Question

In: Advanced Math

In this project we explore how two populations develop when one preys on the other. Clearly...

In this project we explore how two populations develop when one preys on the other. Clearly if there are no predators, the prey population will keep growing, whereas if there are no prey, the predators will go extinct. Suppose x and y denote the populations of the prey and predators respectively.

If y = 0, we will assume that

dx/dt = ax, a > 0.

If y does not equal 0, it is natural to assume that the number of encounters between predators and prey is jointly proportional to x and y. If we further assume a proportion of these encounters leads to the prey being eaten, we have

dx/dt = dx − bxy, a, b > 0.

Similarly, we have

dy/dt = −cy + dxy, c, d > 0.

This system of equations is called Volterra’s predator-prey equations.

Part a) Solve this system of equations to find solutions in the form g(y) = f(x). You will see that we cannot explicitly find y in terms of x, so our solutions are implicit. We can still, however, study these solutions.

Part b) Suppose g(y) = C1, where C1 is a constant. Determine how many solutions there are to this equation by using calculus techniques. Note this may well depend on the value of C1. Do the same thing for f(x) = C2, where C2 is a constant.

Part c) Hence determine the shape of the trajectories in the x, y-plane (do a sketch!), and their directions.

Part d) Clearly the system has a rest point at x = c/d and y = a/b. By making the substitutions x = c/d + X and y = a/b + Y , assuming X and Y are small enough that we can neglect any second order terms in X and Y , show that near the rest point, trajectories are approximately ellipses.

Part e) Finally, sketch graphs of x(t) and y(t) against t on the same axes. To help, show that d^2 * y / dt^2 > 0 whenever dx/dt > 0 are think about what this means in terms of the shapes of the graphs.

Solutions

Expert Solution


Related Solutions

How do we calculate the relative velocities of two bicycles when one is still? When they...
How do we calculate the relative velocities of two bicycles when one is still? When they are traveling in the same direction? When they are traveling in opposite directions? You might find it easier to answer if you randomly assign some numbers to each.
a) How can we decide if we need to use a one-tailed or two-tailed test when...
a) How can we decide if we need to use a one-tailed or two-tailed test when conducting our analyses? b) What are the differences between the two? c) Why do we need to make the choice between a one-tailed or a two-tailed test? Discuss types of research where using the t statistic may be an appropriate alternative to using a z-score
2. Flies inoving two populations are isolated from other populations of flies. Population S lives on...
2. Flies inoving two populations are isolated from other populations of flies. Population S lives on a very small island and has 103 individual population size. Population L lives on a continent and has 108 individual population size. Assuming that the island population was found by a small group of organisms that drifted on a log from the continent to the island. No interchange of alleles were found between populations since that time. a. Would heterozygosity be expected for most...
the project will cost $45,000 to develop. When the system becomes operational, after a one-year development...
the project will cost $45,000 to develop. When the system becomes operational, after a one-year development period, operational costs will be $9,000 during each year of the system's five year useful life. The system will produce benefits of $30,000 in the first year of operation, and this figure will increase by a compound 10% each year. Answer the following questions based on the background information provided and what you have learned from the toolkit: What is the NPV for this...
How can we improve the results of the project when we fail to reject the null...
How can we improve the results of the project when we fail to reject the null hypothesis?
Programming Project #1 – Day Planner In this project we will develop classes to implement a...
Programming Project #1 – Day Planner In this project we will develop classes to implement a Day Planner program. Be sure to develop the code in a step by step manner, finish phase 1 before moving on to phase 2. Phase 1 The class Appointment is essentially a record; an object built from this class will represent an Appointment in a Day Planner . It will contain 5 fields: month (3 character String), day (int), hour (int), minute (int), and...
Do we truly understand how people decide to identify themselves? How we identify other people when...
Do we truly understand how people decide to identify themselves? How we identify other people when it comes to gender roles. Do we even understand the difference between gender and sexuality?
If we assume that there are two economies that are trading parties. When one of them...
If we assume that there are two economies that are trading parties. When one of them goes through an economic recession show graphically how this business cycle is transmitted to the other country.
We are working on a project. Two companies have bid on the project and we are...
We are working on a project. Two companies have bid on the project and we are choosing from two machines. This is a mutually exclusive project, we must pick one machine or the other but cannot pick both. Machine A has an initial cost of $19,500 and a salvage value of $7500 (today's value) at the end of its 12 year life. Machine B has an initial cost of $17,900 and a salvage value of $2300 (today's value) at the...
why and how could we clearly establish POP and POD?
why and how could we clearly establish POP and POD?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT