Math 375T Uhlenbeck Assignment 4 Due April 6, 2005 1. Assume the logistics model is a realistic model for a fishery, and that fishing has been banned so that the population has come close to the carrying capacity of the model. It has been decided to allow fishing, and to allow a certain constant number of fish to be taken from the lake each day (or week or however you want to measure the time). Explain how large this number should be to permit a stable situation to evolve. Explain your number in relation to the growth rate of a small population and the carrying capacity of the model. If the population of the fish is allowed to reach almost zero, and this same amount of fishing is allowed without allowing the population to increase in size first, what happens? Could you explain this to a fisherman? 2. Page 257 #2. 3. Page 257 #3 4. Page 265 #31 5. A population of small animals is assumed to more or less follow the logistics model. A population of large, fierce animals is assumed to die out (at a rate proportional to their number) in the absence of the small animals. The small animals are caught and eaten at a rate proportional to the encounters between the two populations, which is assumed to be jointly proportional to the numbers of each. The fierce animals thrive in proportion to the number of small animals eaten, but they need a "large" number of small animals per head to reproduce. Write out the equations, reduce the number of parameters to as few as possible, and explain how the numbers of small animals each large animal needs to eat to reproduce affects the outcome. Do this by finding the fixed points and their stability. Explain your answer in terms of the original parameters.