An application of the nonlinear system of differential equations in mathematical biology / ecology: to model the predator-prey relationship of a simple eco-system. Suppose in a closed eco-system (i. e. no migration is allowed into or out of the system) there are only 2 types of animals: the predator and the prey. They form a simple food-chain where the predator species hunts the prey species, while the prey grazes vegetation. Let x(t) denotes the population of the prey species, and y(t) denotes the population of the predator species. Then x = a x xy y = c y + xy a, c, , and are positive constants. Note that in the absence of the predators (when y = 0), the prey population would grow exponentially. If the preys are absence (when x = 0), the predator population would decay exponentially to zero due to starvation. One is the origin, and the other is in the first quadrant.