2 Modelling with Difference Equations Flashcards
difference equations
also called recurrence/ iterative relations/ maps.
*models aiming to describe how a pop changes in time
*relationships between quantities that change over discrete intervals of time, e.g. over time t = 0, 1, 2, . . . , where t represents the generation number t
*for generations that do not overlap
E.g difference equations can be used
Insects often have well-defined annual non-overlapping generations
Adults may lay eggs in spring and then die
The eggs hatch out into larvae, which eat and grow and then overwinter in a pupal stage to reach adulthood the following spring. The size of this population can be modelled with a first-order difference equations giving a relationship between the size in generations t and t + 1
working assumptions for difference equations
(i) evolution happens in synchrony
(reproduction/death occurs at same discrete time);
(ii) populations are large and spatially homogeneous (no space, no fluctuations);
(iii) populations grow asexually (no genetics, no
mating).
Simple growth models
Malthusian growth model
assumes that a population of initial size N_0
vary by a factor λ > 0 each year
t=1: N_1=λN_0
t=2 N_2=λ^2N_0
N_{t+1}= λN_t
scalar first-order map
N_t= λ^tN_0
pop size in gen t
This map is of the first order, since
N_{t+1}= λN_t
scalar first-order map
it is relation between N_{t+1} and N_t,
and
is linear since the right-hand-side depends linearly on N_t
.
growth rate
In a small abuse of language, λ is sometimes called “growth rate”. Stricly speaking the growth rate GR
of the quantity Nt is its “fractional change per unit time”, that is
GR ≡ (1/∆t){N_{t+∆t} − N_t}/N_t
for the case with differential equations
r-1 is actually
limiting term 1-(N_t/K)
Simple growth models
Malthusian growth model
N_t
when t → ∞,
referred to as simple “exponential growth”
as the “geometric growth model”, when λ > 1.
when t → ∞,
Nt →
{∞ if λ > 1
{0 if λ < 1
{N_0 if λ = 1.
when λ > 1 the population grows without bounds (“explodes”): there is exponential or geometric growth
when λ < 1 population decreases and goes extinct exponentially (geometrically)
when λ = 1. its size remains constant if λ = 1.
malthusian growth negative
This is generally not realistic! In most applications, growth depends on finite resources: exponential growth occurs initially, when there are plenty of resources, and then slows down as resources become more scarce.
This has motivated modifications of Malthus’ model. The basic idea is to assume that resources are limited and that a population size cannot be maintained above certain carrying capacity
growth limiting term/factor model
Nₜ₊₁= rNₜ + F(Nₜ)
Nₜ₊₁ results from the growth of Nₜ at a rate r that is limited by a growth term F(Nₜ)
Usually F(N_t)=-(r/K) Nₜ²
with r>0
0 < K < ∞ is the carrying capacity.
Growth limiting factor ensures pop size Nₜ positive and doesn’t exceed K when 0<N_0≤K
and 0 < r ≤ 4
initially exponential growth for small pops then increase to limiting capacity 1 - (N_0/k) where slows stops growing decreases
difference equation for the pop model with growth limiting term.
SCALED
Nₜ₊₁= rNₜ[ 1- (Nₜ/K)]
xₜ= Nₜ/K of the carrying capacity used at time t
substituting to find the first-order difference equation
xₜ₊₁= rxₜ [1-xₜ]
0 < r ≤ 4
0<x₀<1
ensures x_t in [0,1] for all t in N_0={1,2,3,…}
xₜ=Nₜ/K is the pop size relative to carrying capacity that it can never exceed
logistic map
xₜ₊₁= rxₜ [1-xₜ]
nonlinear difference equation of first-order
generic form of a scalar first-order linear difference equation, or “one-dimensional first- order linear map”
xₜ₊₁= aₜxₜ +bₜ
for t=0,1,…
*coeffs aₜ and bₜ can be functions of the discrete time t
* equation is scalar (1D) as consist of one variable
*LINEAR since RHS dep only linearly on x_t
*FIRST ORDER because x at t+1 def only of its value at prev iteration
The linear first-order scalar map is autonomous if
a_t = a and b_t = b are independent of t,
and non-autonomous otherwise
The linear first-order scalar map is homogeneous if
b_t=0
inhomogeneous otherwise
x_{t+1}=2x_t +1 is…
linear autonomous and inhomogeneous
x_{t+1}=2t x_t is….
linear non-autonomous and homogeneous
scalar first-order autonomous linear map
x_{t+1} = ax_t + b
with given initial condition x_0 where a,b are constant and t=0,1,…
LINEAR FIRST ORDER MEANS IT CAN BE SOLVED BY THE PRINCIPLE OF SUPERPOSITION
we can find an explicit eq for x_t prediction without iterating
x_{t+1} = ax_t + b
with given initial condition x_0 where a,b are constant and t=0,1,..
SOLVING BY THE PRINCIPLE OF SUPERPOSITION
1) seek general sol of homogeneous eq
x⁽ʰ⁾ₜ₊₁ = ax⁽ʰ⁾ₜ
x⁽ʰ⁾ₜ₊₁ = aᵗ *constant. dep on IC but found later?
