Abstract
By introducing a delayed fractional-order differential equation model, we deal with the dynamics of the stability and Hopf bifurcation of a paddy ecosystem with three main components: rice, weeds, and inorganic fertilizer. In the system, there exists an equilibrium for rice and weeds extinction and an equilibrium for rice extinction or weeds extinction. We obtain sufficient conditions for the stability and Hopf bifurcation by analyzing their characteristic equation. Some numerical simulations validate our theoretical results.
Highlights
Rice is one of the major grain crops in the world
We give a detailed stability analysis of the system equilibria and study the existence of Hopf bifurcation by using the Hopf bifurcation conditions proposed by Xiao et al [26]
To discuss the stability of the other two equilibria, we introduce the polynomial of degree 4 with real coefficients a = (1, a1, a2, a3, a4)
Summary
Rice is one of the major grain crops in the world. China is the largest rice producer and consumer country in the world, where over 60% of the population is staple food for rice. A differential equation model of a paddy ecosystem in fallow season was proposed by Xiang et al [14] They revealed the interaction between weeds and inorganic fertilizer and found that in the system, there exists a stable node, an unstable saddle point, or a saddlenode point. 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. Some researchers have concerned about the existence of Hopf bifurcation of fractional-order models [21,22,23,24,25,26,27]. We establish a factional-order differential equation model with delay for the interaction among the main components of a paddy ecosystem. We give a detailed stability analysis of the system equilibria and study the existence of Hopf bifurcation by using the Hopf bifurcation conditions proposed by Xiao et al [26]
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