Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

Abstract

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and herbivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilibrium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.

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

“…Considering herbivores in paddy fields can eat some weeds and increase fertilizer by excreting feces, Xiang, Wu, and Zhou proposed a differential equations model of the paddy ecosystem in the fallow season to reveal the effects of herbivores on inorganic fertilizer and weeds. The results show that the introduction of some herbivores into a paddy ecosystem in the fallow season can increase the content of inorganic fertilizer and makes it possible for a Hopf bifurcation phenomenon to emerge in the system [2].…”

“…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].…”

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].…”

“…By considering the effects of herbivores on the paddy ecosystem in fallow season, Xiang, Wu, and Zhou found that the content of inorganic fertilizer is improved by putting some herbivores into the paddy ecosystem in fallow season. 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.…”

“…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.…”