2) particular sol of inhomogeneous eq
x⁽ᵖ⁾ₜ₊₁ = ax⁽ᵖ⁾ₜ + b
x⁽ᵖ⁾ₜ = b/(1-a) if a≠1
3)GENERAL SOL
xₜ = x⁽ʰ⁾ₜ + x⁽ᵖ⁾ₜ
Constant= found now
How is the particular solution found?
particular sol of inhomogeneous eq
x⁽ᵖ⁾ₜ₊₁ = ax⁽ᵖ⁾ₜ + b
x⁽ᵖ⁾ₜ₊₁ = ax⁽ᵖ⁾ₜ + b
suppose x⁽ᵖ⁾ₜ = constant C
C=aC+b
C= (1-a)/b
x⁽ᵖ⁾ₜ = b/(1-a) if a≠1
x_{t+1} = ax_t + b
with given initial condition x_0 where a,b are constant and t=0,1,..
**
SOLUTION BY THE PRINCIPLE OF SUPERPOSITION**
xₜ =
{x₀aᵗ + b [(1-aᵗ)/(1-a) ]. if a≠1
{x₀ +bt if a =1
Constant =in homogeneous gen sol found at the end
solution by principle of superposition
In the special case a = 1,
the homogeneous equation becomes
x⁽ʰ⁾ₜ₊₁ = x⁽ʰ⁾ₜ
general sol
x⁽ʰ⁾ₜ₊₁ = c
particular sol
x⁽ᵖ⁾ₜ₊₁ = x⁽ᵖ⁾ₜ + b
x⁽ᵖ⁾ₜ = dt
substitution d(t+1)=dt+b d=b
x⁽ᵖ⁾ₜ = bt
xₜ = c+bt
c=x_0 found
xₜ =
{x₀aᵗ + b [(1-aᵗ)/(1-a) ]. if a≠1
{x₀ +bt if a =1
long term behaviour of map
t → ∞:
x_t →∞. when|a|>1
x_t →0. when|a|<1.
x_t grows/decreases exponentially in t when |a|≠ 1
When a < 0, solution x_t decreases (if −1 < a < 0)
or
increases (if a < −1) and its sign changes, or “switches”, with t.
When a = 1, x_t grows linearly with t.
difference equations
scalar first-order nonlinear map
nonlinear maps
one single variable x
difference equations for which x_{t+1) depends nonlinearly on x_t,x_(t-1)
when x_{t+1} depends nonlinearly only on x_t we have a scalar first order nonlinear map with general form
x_(t+1)=f(x_t)
function generally. nonlinear we focus on autonomous not dep on t
nonlinear cannot solve analytically
example of a scalar first-order nonlinear map
Logistic map
logistic map
x_(t+1)=f(x_t)
f(x_t)= rxₜ [1-xₜ]
xₜ₊₁= rxₜ [1-xₜ]
with 0 < r ≤ 4 as ensures 0<x≤1 we require this as x represents the density of a pop
nonlinear maps can’t be solved exactly but we use linear stability analysis and cobweb by looking for FPs
Definition 2.1 (Fixed point).
The scalar first-order nonlinear map admits a fixed point (or steady state) x∗ if
xₜ = x∗ for all t ∈ N_0 s.t.
x∗ = f(x∗).
*must be physical x_t density of pop means x_t in [0,1] get rid of any not when checking for fixed points mention this in the exam
fixed points of logistic map
x∗ = rx∗(1 − x∗)
solving gives
x=0 and x= 1-(1/r)
physical when r>1
LOGISTIC MAP has one fixed point x*=0 when 0<r≤1
two fixed points when r>1
STABILITY of fixed point
We analyse by linearizing the map about F.P.
Sol is stable if the sequence
{x_0, x_1, x_2, . . . } from initial condition x_0 and the sequence
{x′0, x′1, x′2, . . . } obtained by starting with another initial condition x′_0
are s.t |x_t − x′_t| is small
for all t when |x_0 − x′_0| is small.
When |x_t − x′_t| → 0 for t → ∞, there is asymptotic stability: the map converges to a fixed point x_t → x∗ that is said to be asymptotically stable.
linearize f(x) about x*
introduce y_t
write x_t = x∗ + y_t
y_t# deviations from the fixed point x*
assume x_t deviates slightly from x* and thus |y_t| ≪ x∗
2) sub x_t=x* +y_t into f.p
x∗ + y_{t+1} = f(x∗ + y_t)
3)
Since |y_t| ≪ x∗,
Taylor-expanding about x∗ gives:
f(x∗ + yt) =
f(x∗) + y_{t} f′(x∗)+ higher orders, where
f′(x∗) ≡df/dy|y=0=df/dx|x=x∗
Since x∗ = f(x∗), to
linear order (when all higher order terms y^2_t, y^3_t, . . . are neglected),
thus:
y_{t+1} = y_tf′(x∗)
therefore y_t = [f′(x∗)]^t y_0,
where now f′(x∗) can be positive or negative.
THM 2.1
Theorem 2.1 (Local stability of first-order maps).
