THE STABILITY OF TWO AREA-PRESERVING BIOLOGICAL MODELS

This paper will focus on two area-preserving biological models: a host-parasite model x n +1 = ax n f ( y n ) , y n +1 = bx n y n f ( y n ) , where f is a smooth, monotonically increasing positive function, and a maternalquality single species population model x n +1 = x n g ( y n +1 ) , y n +1 = y n f ( x n ) with f being a smooth, monotonically increasing positive function and g a smooth, monotonically decreasing positive function. By computing the first twist coefficient, we will find sufficient conditions for non-resonant elliptic fixed points to be non-degeneracy, and thus utilize KAM theory to establish the stability of these fixed points.