Many products sold in the US are manufactured overseas. Glow-in-the-dark back scratchers are imported by an international importer, who stores them in their warehouse in Los Angeles and distributes them further to the one retail store that actually sells these things, Mel's Magic Emporium in Chicago. The following assumptions are true of the number of back scratchers in the importer's warehouse (W ) and the number of back scratchers in stock at Mel's store (S). All rates are per-week rates.

The number of items the importer orders from the overseas manufacturer is a steep decreasing sigmoid function of the number of back scratchers in its warehouse. The rate is half of the maximum when there are 1000 back scratchers in the importer's warehouse.

The product is transported by cargo ships, so from the moment the order is placed, it takes 4 weeks for each shipment to get from the overseas manufacturer to the importer's warehouse in Los Angeles.

Because Mel cuts prices when he has too many back scratchers in stock, the rate at which each customer buys them from the store is a non-sigmoid saturating function of the number in stock, with a maximum of 3 per customer. (These things make great gifts!)

The number of customers who visit Mel's store each week does not depend on the number of available back scratchers.

Also, when Mel has a lot of back scratchers in stock, he will order fewer of them. The number of items Mel orders from the importer is proportional to the inverse of the number of back scratchers he has in stock, with a proportionality constant of 50. (Remember: the inverse of X is 1/X.) Mel's store is never completely out of back scratchers.

When Mel calls the importer and places his order, they ship out the back scratchers from their warehouse immediately. However, it takes about three and a half days for the shipment to get from the Los Angeles warehouse to Mel's store in Chicago.

By using the assumptions above, write a differential equation model of the number of back scratchers in the importer's warehouse (W ) and the number of back scratchers in stock at Mel's store (S). You should introduce parameters where necessary