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2. Outbreaks of the spruce budworm have been responsible for some major deforestations in Canada and the United States. The equation x' = r(1 - x/K)x - c * x^2 / (a + x^2) has been used to describe the dynamics of spruce budworm populations, where the variable x denotes the population or density of the insect [16]. One explanation that has been given for the occurrence of outbreaks is based on the multiple bifurcations that occur with this differential equation. a. If a = 0.01, c = 1, and K = 1, we have a family of differential equations parameterized by r, x' = rx(1 - x) - x^2 / (0.01 + x^2). Solve the equation equation rx(1 - x) - x^2 / (0.01 + x^2) = 0 and plot the result in the xr-plane for 0 <= r < 1. b. To find the bifurcation diagram for the spruce budworm equation, reflect the graph obtained in part (1) about the line r = x line. c. Estimate the two bifurcation values from your graph. Explain what happens to the population as r increases. That is, when does an outbreak occur? What happens after an outbreak?

          2. Outbreaks of the spruce budworm have been responsible for some major deforestations in Canada and the United States. The equation
x' = r(1 - x/K)x - c * x^2 / (a + x^2)
has been used to describe the dynamics of spruce budworm populations, where the variable x denotes the population or density of the insect [16]. One explanation that has been given for the occurrence of outbreaks is based on the multiple bifurcations that occur with this differential equation.
a. If a = 0.01, c = 1, and K = 1, we have a family of differential equations parameterized by r,
x' = rx(1 - x) - x^2 / (0.01 + x^2).
Solve the equation equation
rx(1 - x) - x^2 / (0.01 + x^2) = 0
and plot the result in the xr-plane for 0 <= r < 1.
b. To find the bifurcation diagram for the spruce budworm equation, reflect the graph obtained in part (1) about the line r = x line.
c. Estimate the two bifurcation values from your graph. Explain what happens to the population as r increases. That is, when does an outbreak occur? What happens after an outbreak?

2. Outbreaks of the spruce budworm have been responsible for some major deforestations in Canada and the United States. The equation
x' = r(1 - x/K)x - c * x^2 / (a + x^2)
has been used to describe the dynamics of spruce budworm populations, where the variable x denotes the population or density of the insect [16]. One explanation that has been given for the occurrence of outbreaks is based on the multiple bifurcations that occur with this differential equation.
a. If a = 0.01, c = 1, and K = 1, we have a family of differential equations parameterized by r,
x' = rx(1 - x) - x^2 / (0.01 + x^2).
Solve the equation equation
rx(1 - x) - x^2 / (0.01 + x^2) = 0
and plot the result in the xr-plane for 0 <= r < 1.
b. To find the bifurcation diagram for the spruce budworm equation, reflect the graph obtained in part (1) about the line r = x line.
c. Estimate the two bifurcation values from your graph. Explain what happens to the population as r increases. That is, when does an outbreak occur? What happens after an outbreak?

Added by Tina L.

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