Stability analysis of predator-prey model on the case of aerosol-cloud-precipitation interactions

Abstract

A preliminary study has been performed on the analysis of the stability of predator-prey models in the case of aerosol-cloud-precipitation interactions which initiated by Koren-Feingold. The model consists of two coupled non-linear differential equations describing the development of a population of cloud drop concentration and cloud depth for precipitation. Stability analysis of the models was conducted to understand the stability behavior of systems interactions. In this paper, the analysis focused on the model without delay. The first step was done by determining the equilibrium point of the model equations which yielded 1 non-trivial equilibrium point and 4 trivial equilibrium point. Nontrivial equilibrium point (0,0) associated with the steady state or the absence of precipitation while the non-trivial equilibrium point shows the oscillation behavior in the formation of precipitation. The next step is linearizing the equation around the equilibrium point and calculating of eigenvalues of Jacobian matrix. Evaluation of the eigen values of characteristic equation determined the type of stability. There are saddle node, star point, unstable node, stable node and center. The results of numerical computations was simulated in the form of phase portrait to support the theoretical calculation. Phase portraits show the characteristic of populations growth of cloud depth and drop cloud. In the next research, this analysis will compared to delay model to determine the effect of time delay on the equilibrium point of the system.