The physical fixed point x∗
of x_{t+1} = f(x_t), with given initial condition x_0,
is asymptotically stable when |f′(x∗)| < 1
- is unstable when |f′(x∗)| > 1
- is non-hyperbolic when |f′(x∗)| = 1, and the stability of this case has to be discussed separately.
explaining thm 2.1
- if f′(x∗) > 1, there is geometric growth of yt : x∗ is unstable since any small deviation
from it grows exponentially, with |xt − x∗ - Similarly, if f′(x∗) < −1, there is geometric growth with sign switch of yt and thus x∗ is unstable.
- if 0 < f′(x∗) < 1, y_t decreases exponentially: x
∗ is asymptotically stable since any small deviation from it decreases exponentially, with |xt − x∗| → 0 for large t. - similarly, if −1 < f′
(x∗) < 0, there is exponential decay with sign switch of yt and thus
x∗ is asymptotically stable.
These results are valid for small deviations yt around x ∗ (linearization). Are they still valid when nonlinear terms are taken into account?
≫ 1 for large enough t.
period _n bifurcation
linear analysis and
Thm. 2.1 do not say anything about the non-hyperbolic case (when |f’(x∗)| = 1) which is
characterized by a possible change of behaviour arising when f′(x∗) = ±1, which is referred
to as “bifurcation”. In particular, when f′(x∗) = −1 a period-doubling bifurcation (or “period-2 bifurcation”) arises
logistic map stability
logistic map
x_(t+1)=f(x_t)
f(x_t)= rxₜ [1-xₜ]
xₜ₊₁= rxₜ [1-xₜ]
with 0 < r ≤ 4 as ensures 0<x≤1 we require this as x represents the density of a pop
x∗ = rx∗(1 − x∗)
solving gives
x=0 and x= 1-(1/r)
physical when r>1
LOGISTIC MAP has one fixed point x*=0 when 0<r≤1
two fixed points when r>1
|f′(x)| = r|1 − 2x| thus
x∗ = 0 and x∗ = 1 − (1/r)
asymptotically
stable when r < 1 (actually when r ≤ 1) & when 1 < r < 3
at r=3: a period-doubling
bifurcation occurs
yielding period-2 oscillations when 3 < r < 1+√6.
There is then a further period-doubling bifurcation at r = 1 + √6 yielding period-4 oscillatory behaviour for
1 + √6 < r < 3.54409
The periodicity of the solution increases with r
up to r ≈ 3.570: for r ≳ 3.570, the logistic map generally exhibits incoherent patterns that
depend greatly on the initial condition: the dynamics is chaotic some small windows of reg behaviour too
Q3 of Example Sheet 1 is on the Ricker map x
***worked example in notes
COBWEB DIAGRAM
graphical method to qualitatively study nonlinear maps
generates sequence by repeating steps
(i) Draw the curve x_{t+1} = f(x_t) vs x_t and the straight line x_{t+1} = x_t
, for t = 0, 1, . . . .
The graph of f(xt) and the line intersect at the fixed point(s) x∗
.
(ii) Start from x_0 and find x_1 = f(x_0)
(iii) On the horizontal axis x_1 is obtained by reflection through the line x_{t+1} = x_t
(iv) From x_1, find the next iterate x_2 = f(x_1)
*check slope of f(x) at the origin is greater than 1 unstable
stable if after enough iterations (t large enough) the sequence of iterates converges towards x
*ie stable if cobweb spirals towards the asymptotically
stable fixed point
*periodic oscillations if after a few iterations we have simple closed geometric motifs (a rectangle?)
*chaotic if sequence dep on initial condition and exhibits no coherent pattern
*if we keep increasing r perioid changes
Level 5 properties of the logistic map
intro
we solve x∗ = f(x∗), with f(x) = rx(1 − x) where 0 < r ≤ 4, and find that it has two fixed points,
x∗_0 = 0 and x∗_1 =1− (1/r)
Since we require that
0 ≤ x_t ≤ 1 for all t, x∗_0 = 0 is always physical while
x∗_1 = 1 − (1/r) is
physical only when r ≥ 1.
What is the stability of these fixed points as a function of r? determine when
|f′(x∗_0)| < 1 &|f′(x∗_1)| < 1
Level 5 properties of the logistic map
stability of fp
f′(x∗_0) = r, the fixed point x∗_0 = 0 is asymptotically stable when r < 1 (actually
when r ≤ 1) and unstable when r > 1.
Similarly, since |f′(x∗_1)| = |2 − r|, the fixed point
x∗_1 = 1−(1/r) is asymptotically stable when 1 < r < 3 (actually when 1 < r ≤ 3) and unstable when r > 3.
neither fps stable when r >3: f′(x*_1) = −1 when r = 3 and x∗_1 is thus non-hyperbolic
Level 5 properties of the logistic map
period-doubling bifurcation
also called “period-2 bifurcation” or “flip bifurcation”) occurs, and the dynamics thus changes and is characterized by an oscillatory behaviour
3 < r ≲ 3.57
for 3 < r < 1 + √ 6 the dynamics of the logistic map is periodic of period two: after a transient, the iterates oscillates between two specific values which form what is called a “period-2 orbit
the fixed points of p th functional power composition
f^p(x) ≡ [f ◦ … ◦ f] p times
(x) = f(f(f(. . .)))(x)
for the logistic
map, we have f^2(x) = f ◦ f = f(f(x)) = rf(x)(1−f(x)) = r^2 x(1−x)(1−rx(1−x)),
Definition 2.2 (Periodic points and orbits, stability)
A periodic point x∗ₖ)
of x_{t+1} = f(x_t), with given initial condition x_0, (2.4)
of minimal period p > 1 (positive integer) is such that fᵖ(x∗ₖ) = x∗ₖ
and fᵠ(x∗ₖ) ̸= x∗ₖ for
q = 1, 2, . . . , p − 1 and k = 1, 2, . . . , p.
