Week Ten: Linear Stability Analysis

Question 1

What determines stability at an equilibrium in dynamical systems?

Answer 1

The sign of the Jacobian eigenvalues (or derivative in 1D). Negative → stable, positive → unstable, zero/imaginary → marginal.

Question 2

What does a negative eigenvalue mean?

Answer 2

Perturbations decay back to equilibrium → stable.

Question 3

What does a positive eigenvalue mean?

Answer 3

Perturbations grow away from equilibrium → unstable.

Question 4

What is the stability of the equilibrium when both populations are zero?

Answer 4

Always unstable — any population introduced grows.

Question 5

What does invasion analysis test in Lotka–Volterra competition?

Answer 5

Whether a species at zero can increase when introduced. If yes → equilibrium unstable; if no → stable.

Question 6

When is the single‑species equilibrium stable against invasion?

Answer 6

I- If the competitor’s coefficient alpha >1, invasion fails → stable. If alpha <1, invasion succeeds → unstable.
-alpha <1: competitor is weak, so it can grow when rare → invasion succeeds.
alpha >1: competitor is strong, but the resident suppresses it enough that it cannot grow → invasion fails.

Question 7

How do we approximate dynamics near equilibrium using Taylor expansion?

Answer 7

• Expand f(n) around n^. Since f(n^)=0, leading term is f’(n^)(n-n^).

Question 8

Why can we ignore higher‑order terms near equilibrium?

Answer 8

• Because Delta n is small, so (Delta n)^2,(Delta n)^3 are negligible close to equilibrium.

Question 9

What does the sign of f’(n^*) tell us?

Answer 9

Negative slope → stable equilibrium. Positive slope → unstable equilibrium.

Question 10

Stability of equilibrium at N=0 in logistic growth?

Answer 10

Unstable — derivative positive, population grows if introduced.

Question 11

Stability of equilibrium at N=K in logistic growth?

Answer 11

Stable — derivative negative, population returns to carrying capacity

Question 12

How do we extend stability analysis from one species to multiple species?

Answer 12

Use the Jacobian matrix of partial derivatives. Stability depends on the signs of its eigenvalues, not just the diagonal entries.

Question 13

What does the Jacobian represent in multi‑species systems?

Answer 13

It shows how each population’s growth rate changes with respect to its own density and the other species’ density at equilibrium.

Question 14

What do eigenvectors of the Jacobian represent?

Answer 14

The directions in phase space along which perturbations evolve. Perturbations align with eigenvectors.

Question 15

What do eigenvalues of the Jacobian tell us?

Answer 15

The rate of change along each eigenvector. Negative real parts → stable; positive real parts → unstable; imaginary parts → oscillations.

Question 16

How are eigenvalues found for a 2×2 Jacobian?

Answer 16

Solve the characteristic polynomial:
lambda ^2-(a+d)lambda +(ad-bc)=0

Question 17

What is the trace–determinant test for 2×2 systems?

Answer 17

Trace = a+d
Determinant = ad-bc
D>0, T<0 = stable
D>0, T>0 = unstable
D<0 = unstable
D>0, T=0 = stable

Question 18

What happens if the eigenvalues are imaginary values?

Answer 18

No decay or growth and syability is marginal