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