The set of iterates
{x∗_1, x∗_2, . . . , x∗_p}, where
x∗_2 = f(x∗1),
x∗_3 = f(x∗2) = f^2(x∗1), . . . , x∗p =f^{p−1}(x∗_1), is a periodic orbit of period p (or “p-cycle”)of (2.4)
Periodicity of the p-cycles implies fᵖ⁺ʳ(x∗_1) = fʳ(fᵖ(x∗1)
=x∗1) = fr(x∗1) = x∗r+1, for r = 0, 1, . . . , p − 1.
For the logistic map (2.4), the period-2 orbit is denoted by {x∗−, x∗+}
{z }
Theorem 2.2 (Stability of periodic orbits)
When is a p-cycle stable?
A periodic orbit {x∗₁, x∗₂, . . . , x∗ₚ} of period p
is asymptotically stable if
|d/dx fᵖ(x∗ₖ) | < 1 for some k = 1, 2, . . . , p,
and the periodic orbit
is unstable if |d/dx fᵖ(x∗ₖ)|> 1 for some k.
logistic map example
sheet 1
summary of behaviour
* The period-2 orbit loses its stability at r = r_2 = 1 + √6, when (d/dx)f ^2|{x=x∗±} = −1
and x∗± become non-hyperbolic fixed points of f^2(x).
* Hence f^2(x) undergoes a period-doubling bifurcation at r2 = 1 + √6. Similarly as before, this leads to the existence period-4 orbits.
* These period-4 orbits are stable for r2 < r < r3 and are followed by another period-doubling bifurcation at some value r = r3
* When r is raised further, there is an increasing sequence of parameters values r1 < r2 <
r3 < . . . < rn for which stable orbits of period 2n
succeed to each other.
* The sequence r1 < r2 < r3 < . . . < rn converges to the number r∞ ≈ 3.570
which periodicity is generally lost and the logistic map exhibits chaotic behaviour. When this happens, the sequence of iterates {x0, x1, x2, . . . } exhibits no coherent patterns and any small change in the initial condition generates a completely different sequence
Linear planar first-order maps
homogeneous planar (or “two-dimensional”)
first-order maps, i.e. first-order difference equations of two variables.
general form of homogeneous planar linear maps
xₜ₊₁ = a xₜ + b yₜ
yₜ₊₁ = c xₜ + d yₜ
for constant a, b, c and d, which are not all zero, t ∈ N₀, and given initial condition (x₀, y₀).
here xₜ₊₁ belongs on both xₜ and yₜ etc
FIXED POINT of a planar linear map
By letting t → ∞
substituting xₜ and xₜ₊₁ by
x∗ = lim t→∞ xₜ
and yₜ and yt+1 by
y∗ = lim t→∞ yₜ
Giving
x∗ = a x∗ + b y∗
and
y∗ = c x∗ + d y∗
to be solved simultaneously
We assume invertibility of matrix
(a − 1)(d − 1) − bc ≠0,
invertibility of matrix assumption:
FIXED POINT of a planar linear map
Invertibility of the matrix A-I
[a b] - [1 0]
[c d] [0 1]
(a − 1)(d − 1) − bc ≠0,
This guarantees that the unique stationary solution is (x∗, y∗ ) = (0, 0).
In general, the “origin” (x
∗, y∗) = (0, 0) is therefore the only fixed point
Linear planar first-order maps:
Looking at stability of (x
∗, y∗) and the dynamics in its vicinity
using matrix form
xₜ₊₁ = A Xₜ
with
xₜ = (xₜ,yₜ)^T
≡
(xₜ)
(yₜ)
A=
[a b]
[c d]
with given initial condition x₀= (x₀, y₀)^T
To study dynamics using matrices we use a similarity transformation using a
change of basis
A → B A transformed to B
B= P⁻¹AP ,
where P is a 2 × 2 non-singular (invertible) “change of basis matrix” and B is the “similarity transformation” of A.
