Modeling Virtual Stability With a Population Simulation

1
Modeling Virtual Stability With a Population
Simulation

• Burton Voorhees Joseph Senez
• Center for Science
• Athabasca University
• Supported by NSERC Discovery Grant OGP 0024817,
  NSERC Undergraduate Student Research
  Assistantships, and grants from the Athabasca
  University Research Fund

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Contributors Todd Keeler Rhyan Arthur
National Science and Engineering Research
Council of Canada Undergraduate Summer Research
Assistants Martin Connors Professor of Physics,
Center for Science, Athabasca University
3
The Assumption of Stability
The standard assumption Complex systems will be
found in states that are stable, or at least
metastable, with only occasional brief periods of
transition between such states. The Principle
of Selective Retention Stable configurations
are retained, unstable ones are eliminated.
Heylighen
4
Discrete State Space Models
Discrete models of complex systems generally
assume that the system state space is partitioned
into basins of attraction, determined by system
dynamics. Each basin represents a coarse-grained
state of relative stability. State space
trajectories remain in a given basin unless
driven to another by system dynamics or until
perturbed sufficiently by noise.
5
Discrete State Space Models
Robert May outlined the intuition behind this
idea, arguing that only small regions in a system
parameter space can provide long term stability.
Note, however, that the system state space is not
the same as its parameter space, but state space
dynamics can lead to dynamical changes in
parameter values.
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Discrete State Space Models
Stable states are determined by a fitness
landscape the instantaneous system state remains
near a fitness peak. Applied specifically to
models of evolution, a species is characterized
by a set of phenotypic parameters and parameter
values are expected to cluster in relatively
small regions falling within constraints imposed
by selective fitness barriers defining this peak.

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Course Graining
State space trajectories are represented as a
discrete time series of transitions on a finite
set of states. In this course-grain version of
the continuous representation each basin of
attraction corresponds to a distinct state in the
discrete model.
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Control
Course-grained trajectories of complex adaptive
systems are not random. They represent adaptive
responses to environmental contingencies, or (in
systems with a cognitive component) goal directed
action sequences. In either case, the
course-grained system trajectory is determined by
adaptive control mechanisms.
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Control and Transition
Control studies are especially important when
human value judgments are associated to the
possible coarse-grained states of a complex
system. Both theoretical models and
empirically studied exemplary cases show that
catastrophic jumps between attractor basins do
occur, and that such jumps may be exceptionally
difficult to reverse.
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Control and Transition
Typically, a coarse-grained state appears
relatively unchanged over time, while system
parameter values change slowly in a way that
drives the system trajectory toward an attractor
basin boundary, or weakens the strength of the
attractor, leaving the system vulnerable to small
fluctuations that move it to a new attractor
basin. A sudden jump of coarse-grained state
occurs and, due to hysteresis effects, returning
system parameters to earlier values does not
reverse the jump.
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Example Ecology
The pristine state of most shallow lakes is
probably one of clear water and a rich submerged
vegetation. Nutrient loading has changed this
situation in many cases. Remarkably, water
clarity seems to be hardly affected by increased
nutrient concentrations until a critical
threshold is passed, at which the lake shifts
abruptly from clear to turbid. With this
increase in turbidity, submerged plants largely
disappear. Reduction of nutrient concentration
is often insufficient to restore the vegetated
clear state. Indeed, the restoration of clear
water happens at substantially lower nutrient
levels than those at which the collapse of the
vegetation occurred. M. Scheffer, S. Carpenter,
J.A. Foley, C. Folke, B. Walker (2001)
Catastrophic shifts in ecosystems. Nature 413
591-596
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Psychology/Neurophysiology
The core idea is to interpret mental
representations as more or less stable
attractors and states encoding the sensory
information about the stimuli as initial
conditions. Empirical results from experiments
on bistable perception suggest that transitions
between the two possible perspectives on the
Necker cube illusion transit through an unstable
saddle between two relatively stable
attractors. In this case, it is a matter of the
projection of a three dimensional object from a
two dimensional image. The unstable state is the
objective perception of this two dimensional
image, without the projection of a third
dimension. Anybody who makes the attempt will
find that it requires effort to maintain this
perceptionwe have learned to automatically see
in three dimensions. J. Kornmeier, M. Bach,
H. Atmanspacher (2004) Correlates of perceptive
instabilities in event-related potentials.
International Journal of Bifurcation and Chaos
14(2) 727-736.
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Examples such as these illustrate the centrality
of stability and instability for every area of
complex systems theorizing.
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Ashbys Law of Requisite Variety
In order to maintain systemic integrity in a
fluctuating environment the variety of responses
available to a system must be at least as great
as the variety in the spectrum of environmental
perturbations. There is an additional factor
not taken into account in this criteria.
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The Importance of Instability
In terms of management and control, it is not
only a matter of maintaining a system in a
desired state, but of managing state transitions.
It is not only a matter of maintaining
sufficient variety in a set of possible
responses, but also of being able to switch
between responses in a timely manner.
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The Importance of Instability
This implies the existence of a trade-off between
stability and flexibility. It is easy to change
an unstable state, difficult to change a stable
one if a possible behavioral response state is
stable, change consumes time and energy if not
stable, it is easy to change, but energy is
required to maintain it.
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The Importance of Instability
If behavioral responses at the physiological,
neural, and habitual levels are relatively stable
attractors, then avoidance of commitment to an
immediate response is like remaining on an
unstable boundary between possible attracting
behaviors. Maintaining such a state requires
effort and so exacts a cost. What is purchased
by the energy expended is increased behavioral
flexibility in the face of uncertainty.
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Example Standing
The standing posture is learned in early
childhood and automated as an unstable state,
maintained by a process of proprioceptive
feedback and small muscular adjustments. The
resulting flexibility shows up in the ease of
walking. If standing were stable, the stability
would act as an attractive force maintaining the
state. Every step would require effort to
overcome the stability and would feel like
walking uphill. As it is, taking a step is a
controlled fall.
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Virtual stability

