Question

4. (15 points) Consider the predator prey model given by $x_1' = alpha x_1 - eta x_1 x_2$ $x_2' = delta x_1 x_2 - gamma x_2$ Note that this is a system of non-linear ODEs, and so we do not have the tools to solve it. However, we do have the skills to study some of its behavior. In this somewhat basic model, we assume prey has unlimited resources, represented by $alpha x_1$, and that predator eats prey at a rate proportional to the number of encounters between predator and prey, represented by $eta x_1 x_2$. $delta x_1 x_2$ represents the growth of the predator population, while $gamma x_2$ represents the loss of predator population to emigration from the habitat or death. (a) Put the system into matrix equation form. (Hint: for this equation, you will have functions $x_1, x_2$ in your matrix.) (b) Find the equilibrium solutions to the system. Interpret what the results of each equilibrium means in terms of the actual population dynamics at play. (c) The attached figure represents a set of solution curves to the predator prey model described. Explain why each solution is a closed curve, not a line in the plane as we've seen with linear equations. (Hint: your explanation should be in terms of what the variables represent.)

          4. (15 points) Consider the predator prey model given by
$x_1' = alpha x_1 - eta x_1 x_2$
$x_2' = delta x_1 x_2 - gamma x_2$
Note that this is a system of non-linear ODEs, and so we do not have the tools to solve it. However,
we do have the skills to study some of its behavior. In this somewhat basic model, we assume prey
has unlimited resources, represented by $alpha x_1$, and that predator eats prey at a rate proportional to the
number of encounters between predator and prey, represented by $eta x_1 x_2$. $delta x_1 x_2$ represents the growth
of the predator population, while $gamma x_2$ represents the loss of predator population to emigration from the
habitat or death.
(a) Put the system into matrix equation form. (Hint: for this equation, you will have functions $x_1, x_2$
in your matrix.)
(b) Find the equilibrium solutions to the system. Interpret what the results of each equilibrium means
in terms of the actual population dynamics at play.
(c) The attached figure represents a set of solution curves to the predator prey model described.
Explain why each solution is a closed curve, not a line in the plane as we've seen with linear
equations. (Hint: your explanation should be in terms of what the variables represent.)

4. (15 points) Consider the predator prey model given by
x1' = alpha x1 - eta x1 x2
x2' = delta x1 x2 - gamma x2
Note that this is a system of non-linear ODEs, and so we do not have the tools to solve it. However,
we do have the skills to study some of its behavior. In this somewhat basic model, we assume prey
has unlimited resources, represented by alpha x1, and that predator eats prey at a rate proportional to the
number of encounters between predator and prey, represented by eta x1 x2. delta x1 x2 represents the growth
of the predator population, while gamma x2 represents the loss of predator population to emigration from the
habitat or death.
(a) Put the system into matrix equation form. (Hint: for this equation, you will have functions x1, x2
in your matrix.)
(b) Find the equilibrium solutions to the system. Interpret what the results of each equilibrium means
in terms of the actual population dynamics at play.
(c) The attached figure represents a set of solution curves to the predator prey model described.
Explain why each solution is a closed curve, not a line in the plane as we've seen with linear
equations. (Hint: your explanation should be in terms of what the variables represent.)

Added by Michelle R.

Question

Please give Ace some feedback