Take Home Assignment #1-Due 9/8/17 Learning Targets: cM If you need help startin
ID: 2895309 • Letter: T
Question
Take Home Assignment #1-Due 9/8/17 Learning Targets: cM If you need help starting this problem, you are encouraged to seek help from Dr. Dunmyre or from the learning assistants. If you work with another student, you must indicate who you worked with and what the results of the collaboration were. In any case, the work you submit must be entirely your own. 1. Consider the following systems of rate of change equations: System A System B dt db = 301 1 dt + 25ry 10 dt In both of these systems, a, b, r and y refer to the number different species at time t. In particular, in one of these systems the prey are large animals and the predators are small animals, such as piranhas and humans. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey, such as whales and krill. For each system, determine the following: i) Which variable corresponds to the prey, and which variable corresponds to the predator? Fully justify your answer. It is not enough to claim one variable is predator/prey and that the other is prey/predator by process of elimination ii) Does the system have large prey or small prey? Large predators or snall predators? Explain the reasoning behind you ur decision. Again, process of elimination is not a valid explanation for this problem.Explanation / Answer
(i) If the 2 species are represented by 'a' and 'b', the prey has a -ve 'ab' term (the interaction term), because interactions between prey and predator ('ab') will reduce the rate of growth of prey. Predator has a +ve 'ab' term, because interactions between prey and predator ('ab') will increase the rate of growth of predator.
In system A: da/dt contains a +ve 'ab' term, hence 'a' is predator. db/dt contains a -ve 'ab' term, hence 'b' is prey.
In system B: dx/dt contains a -ve 'xy' term, hence 'x' is prey. dy/dt contains a +ve 'xy' term, hence 'y' is predator.
(ii) Again, the interaction term gives a hint towards the size of prey and predator. If the 'ab' term in the da/dt equation has a small co-efficient, it means that the 'a' population is not much affected (+vely or -vely, depending on the sign of the 'ab' term) by 'b', hence 'a' must be large in size compared to 'b'. If the 'ab' term in the da/dt equation has a large co-efficient, then it means that the 'a' population in affected severely (+vely or -vely, depending on the sign of the 'ab' term) by the 'b' population.
In system A, 'a' is predator and the co-efficient of the 'ab' term in the equation da/dt is 1/20 (it is very small), so it must be a large predator. The equation db/dt has a large 'ab' term, so it is affected significantly by 'a', so it must be a small prey.
In system B, 'x' is prey and the co-efficient of the 'xy' term in the equation dx/dt is -1/100 (it is very small), so it must be a large prey. The equation dy/dt has a large 'xy' term, so it is affected significantly by 'x', so it must be a small predator.
Notes: In system A, da/dt has a -ve 'a' co-efficient, so the predator decays exponentially in the absence of prey. db/dt, however, has the logistic term 3b(1 - b/10), meaning that the prey attains equilibrium in the absence of predator. In system B, dx/dt has a +ve 'x' co-efficient, so the prey grows exponentially in the absence of predator, and does not attain a logistic equilibrium. dy/dt, however, has the logistic term 15y(1 - y/17), meaning that the predator attains equilibrium in the absence of prey.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.