• A state in which a system employs self-monitoring
  and adaptive control in order to maintain itself
  in a configuration that would otherwise be
  unstable.
• A degree of instability is maintained, at a
  certain cost, in order be able to quickly
  adapt/move to a desired state.

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Virtual Stability

• Not the same as stability or metastability
• Stability There is a single global attractor, or
  the system is deterministic and once an attractor
  basin is entered system trajectories remain
  there. If there is noise, the system is stable
  against small perturbations.
• Metastability A system has multiple attractor
  basins with fractal boundaries containing chaotic
  saddles. Basin boundary dimensions are close to
  the dimension of the full state space. Even a
  small amount of noise can produce transitions
  between attractor basins.
• Virtual Stability Through processes of
  self-monitoring and adaptive control a system
  maintains itself on a boundary between two or
  more attractor basins.

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Condition for Virtual Stability
Meta-level control functions direct the
expenditure of energy to maintain an unstable
trajectory, or an unstable state. At a minimum,
this requires that a system have the capacity to
monitor its momentary state and produce adaptive
responses at a frequency high enough that only
small (i.e., inexpensive) corrective actions are
required.
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The Popsim program

• The advantages/disadvantages of virtual stability
  are explored by comparing a population of three
  types of individuals (A, B or C) in a varying
  sequences of three different environments (A, B,
  or N).
• Stability and instability are modeled with
  probability. Populations A and B are stable,
  with only a small chance of transition from A to
  B or vice versa. Members of population C are
  virtually stable--they can easily make
  transitions to both A and B. When members of
  population C are acting as A, they are said to be
  in the state AC, when they are acting as B they
  are said to be in the state BC.
• The program seeks conditions and sequences of
  environments that will favor the stable A and B
  populations OR the virtually stable C population.

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States Environments
Stable
Virtually Stable
Env A
State A low death rate
State B high death rate
State AC low death rate
State BC high death rate
State C very high death rate
Stable
Virtually Stable
Env B
State A high death rate
State B low death rate
State AC high death rate
State BC low death rate
State C very high death rate
Stable
Virtually Stable
Env N
State A medium death rate
State B medium death rate
State AC medium death rate
State BC medium death rate
State C very high death rate
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A Simulation

• The simulation consists of exposing the
  populations to a sequence of environments
  (AANBBAANNABBBNA) and determining whether the
  AB populations or the C population eventually
  dominate.
• For each entry in the sequence, the populations
  are put through a long run in the specified
  environment.

Original Population
New Population
New Population
New Population
New Population
New Population

Long run in Env. A
Long run in Env. A
Long run in Env. N
Long run in Env. B
Long run in Env. B
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What is a long run?

• Each environment has a corresponding transition
  matrix which is used to determine what state
  changes an individual will undergo during a long
  run.
• Every individual goes through a certain number of
  short runs during one long run. For every short
  run, the individuals state will change (or
  remain the same) according to the probabilities
  specified in the environments transition matrix.

