Math 142 Test3 S22
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Test 1
= x(9 − x − 3g) dt(dy) = g(−6 + 2x)
(a) (5 oints) what is the relationship between x and g?How does x grow in the absence of g?How does g grow in the absence of x?
(b) (5 points) sketch the nullclines and direction arrows of the system.
(c) (4 points) Find the eigenvalues of the interior critical point.
(d) (7 points) sketch the general solution. Be detailed ·
(e) (4 points) sketch x(t) and g(t) if x(0) = 1 g(0) = 5 ·
Test 2
2 . (8 points) Determine whether the polynomial has any roots with positive real parts.
λ4 + 4λ3 + 5λ2 + 6λ + 2 = 0.
3. suppose ①’g’2 satisfy the competing species equations
= ①(6 - 2① - 3g - 2)
= g(7 - 2① - 3g - 22)
dt(dz) = 2(5 - 2① - g - 22)
(a) ( points) Find the critical point (0’g*’2*) where g*’2* 米 0 and sketch the nullclines and direction arrows in the g2-plane.
(b) (4 points) Give a detailed sketch of the general solution of the system in the g2-plane.
(c) (6 points) Determine if the critical point(0, y*, z*) is stable.
(d) points) Determine if there is a stable critical point of the form (x*, 0, 0), where x* > 0
Test 3
= − u3 + 3u2 − 2u.
(a) (4 points) Find a phase-plane system for traveling wave solutions
u(x, t) = U (x − ct) ·
(b)5 points) sketch the nullclines and direction arrows of the phase-plane system.
(c) (5 points) Find all critical points and the Jacobian/eigenvalues at each critical point.
(d)4 points) Find a value cmin such that the system has no spiral sinks for c ≥ cmin
(e) (5 points) sketch the phase-plane solution in the UV-plane for c < cmin
(f) (5 points) sketch the phase-plane solution in the UV-plane for c > cmin ·
(g) (6 points) sketch all possible traveling wave solutions for c > cmin ·
(h) (4 points) Explain the meaning of each of your traveling wave solutions above, in terms of what happens to the population u(x, t) over time.
(i) (4 points) Given your answers on the previous page, what type of population or ecological situation could this model? (Hint: it comes from a type of population model that was on a previous homework. )
(j) (3 points) Are there any physically relevant traveling waves for c < cmin? Explain your reasoning for why or why not.
2023-07-28