ABSTRACT
Stability in a nonlinear population model is often established by examining the eigenvalues of its linearized dynamics. This method gives only stability relative to infinitesimal perturbations of the initial state. But populations in the real world are subjected to large perturbations. It is therefore essential that a population model should be stable relative to finite perturbations of its initial state. The direct method of Liapunov is a powerful analytical method for establishing stability relative to finite perturbations. This method requires the construction of a continuous and positive definite function which will be denoted by V(N). The time derivative of V(N) along every solution of the system in a finite region is nonpositive. The chapter examines how the Poincare transformation can be applied to a class of nonlinear population models. The Poincare transformation is applied repeatedly until every component in the transformed model is self-regulating or is neutrally stable on its own.
