  This paper aims to introduce a concept of an equilibrium point of a dynamical
system which will call it almost global asymptotically stable. A biological
prey-predator model is also analyzed with a modification function growth in
prey species. The conditions of the local stable and existence of all its
equilibria are given. After that the model is extended to an optimal control
problem to obtain an optimal harvesting strategy. The discrete time version of
Pontryagin's maximum principle is applied to solve the optimality problem. The
characterization of the optimal harvesting variable and the adjoint variables
are derived. Finally numerical simulations of various set of values of
parameters are provided to confirm the theoretical findings.