Matrices A and B are thus called similar:
same trace (tr(A) = tr(B)),
same det (det(A) = det(B))
same eigenvalues
B= P⁻¹AP ,
A=
B= P⁻¹AP
A= PBP⁻¹ ,
We choose B s.t. easy to compute powers Bᵗ for t=1,2,…
using B= P⁻¹AP ,
P⁻¹AᵗP=( P⁻¹AP)ᵗ = Bᵗ
easier to compute if B diagonal matrix
Linear planar first-order maps: GENERAL SOLUTION
using xₜ₊₁ = A Xₜ
with
xₜ = (xₜ,yₜ)^T
≡
(xₜ)
(yₜ)
A=
[a b]
[c d]
with given initial condition x₀= (x₀, y₀)^T
To solve, compute iterates:
x₁=Ax₀
x₂=Ax₁=A²x₀
**x₃=Ax₂=A²x₁=A³x₀
to give a GENERAL SOLUTION
xₜ= Aᵗx₀ for t∈N₀
(t is a power not transpose)
Linear planar first-order maps: GENERAL SOLUTION in form B
xₜ= Aᵗx₀ for t∈N₀
(t is a power not transpose)
using the linear transformation:
xₜ= =Puₜ=
P( vₜ)
(wₜ)
for components of uₜ
Since
x₀=Pu₀ we have
Puₜ=xₜ=Aᵗx₀=AᵗPu₀
So
**uₜ=P⁻¹AᵗPu₀
Thus we have
(Using B= P⁻¹AP , **xₜ=Puₜ and u₀=P⁻¹x₀ )
uₜ=Bᵗu₀ →
**xₜ=PBᵗP⁻¹x₀
Linear map summary re-written
Matrix
P⁻¹xₜ₊₁
=P⁻¹AP P⁻¹xₜ
giving
uₜ₊₁= Buₜ
with u₀=P⁻¹x₀
hence linear maps have the same properties and the same fixed points
x=u=0 and same dynamics
stability from matrix A and B:
The eigenvalues λ± of A and B are real and distinct: The main features of the map’s dynamics about the origin x∗ = u∗ = 0
The eigenvalues λ± of A and B are real and distinct:
(a) If both λ± are positive:
- The origin is a stable node (attractor) if both λ+ < 1 and λ− < 1. x∗ = u∗ = 0 attracts the iterates along all directions in the x − y or
v − w plane.
- The origin is an unstable node (repeller) if both λ+ > 1 and λ− > 1. In
this case x∗ = u∗ = 0 repels the iterates in any directions in the x − y or
v − w plane. - The origin is an unstable hyperbolic saddle (saddle point) if λ+ > 1 and
λ− < 1. x∗ = u∗ = 0 repels and attracts the iterates along the eigenvector associated with λ+ and λ−, respectively.
(b) If λ+ is negative and λ− is positive or negative:
- The origin is a stable node with reflection (attractor) if |λ+| < 1 and
|λ−| < 1. x∗ = u∗ = 0 attracts the iterates along all directions.
- The origin is an **unstable node with reflection **(repeller) if |λ+| > 1 and
|λ−| > 1. x∗ = u∗ = 0 repels the iterates along any directions. - The origin is an unstable hyperbolic saddle with reflection if |λ+| > 1 and
|λ−| < 1, or if |λ+| < 1 and |λ−| > 1. In the former case, x∗ = u∗ = 0
respectively repels and attracts the iterates along the eigenvectors associated with λ+ and λ−. When |λ+| < 1 and |λ−| > 1, x∗ = u∗ = 0 respectively
stability from matrix A and B:
If A and B have a single real eigenvalue λ: The main features of the map’s dynamics about the origin x∗ = u∗ = 0
The origin is a stable node (attractor) if |λ| < 1, without reflection if λ > 0
and with reflection if λ < 0.
(d) The origin is a unstable node (repeller) if |λ| > 1, without reflection if λ > 0
and with reflection if λ < 0
stability from matrix A and B: If the eigenvalues of A and B are complex conjugates
If the eigenvalues of A and B are complex conjugates λ± = α ± iβ = Re±iθ, where
α, β ∈ R \ {0}, where R =sqrt(α² + β²) and ±θ = arg(λ±) are respectively the modulus and argument of λ±:
(e) If R > 1: the map’s iterates spiral outwards and the origin is an unstable spiral
(also called “unstable focus”)
Example: We consider the linear map xₜ₊₁ = 2xₜ − yₜ; yₜ₊₁ = −yₜ/2.
fixed point stability?
.The corresponding
matrix is A =
[2 −1]
[0 −1/2]
eigenvalues λ₁= 2, λ₂ = −1/2.
For this map, the
fixed point (x∗, y∗) = (0, 0) is therefore unstable: it is a hyperbolic saddle with reflection.
Case:
A has 2 real distinct eigenvalues and eigenvectors λ±
Use eigenvectors as columns to transform
P= [v+ v-]
gives
Bᵗ =
[ λ₊ᵗ 0]
[0 λ₋ᵗ]
=P⁻¹AᵗP
Thus
Aᵗ =
P [ λ₊ᵗ 0] P⁻¹
[0 λ₋ᵗ]
(b) Degenerate case where has 1 real eigenvalue λ±= λ
Either can be diagonalized
Bᵗ =
[ λᵗ 0]
[0 λᵗ]
=P⁻¹AᵗP
Thus
Aᵗ =
P [ λᵗ 0] P⁻¹
[0 λᵗ]
Or cannot be diagonalized
B=
[λ 1]
[0 λ] with P in Jordan form
Bᵗ =
[ λᵗ tλᵗ⁻¹]
[0 λᵗ]
=P⁻¹AᵗP
hence
Aᵗ =
P [ λᵗ tλᵗ⁻¹] P⁻¹
[0 λᵗ]
Case: (c) has complex conjugate eigenvalues
λ± = α ± iβ (β > 0)
eigenvectors
v± = vᵣ ± ivᵢ
What will matrix B be?
