer in a paddy ecosystem. They discussed the existence and stability of equilibria in a paddy ecosystem.…”

“…The three models mentioned have been restricted to integer-order (delay) differential equations [12][13][14]. In recent more than 20 years, the research of fractional-order differential equations has been the concern of many scholars.…”