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Sample Transition Matrix
A
B
AC
BC
C
D
0.9
0.049
0
0
0
0
0.05
0.881
0
0
0
0
0
0
0.55
0.15
0.4
0
T

0
0
0.1
0.35
0.25
0
0
0
0.3
0.43
0.25
0
0.05
0.07
0.05
0.07
0.1
1

• An individual in state AC, for example, will use
  the probabilities in column AC to determine where
  they will end up. In this case they have a 0
  chance of ending up in A, 0 chance of ending up
  in B, a 55 chance of remaining in AC, a 10
  chance of ending up in BC, a 30 chance of ending
  up in C, and a 5 chance of dying.

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Transition Matrix A

• Since state A and state AC are favored, eA is
  less than eB and eAC is less than eBC. Also, dA
  is greater than dB.

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Transition Matrix N

• e eB eA and eBC eAC, dB dA 0.5.

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Simulation parameters

• All of the symbols in the preceding transition
  matrices represent parameters which can be
  adjusted.
• In addition to these, there are several other
  important parameters which can be configured
• of short runs per long run this is the number
  of times an individual will have the possibility
  of changing population type per long run.
• Minimum of short runs per long run if an
  individual enters into a preferred state, it can
  skip the rest of its short runs if has completed
  the minimum of short runs. For example, if the
  of short runs is 6 and the min. of short runs
  is 2, then if an individual is in the preferred
  state after the 2nd short run (or subsequently),
  they will skip the rest of their short runs.
  However, if the min of short runs was 6, then
  they would have to go through the rest of their
  short runs.
• Continued mortality this is a true/false flag.
  When true, it offers a variation on the above
  theme if an individual enters the preferred
  state it will remain there, except that it can
  still die (according to the death rate for the
  preferred state).
• Environmental percentages These determine what
  of the long runs will be in A, what will be in
  B, and what will be in N.

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Example of a long run in environment BThe
starting population consists of 100 individuals.
Each undergoes a number of short runs (ex. 6)
where they will use TB to determine any state
transitions.
A
B
AC
BC
C
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Every single individual in the population
undergoes a series of 6 short runs per long run,
unless it dies before its short runs are
completed.

Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Stay in state A
0.421
0.743
(One short run)
Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Stay in state A
0.623
0.523
Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Switch to state B
0.821
0.975
Switch states according to matrix column 1
Switch states according to matrix column 2
Random
Random
Stay in state A
Stay in state B
0.304
0.314
Switch states according to matrix column 1
Switch states according to matrix column 2
Random
Random
Stay in state A
0.502
0.992
Death
Switch states according to matrix column 1
Random
Stay in state A
0.502
End of short runs
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Alternatively, the simulation can be set up so
that the individual can exit from the short runs
early if it enters into the preferred state for
that environment after a certain minimum number
of short runs has been completed (in this example
the of short runs is 6 and the minimum number
of short runs is 1).

Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Stay in state A
0.421
0.743
(One short run)
Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Stay in state A
0.623
0.523
Switch states according to matrix column 1
Switch states according to matrix column 1
Random
Random
Stay in state A
Switch to state B
0.821
0.975
Switch states according to matrix column 1
Switch states according to matrix column 2
Random
Random
Stay in state A
Stay in state B
0.304
0.314
Switch states according to matrix column 1
Entered into preferred state end of short runs
Random
Stay in state A
0.502
Switch states according to matrix column 1
Random
Stay in state A
0.502
End of short runs
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Once the series of short runs has been completed
for each individual in the population, the long
run is complete. The overall effect of the long
run might be as follows
A
A
B
B
AC
AC
BC
BC
C
C
D
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If these long runs were repeated, eventually
everyone would die. Therefore, after every long
run the dead individuals are redistributed, and
get added to the live states. This is done in a
way which does not alter the proportions between
the states (if 30 of the live individuals are in
state A, 30 of the dead individuals will be
added to state A).
A
A
B
B
AC
AC
BC
BC
C
C
D
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Representation of Results
After the entire sequence of long runs, either
the C populations or the A/B populations will
tend dominate (In some situations where they are
equally favoured). The history of the sequence
of long runs is displayed in barycentric
coordinates in the 2-simplex. Vertices are
labeled by the three populations and points are
plotted with the relative distance from the base
opposite a vertex to the vertex equal to the
proportion of the corresponding population (ie.
The closer a point is to vertex C, the higher
proportion C is of the total).
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Example
A
(a0.6, b0.3, c0.1)
0.1
0.3
0.6
B
C
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Some Results
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Testing the Effect of Number of Short Runs

• The short runs are the number of times an
  individual will attempt to shift states per long
  run.
• All other parameters were kept constant while the
  of short runs was modified. The values of the
  other parameters are shown on the screenshot to
  the right.