B=
[ α β]
[−β α
P= [vᵣ vᵢ]
R= sqrt(α² + β²) = |λ±|
tan θ = β/α
gives
Bᵗ=
Rᵗ
[ costθ sin tθ]
[− sin tθ costθ]
=P
(R represents distance from origin, Rᵗ thus affect behaviour which will give spiral inwards<1 or outwards >1)
hence
Aᵗ=
RᵗP[costθ sin tθ]P⁻¹
[− sin tθ costθ]
Behaviour of x_t found from y_t
Behaviour of xₜ obtained from yₜ
continued behaviour
Behaviour of xₜ obtained from yₜ
yₜ= P⁻¹xₜ =P⁻¹Aᵗx₀= P⁻¹AᵗP y₀
giving
xₜ =Pyₜ= PBᵗy₀=PBᵗP⁻¹x₀
It therefore suffices to study how changes in time to understand the behaviour of and whether its fixed point (0,0) is stable or unstable:
as t→∞
yₜ → (
0)
(0)
xₜ→
(0)
(0)
only if |λ+| < 1 and |λ−| < 1
⇒ if |λ+| < 1 and |λ−| < 1, (0, 0) is asymptotically stable
*as t→∞
∥yₜ∥ → ∞, ∥xₜ∥→ ∞ if at least one eigenvalue is |λ±| > 1
⇒ if |λ+| > 1 and/or |λ−| > 1, (0, 0) is unstable
summary eigenvalues stability
(0,0) can be a node that is asymptotically stable (attractor, if ) or unstable (repeller,
if both|λ|<1 ), with or without reflection (reflection: when at least one eigenvalue is <0)
(0,0) can also be a saddle with or without reflection, that’s when the origin is an attractor in one direction and a repeller in another direction (e.g. when |λ+|>1 |λ-|<1) (one eigen direction along which stable and another which unstable;overall origin will be unstable but two different directions)
(0,0) can be a spiral (also called “focus”) if eigenvalues have nonzero imaginary part, i.e. im(λ±) not equal to 0 . The spiral (focus) is stable if R= |λ±|<1and unstable if R>1. R is the modulus of the eigenvalues
saddle with reflection
if one eigenvalue +ve sign and another -ve sign
Nonlinear planar first-order maps
general form
h(xₜ)
xₜ₊₁ = f(xₜ,yₜ)
yₜ₊₁ = g(xₜ,yₜ)
in matrix form
xₜ₊₁ = h(xₜ)
where xₜ ≡(xₜ,yₜ)^T
[ f(xₜ,yₜ)]
[ g(xₜ,yₜ)] t ∈ N₀ and the initial condition x₀ = (y₀,x₀)^T is given
functs f and g assumed to be known C¹ nonlinear functions of x and y . Just as in the scalar case, nonlinear planar maps can gen not be solved
Definition 2.3 (Fixed points of planar first-order maps).
The column vector x∗ =
(x∗, y∗)^T is a fixed point of the planar map if x∗
and y∗ satisfy
x∗ = f(x∗, y*) and
y∗ = g(x∗, y∗),
or, in matrix form: x∗ = hₜ(x∗).
found by solving simultaneous eq:0,1 or many sols/fixed points
f.ps need to be physical
For instance, if the map
decribes how the number of individuals of two species in a population changes over time (or how the density or fraction of each species making up the population composition evolves with t),
NON NEGATIVE
we need xₜ ≥ 0, yₜ≥ 0 for all t ≥ 0, and in this case physical fixed points are solutions of such that x∗ ≥ 0 and y∗ ≥ 0.
Definition 2.4 (Local stability)
A fixed point x∗ of hₜ
is stable if, for any ϵ > 0, there is a δ > 0 such that for any initial condition x₀ such that |x₀ − x∗|<δ, the iterates of x₀ satisfy satisfy |hₜ − x∗| < ϵ for all t ∈ N₀ .
A fixed point is said to be unstable if it is not
stable.
A fixed point x∗ is asymptotically stable if it is stable and, in addition, there is a value γ > 0 such that hₜ → x∗ as t → ∞ for all x₀ satisfying |x₀ − x∗| < γ.
< δ, the iterates of
Determine local stability of and dynamics about that (x fixed point by linearization:
1)Linearisation
xₜ= x∗+ uₜ
where uₜ are small deviations from x∗
2) Substitute into the map to determine how changes with t
giving taylor expansion abouy fixed point x,y to linear order in vₜ,wₜ
matrix form
uₜ₊₁ =J∗uₜ
with J∗ =
[fₓ(x∗,y∗) fᵧ(x∗,y∗)]
[gₓ(x∗,y∗) gᵧ(x∗,y∗)]
maps jacobian evaluated at FP with partial derivs
3) The sole fixed point of the linearized map uₜ₊₁ =J∗uₜ is u∗ =
(0)
(0)
notion of stability
ie. starting near fp and iterations converge to it it is not only stable but asymptotically stable
stability of u∗
Determined by the eigenvalues λ± of J∗
solving the characteristic equation
det(J∗-λI)=0 where I is identity matrix to get λ±
uₜ when t → ∞
matrix form
uₜ₊₁ =J∗uₜ
with J∗ =
[fₓ(x∗,y∗) fᵧ(x∗,y∗)]
[gₓ(x∗,y∗) gᵧ(x∗,y∗)]
maps jacobian evaluated at FP with partial derivs
3) The sole fixed point of the linearized map uₜ₊₁ =J∗uₜ is u∗ =
(0)
(0)
- uₜ → 0 and xₜ → x∗ when t → ∞ if all eigenvalues |λ±| < 1
(DEVIATIONS uₜ FROM x∗ EVENTUALLY VANISH => IS ASYMPTOTICALLY STABLE) - ∥uₜ ∥ and ∥xₜ ∥ → ∞ when t → ∞ if at least one eigenvalue |λ±| > 1
(DEVIATIONS FROM x∗ GROW => IS UNSTABLE)
Theorem 2.3 (Local stability of first-order planar maps).
