Effect of time delay on dynamic of plant competition under allelopathy

In this paper, a mathematical model is applied to investigate the effect of density of allelochemicals on plant population involving delay. The model is constructed using a system of nonlinear delay differential equations. In the model, there are three state variables: plant populations (P-1 & P-2) and density of allelochemical (T). It is assumed that in the presence of density of allelochemicals, plant populations are affected. It further delays the conversion of resource into the density of allelochemicals and hence affects the plant population adversely. This effect is observed by using delay in the competition model. Equilibrium points are calculated. Using the Routh-Hurwitz theorem, certain explicit formulations defining the stability and path of the Hopf-bifurcation periodic solutions at the positive equilibrium are obtained. Further, the direction of these bifurcating periodic solutions is also studied. Numerical simulation and graphical support are provided for analytical findings using MATLAB.

Automated summary

This paper applies a mathematical model to investigate the effect of delay in allelochemical density on plant population. The constructed nonlinear delay differential equation system includes three state variables: plant populations (P-1 & P-2) and allelochemical density (T). It is assumed that the presence of allelochemical density affects plant populations, by delaying the conversion of resources and adversely affecting plant population. The study calculates equilibrium points and uses the Routh-Hurwitz theorem to obtain explicit formulations defining stability and the path of Hopf-bifurcation periodic solutions at positive equilibrium. Numerical simulation and graphical support using MATLAB are provided to validate the analytical findings.