Please help me check my works, thank you.
a) The model described by equation (5) 4 (6) is Lotka- Vetter, model and it is not very realistic. It does not consider any competition among frey predators. As a result population may grow infinitely without any resource limits. the model has no asympotatie stability b ) 10 = 10a -100 b (1 ) 10 = -10d + 100c ( 2 ) -8 = - 10d +10c - (3 ) 100 = 109 - 10b - 14) Solving (1) & (4 ) 10 = 10a - 100b 100 = 10a- 10b - 90 = - gob =) b = 1 = ) a = 11 Solving ( 2) & (3 ) 10 = - 101 + 100 c - 8 - 10d + 10c () 18 = go c 7 1= 18 90 1= 1/S d = 1 Hence, a = 11, b = 1, (= 1/5 , d = 1(4 $8 me had 10oo predators and no hey then the predator population would see a exponential decay. The term dy represents the loss of predators. ( d) Population equilibrium occurs in the model when neither of the population level is changing, ce. both of the derivatives are equal to zero. x ( 9 - by ) = 0 - 4 ( d - CX ) = 0 Above system of equation yields two solutions: Ey=0 , x = 03 and { y = 2 , x = d? The first solution represents the extinction of two species. In second solution, the levels of population at which this equilibrium is achieved depend on the chosen values of parameters a, b, Lid.\f4 Predator-Prey Model Ellen is a keen ecologist studying life on a planet far, far away. She found two species on the planet, one predator and the other prey. You found her journal back on Earth and are trying to decipher life on this other planet. In Ellen's journal, she states the following observations about the predator and prey purula'r'ms o The prey population grows at a rate proportional to the current prey population. a For a given prey population, the rate of increase of prey population decreases linearly with the predator population. As the predator population inu'eases, it is harder for the prey population to grow. a The predator population also grows at a rate proportional to the current predator population. 0 For a xed predator population, predator population increases linearly with prey population. As the amount of prey increases, predators have more food and their population increases faster. Let's call these species J: (prey) and y (predator) to keep things simple. A friend suggests using the following model for predator and prey populations: gait) = {a by)x (5) it rst} = (exdry (6) a} Conunent on the validity of the model described in Equations 5 and 6. b) You nd a data entry in Ellen's journal measuring the growth rates of predator and prey populations. At a predator population of 10 and prey population of 10, both populations increase at 1W5. With a single prey remaining, the predators starve and their population decreases at Sis ata population of 10 predators. At population levels of 1 predator and 10 prey, the prey population rapidly increases at 100fs. Using these observations, nd the model parameters a, b, c and d. c} If we had 1000 predators on the planet and no prey. What will happen to the predator population? d} At what levels will the predator and prey populations not change any more. As we have seen in class, this set of population levels is called an equilibrium e) We want to analyze the model at a set of population levels. We don't know how to solve the model described in Equations 5 and 6. We can, however, linearize the model around a set of population levels and analyze the population changes in that vicinity. If the predator population is 10 and the prey population is 1, linearize the system around these populations and express the new, linearized system in the form are m