Let x∗ = (x∗, y∗ ) be a fixed
point of the map and J∗ its Jacobian whose eigenvalues are λ±. If x∗ is hyperbolic, that is if |λ−|≠ 1 and |λ+|≠ 1, then
- x∗ is asymptotically stable if all eigenvalues λ± of J∗ are |λ±| < 1;
- x ∗ is unstable if at least one eigenvalue of J∗ is |λ±| > 1;
- The nature of x∗ and the dynamics in its vicinity depend on the modulus, imaginary/real part, and sign of λ±, and is given by the classification (a)-(f) obtained from the linear stability analysis: x∗ is a stable or unstable node
(with or without reflection), or a saddle (with or without reflection) [cases (a)-(d)],
or a stable or unstable spiral (focus) [cases (e, f)].
THEOREM DOES NOT SAY ANYTHING ABOUT THE STABILITY OF A NON-HYPERBOLIC FIXED POINT
Jury conditions for first-order planar maps
The FP x∗ is asymptotivally stable IFF
The FP x∗ is asymptotivally stable IFF
|tr(J∗)| < 1 + det(J∗) and
det(J∗) < 1
or, equivalently, if and only if |tr(J∗)| < 1 + det(J∗) < 2.
tr(J∗) =
det(J∗)
tr(J∗) = λ₊ + λ₋ = fₓ(x∗) + gᵧ(x∗)
det(J∗) = λ₊λ₋ = fₓ(x∗)gᵧ(x∗) − fᵧ(x∗)gₓ(x∗)
characteristic equation
det(J∗-λI)=0 where I is identity matrix to get λ±
λ± = [tr(J∗) ± sqrt((tr(J∗))^2 - 4det(J∗))
Over 2?
Jury conditions for first-order planar maps
The FP x∗ is s unstable IFF
tr(J∗) > 1 + det(J∗), tr(J∗) < −1 − det(J∗), or det(J∗) > 1
jury conditions
valid only for maps, differ for differential equations
These conditions are only for maps,
there are different conditions for
ordinary differential equations (ODEs)
Jury conditions for first-order planar maps
± have an imaginary part and the dynamics is oscillatory when
λ± have an imaginary part and the dynamics is oscillatory when
4 det(J∗) > [tr(J∗)]^2
.
Example: The nonlinear map x_{t+1} = x_t(y_t − 2); y_{t+1} = 3y_t − x_t
fixed points?
has two fixed points:
(x₀∗, y₀∗) = (0, 0) and
(x₁∗, y₁∗) = (6, 3). The Jacobian matrices associated with the former and the latter are respectively
J₀∗=
[−2 0]
[−1 3]
and
J₁∗ =
[1 6]
[−1 3]
The eigenvalues for J₀∗ are −2 and 3 and J₁∗ 2±i√5.
Hence, according to Thm. 2.3 and the classification (a)-(f)
(x₀∗, y₀∗) is an unstable node (with reflection) and
(x₁∗, y₁∗) is an unstable spiral (focus)
diagram for stability
showing complex and real eigenvalues
(x∗,y∗) asymptotically
stable inside hatched
triangle, unstable
outside of it
is asymptotically stable in )
triangular hatched region, and unstable outside the triangle.
is a spiral (focus) above the
dashed parabola: unstable above the line and stable within the triangle
Extra MATH5567M topic: host-parasitoid & Nicholson-Bailey
Models
Host-parasitoid models (HPMs) address the life cycle of two interacting species, the host and the parasitoid interacting DISRETELY IN TIME
NEEDS ONE TO REPRODUCE and PROSPER Host repro is hindered by parasite
evolution of host-parasitoids modelled with planar maps such as the Nicholson-Bailey’s map
Videos on ex sheet 1
host-parasitoid & Nicholson-Bailey models
ASSUMPTIONS
(i) generations are supposed to be non-overlapping; working in DISCRETE time and reference to GENERATIONS
(ii) in the absence of parasitoids, the population of host grows exponentially with a “growth rate” k > 0 ( k is the average number of offspring of an unparasitized host surviving to next generation, if k>1 pop explodes);
(iii) we assume that the average number of viable eggs laid by a single parasitoid is c > 0 (“parasitoids reproduction rate”).
host-parasitoid & Nicholson-Bailey models
Model map
Hₜ~ # hosts in a generation t
Pₜ~ # parasitoids in a generation t
Hₜ₊₁=kf(Hₜ,Pₜ) Hₜ
Pₜ₊₁=c[1- f(Hₜ,Pₜ)]Hₜ
- only non-parasitized hosts give rise to next gen of hosts
*Parasitized hosts give rise to next generation of
parasitoids
*f(Hₜ,Pₜ) is the fraction of non-parasitized hosts
*k is their growth rate (others dont repro)
* given initial condition H₀ > 0, P₀ ≥ 0
host-parasitoid & Nicholson-Bailey models
what we expect?
