Question

Problem 1 In simple predator-prey models, sinusoidal functions can be used to model the oscillating populations of two species of animals in the same environment. As the population of the predator species increases, the population of the prey species will decrease. If the number of prey gets too low, the population of the predator species will suffer from limited resources and start to decline. In this problem we will be modeling one population of rabbits (the prey) and one population of foxes (the predator). In week 30, the number of rabbits was observed to be at its maximum of 3500 rabbits. The minimum number of rabbits observed was 500 rabbits at week 90. Let y = R(t) give the number of rabbits in week t of an environmental study. (a) (7 pts) What is the period, amplitude and midline of R(t)? (b) (3 pts) Which of the following general formulas could we use as our model for R(t)? R(t) = A · sin(Bt) + k or R(t) = A · cos(Bt) + k Explain your reasoning. (c) (5 pts) Use your answers from parts (a) and (b) to determine the values of A, B, and k and to write out a complete formula for R(t). (d) (5 pts) The fox population can be modeled by the function F (t) = −80 cos ( π (t − 10))+ 60 120. What can you say about the fox population just by looking at the formula? Be as detailed as possible. (e) (5 pts) Graph both populations, R(t) and F(t), from week 0 to week 450. Describe how each population is changing at week 70 and at week 130.

Answer 1

& At t= 30 weeks , y = 3500 A umanina & ut to go week, y = 500 minina a coso initially attacks manina & then mihina since general formula - R(t)= Acos Bt + K as Amplitude 35 at manima, r(t) = 3500 & sti cos Bt = 1 - 3500 - A+K 0 lat minina, R(t) = 500 acast 2 500=-Atk 6 2 A= 1500 K= 2000 period of cos o= 20 & here difference sho na Also & min = 6oweeks nB = 218 = 1 R(t) = 150000 s/11t) + 2000 / F(t)=-80 cos (Alt-8100) + 60120 For population varies periodically

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