Symbiosis is the interaction of two species whose survival depends upon their mutual cooperation. As an example we might take one species to be the common honey bee and the other to be the clover plant; the bee feeds on the nectar of the plant and simultaneously pollinates the plant. In this project, we will use difference equations to study the change in their populations over time. We begin our observations by counting the populations of these two species in the Aton prairie preserve in 1990. Let X, and Yn denote the populations of the bees and the clover, respectively, after n years. Problem 1. Assume that the bees have no other source of food besides the clover and that the clover have no other source of pollination besides the bee. Explain wliy a reasonable model for the changes in population from year to year is AX, = -aXn+bX, Yn AY = -cYn + DX,Y, for some positive constants a, b, c, d. You should be able to describe in words what the constants a, b, c, d represent. Problem 2. For this particular prairie field, the constants a = 0.2, b = 0.001, c = 0.3.d = 0.002 have been estimated from observations of a few representative plants and bees. It was observed in 1990 that Xo = 200 and Yo = 300. Use the model to estimate the populations which should have been attained in 1998. Estimate the populations in 2008. Symbiosis is the interaction of two species whose survival depends upon their mutual cooperation. As an example we might take one species to be the common honey bee and the other to be the clover plant; the bee feeds on the nectar of the plant and simultaneously pollinates the plant. In this project, we will use difference equations to study the change in their populations over time. We begin our observations by counting the populations of these two species in the Aton prairie preserve in 1990. Let X, and Yn denote the populations of the bees and the clover, respectively, after n years. Problem 1. Assume that the bees have no other source of food besides the clover and that the clover have no other source of pollination besides the bee. Explain wliy a reasonable model for the changes in population from year to year is AX, = -aXn+bX, Yn AY = -cYn + DX,Y, for some positive constants a, b, c, d. You should be able to describe in words what the constants a, b, c, d represent. Problem 2. For this particular prairie field, the constants a = 0.2, b = 0.001, c = 0.3.d = 0.002 have been estimated from observations of a few representative plants and bees. It was observed in 1990 that Xo = 200 and Yo = 300. Use the model to estimate the populations which should have been attained in 1998. Estimate the populations in 2008