1. (10 points) The Devil's Hole pupfish (Cyprinodon diabolis) is a fish that lives in a flooded cavern in Nevada. It is critically endangered. An important part of the pupfish's diet consists of an alga, Spirogyra. You are working on pupfish conservation efforts. Write a model for the mature pupfish (M), juvenile pupfish (J), and carp (C) populations based on the following descriptions. Note that time is measured in months. Spirogyra growth is limited by space, nutrients and sunlight. We estimate that up to 25,000 Spirogyra can be supported by the cavern ecosystem. Spirogyra have a per capita growth rate of 0.75. . When a mature pupfish encounters a strand of Spirogyra, the likelihood that the pupfish eats the Spirogyra is 0.25. . The rate at which an individual juvenile pupfish eats Spirogyra is proportional to the Spirogyra population with proportionality constant 0.3 . Mature Devil's hole pupfish live an average of 1 year. We may assume that the natural per capita death rate for mature pupfish is 1/12. . Each month, the number of eggs laid by a single mature pupfish is proportional to the amount of Spirogyra it consumes, with a proportionality constant of 0.02. Since the eggs hatch within a few days, we can assume that all the eggs immediately become juveniles. While pupfish eggs can hatch within a few days, few larvae survive to adulthood. The annual per capita death rate for juveniles is estimated to be 0.9. . As part of the conservation effort, we introduce 3 healthy pupfish larvae into the cavern every month. . It takes pupfish larvae roughly 2 months to mature. We may assume that 5% of the pupfish larvae become mature pupfish each month. M' = J' =