Hₜ~ # hosts in a generation t
Pₜ~ # parasitoids in a generation t
Hₜ₊₁=kf(Hₜ,Pₜ) Hₜ
Pₜ₊₁=c[1- f(Hₜ,Pₜ)]Hₜ
- expect f to be expected to be decreasing funct of Pₜ (more Parasites less non parasitized)
*for non constant f, there is a nonlinear coupling between Hₜ and Pₜ
Classical Nicholson-Bailey model (NBM) assumes that the host-parasites encounters are
totally random=
what f we use?
Hₜ~ # hosts in a generation t
Pₜ~ # parasitoids in a generation t
Hₜ₊₁=kf(Hₜ,Pₜ) Hₜ
Pₜ₊₁=c[1- f(Hₜ,Pₜ)]Hₜ
f(Hₜ,Pₜ)= exp(−aPₜ) a is a positive parameter (searching efficiency constant)
Hₜ₊₁=kHₜ exp(−aPₜ) ≡ g₁(Hₜ,Pₜ)
Pₜ₊₁=cHₜ [1- exp(−aPₜ))] ≡g₂(Hₜ,Pₜ)
NMB can’d be solved we find fixed points and linear stability analysis
Nicholson-Bailey model (NBM)
fixed points and linear
Fixed points: (H∗, P∗)
Physical solutions only of:
H∗ =g₁(H∗, P∗) ≥ 0
P∗=g₂(H∗, P∗)≥ 0
are s.t
H∗=kH∗ exp(−aP∗)
P∗=cH∗ [1- exp(−aP∗))]
2 FP’s
(H₁∗, P₁∗) = (0,0) EXTINCTION of hosts and parasites always physical
(H₂∗, P₂∗) = ([k lnk]/[ac(k-1)],[ln k]/a) COEXISTENCE of hosts and parasites when k>1 (PHYSICAL ONLY, ensures P₂∗>0 unphysical when k<1)
Nicholson-Bailey model (NBM)
(H₁∗, P₁∗) = (0,0) EXTINCTION of hosts and parasites always physical
(H₂∗, P₂∗) = ([k lnk]/[ac(k-1)],[ln k]/a) COEXISTENCE of hosts and parasites when k>1 (PHYSICAL ONLY, ensures P₂∗>0 unphysical when k<1)
linear stability analysis
Linear stability analysis: eigenvalues of Jacobian at each fixed point
J(H,P)=
[∂ₕg₁ ∂ₚ g₁]
[∂ₕg₂ ∂ₚ g₂]
=
[k e⁻ᵃᴾ -akHe⁻ᵃᴾ]
[c(1-e⁻ᵃᴾ) acHe⁻ᵃᴾ]
evaluated at:
J(H₂∗, P₂∗)=
[1 -k(ln k)/[c(k-1)]]
[c(k-1)/k (lnk)/(k-1) ]
trace = 1 + (ln k)/(k-1)
det = (k/(k-1))ln k
Jury condition:
(H₂∗, P₂∗) is asymptotically stable if
|trace| < 1 + det
and
det< 1
unstable otherwise
det>1 here so
(H₂∗, P₂∗) is unstable
(det >1 because looking at h(k)=klnk - (k-1) this funct is h(k)>0 for k>1 so klnk> k-1)
det>1 means eigenvalues of λ± of J∗2 such that λ₊λ₋ = det(J∗
2) > 1
⇒ |λ₊| > 1 and/or |λ₋| > 1;
since the modulus of at least one eigenvalue > 1 is indeed unstable. It can actually be shown to be an unstable focus (spiral from non zero imaginary part)
oscillations of growing amplitude around (H₂∗, P₂∗)
NMB
oscillations of growing amplitude around (H₂∗, P₂∗)
No stable coexistence in NBM
- Oscillations of growing amplitude
- Unrealistic feature in the long run
- Used as an approximate model
for short times
DIAGRAM: (as n grows oscillations grow in time for both H and P)
The numerical solution of the NBM
shows that the level of parasitoids is regularly very
low => (geometric) growth of hosts followed by
steep decrease and then even steeper growth
This is an unrealistic feature.
trajectory in the phase
plane spirals outwards=>
(unrealistic) unbounded
oscillations
better to only describe the first few generations how they oscillate together
after a longer term spirals outwards better to:
Generalization to make NBM more realistic:
introduce a capacity K to limit the growth
Generalization to make NBM more realistic:
introduce a capacity K to limit the growth
Hₜ₊₁= kHₜ₊e−ᵃᴾ-ᵗ eʳ⁽¹−ᴴ-ᵗ/ᴷ⁾
introducing carrying capacity k
prevents this growing amplitude of oscillations in generations
for longer times
growth-limiting factor exp[r(1 − Hₜ/K)],
