Journal of Computations & Modelling

Limits on the use of Qualitative stability for Non-Linear Systems

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• Abstract

  This paper describes the use of qualitative stability applied to nonlinear systems with reference to ecological models. The principles of qualitative stability are outlined for linearised systems. The requirements necessary for this special condition are given with an example. A modified Jacobian matrix is evaluated using a Hessian expansion for the nonlinear terms following the method of Saleh and Davidsen. Some observations are made about the possibility of achieving Qualitative stability for the nonlinear case. A predator prey example is examined showing that although the linearised system is qualitatively stable the nonlinear case is not necessarily stable away from the equilibrium. The time varying eigenvalues show periods of both instability and stability which converge to stable behaviour as the initial conditions are made closer to the equilibrium values. Analysis displayed here illustrates the possibility of finding a region of the nonlinear system that is qualitatively stable.