All

Question 1

What is a fixed point?

Answer 1

A population state where all time derivatives are zero.

Question 2

What is linear stability?

Answer 2

Behaviour of small perturbations around an equilibrium.

Question 3

What is the Jacobian matrix?

Answer 3

Matrix of partial derivatives of the system.

Question 4

What is non-dimensionalisation?

Answer 4

Rescaling variables to remove units.

Question 5

What is logistic growth? and equation

Answer 5

Growth limited by carrying capacity. rc(1-c/k)

Question 6

What is the law of mass action?

Answer 6

A chemical reaction proceeds at a rate proportional to the concentration of the participating reactants.

Question 7

What is a delay differential equation?

Answer 7

An equation involving past states.

Question 8

What is a characteristic equation?

Answer 8

Equation determining perturbation growth rates.

Question 9

What is a Hopf bifurcation?

Answer 9

Transition to oscillations.

Question 10

What is a Turing instability?

Answer 10

Diffusion-driven pattern formation.

Question 11

What is a nullcline?

Answer 11

Set where a derivative is zero.

Question 12

What is carrying capacity?

Answer 12

Maximum sustainable population.

Question 13

What is flux?

Answer 13

Movement across space.

Question 14

What is spatially uniform state?

Answer 14

Constant in space.

Question 15

What do we need for DDI?

Answer 15

U* linearly stable for spatially homogeneous perturbations. k = 0
system is unstable to spatially varying perturbations

Question 16

DDI stable to spatially homogeneous perturbations equations

Answer 16

f_u + g_v < 0
f_ug_v -f_vg_u > 0

Question 17

DDI unstable to spatially varying perturbations equations

Answer 17

D_u g_v + D_v f_u > 0
4D_uD_v (f_ug_v -f_vg_u) < (D_u g_v + D_v f_u)^2

Question 18

What is critical delay?

Answer 18

Delay threshold for instability. make imaginary mu = 0

Question 19

What is phase plane?

Answer 19

Graphical representation of dynamics.

Question 20

What is feasibility?

Answer 20

Non-negative populations.

Question 21

What is saturation?

Answer 21

Response limited at high density.

Question 22

What is stationary wave?

Answer 22

Wave with zero speed.

Question 23

What is inhibition?

Answer 23

Growth suppression by interactions.

Question 24

“Provide biological explanations for the terms”

Answer 24

Go term-by-term: sign → process → power → saturation or interaction.

Question 25

“Using the Law of Mass Transport”

Answer 25

Write conservation law → identify fluxes → add reactions → justify each term.

Question 26

“Show that the system can be reduced to one ODE”

Answer 26

Use conservation laws or constraints to eliminate variables.

Question 27

“Non-dimensionalise the equations”

Answer 27

Choose scales → rescale variables → substitute → define dimensionless parameters.

Question 28

“Find all fixed points”

Answer 28

Set all time derivatives to zero → solve algebraic system.

Question 29

“Determine linear stability”

Answer 29

Compute Jacobian → evaluate at equilibria → find eigenvalues.

Question 30

“Graphically determine fixed points”

Answer 30

Plot nullclines → identify intersections.

Question 31

“Discuss stability graphically”

Answer 31

Use phase plane and nullcline slopes.

Question 32

“Show that there are no instabilities”

Answer 32

Prove all eigenvalues have negative real part.

Question 33

“Consider small perturbations”

Answer 33

Linearise system around equilibrium.

Question 34

“Assume exponential solutions”

Answer 34

Substitute exp(λt) into linearised equation.

Question 35

“Derive the characteristic equation”

Answer 35

Substitute exponential ansatz into linearised model.

Question 36

“Find the critical delay”

Answer 36

Assume purely imaginary roots → separate real and imaginary parts.

Question 37

“Write down the PDEs”

Answer 37

State conservation law then specify flux and reaction terms.

Question 38

“Justify each term”

Answer 38

Explain biological process represented.

Question 39

“Random movement”

Answer 39

Model with diffusion.

Question 40

“Attraction saturates”

Answer 40

Use bounded functional response.

Question 41

“Growth is logistic”

Answer 41

Use r n (1 − n/K) form.

Question 42

“Find spatially uniform steady states”

Answer 42

Drop spatial derivatives → solve ODE steady states.

Question 43

“Transform to moving frame”

Answer 43

Define ξ = x − ct.

Question 44

“Derive wave speed”

Answer 44

Integrate ODE using boundary conditions.

Question 45

“Plot f(c)”

Answer 45

Sketch piecewise reaction term.

Question 46

“Wave connects steady states”

Answer 46

Use boundary conditions at ±∞.

Question 47

“Explain parameter meaning”

Answer 47

Relate each parameter to biological rate.

Question 48

“Reduce parameters”

Answer 48

Use nondimensionalisation.

Question 49

“Explain why instability occurs”

Answer 49

Interpret eigenvalues biologically.

Question 50

“Show total population conserved”

Answer 50

Sum equations and simplify.

Question 51

“Explain quadratic dependence”

Answer 51

Relate to pairwise interactions.

Question 52

“Explain cubic dependence”

Answer 52

Higher-order effects.

Question 53

“Linearise PDE system”

Answer 53

Expand around uniform steady state.

Question 54

“Identify unstable modes”

Answer 54

Find k with positive λ.

Question 55

Using lambda stable is

Answer 55

if lambda is negative, fixed point is stable

Question 56

Using lambda unstable is

Answer 56

if lambda is positive, fixed point is unstable

Question 57

If determinant is negative

Answer 57

fixed point is unstable

Question 58

If det is positive and tr is positive

Answer 58

fixed point is unstable

Question 59

If det is positive and trace is negative

Answer 59

fixed point is stable

Question 60

(trace)^2 > 4(det) b^2 > 4ac

Answer 60

node

Question 61

(trace)^2 < 4(det) b^2 < 4ac

Answer 61

spiral

Question 62

“Show wave travels right”

Answer 62

Choose sign of c.

Question 63

“Interpret delay biologically”

Answer 63

Relate to maturation or feedback lag.

Question 64

“Show delay induces oscillations”

Answer 64

Complex eigenvalues cross imaginary axis.

Question 65

if f'(x) > 0

Answer 65

fixed point is unstable

Question 66

if f'(x) < 0

Answer 66

fixed point is stable

Question 67

Hill function

Answer 67

(x^n) / (x^n + k^n)

Question 68

When would you use a hill function

Answer 68

Non linear feedback

Question 69

Taylor expansion f(x +x1)

Answer 69

f(x + x1) = f(x) + f'(x)x1 + hot