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Number of Short Runs 1
A
C
B
40
Number of Short Runs 10
A
C
B
41
Number of Short Runs 13
A
C
B
42
Number of Short Runs 20
A
C
B
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Testing the Effect of Minimum Short Runs

• The minimum short runs is used to determine how
  many short runs an individual must go through
  before remaining in the preferred state.
• All other parameters were kept constant while the
  min. short runs was modified. The values of the
  other parameters are shown on the screenshot to
  the right.
• The results are displayed in barycentric
  coordinates on the following sheets, with the A
  population as the top vertex, the B population as
  the left vertex, and the C populations as the
  right vertex.

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Minimum Short Runs 1
45
Minimum Short Runs 5
46
Minimum Short Runs 9
47
Minimum Short Runs 11
48
Minimum Short Runs 15
49
Minimum Short Runs 20
50
Testing the Effect of Directionality Power

• The directionality power is used to determine dA
  and dB in the transition matrices.
• All other parameters were kept constant while the
  directionality power was modified. The values of
  the other parameters are shown on the screenshot
  to the right.
• The results are displayed in barycentric
  coordinates on the following sheets, with the A
  population as the top vertex, the B population as
  the left vertex, and the C populations as the
  right vertex.

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Directionality Power 1.0
A
C
B
52
Directionality Power 5.0
A
C
B
53
Directionality Power 10.0
A
C
B
54
Directionality Power 15.0
A
C
B
55
Directionality Power 20.0
A
C
B
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Testing the Effect of m

• m is the percentage of individuals in the c
  state which will stay in it (not shift to ac, bc,
  or death) after one short run.
• All other parameters were kept constant while m
  was modified. The values of the other parameters
  are shown on the screenshot to the right.
• The results are displayed in barycentric
  coordinates on the following sheets, with the A
  population as the top vertex, the B population as
  the left vertex, and the C populations as the
  right vertex.

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m 0.1
58
m 0.25
59
m 0.4
60
m 0.5
61
m 0.7
62
Testing the Effect of rAB

• rAB is the percentage of individuals in the a or
  b state which will not switch states after one
  short run.
• All other parameters were kept constant while rAB
  was modified. The values of the other parameters
  are shown on the screenshot to the right.
• The results are displayed in barycentric
  coordinates on the following sheets, with the A
  population as the top vertex, the B population as
  the left vertex, and the C populations as the
  right vertex.

63
rAB 0.10
64
rAB 0.60
65
rAB 0.70
66
rAB 0.80
67
rAB 0.90
68
rAB 0.95
69
rAB 0.9875
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Testing the Effects of Environment Percentages

• The environment percentages determine which
  environment will be used for a particular long
  run. If A 0.4, B0.4, and N0.2, 40 of the
  long runs will be in the A environment, 40 will
  be in the B environment, and 20 will be in the N
  environment.
• To determine the effect of modifying these
  percentages, a program was created which would
  test thousands of different sets of percentages
  by running a simulation for each. The results of
  each simulation (on every set of percentages)
  were put into three different categories
  converges to C, converges to A and B, or
  converges to neither repeatedly. All OTHER
  parameters are kept constant when running these
  simulations.

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Testing the Effects of Environment Percentages

• This overall program implements an environment
  test. One environment test involves up to 5150
  simulations with different environmental
  percentages. To see if modifying the
  environmental percentages has a predictable
  effect, several environment tests were conducted,
  each of which modified OTHER parameters.
• The environmental percentages are given by 3
  parameters, hence the results can be represented
  as a barycentric point in the same way that
  populations are represented. Thus, all of the
  environmental percentages for a particular
  environment test can be displayed as points on a
  barycentric graph, color coded according to which
  category they belong to. Alternatively, only
  those percentages which fall into a particular
  category can be displayed. The barycentric graph
  has the N percentage equal 1 at the top vertex,
  the A percentage equal 1 at the left vertex, and
  the B percentage equal 1 at the right vertex.

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Displaying different categories as different
colors

• Red points are environmental percentages for
  which the simulation converged to A or B. Blue
  points are those for which the simulation
  converged to C. Purple points (difficult to see,
  on the boundary between red and blue points) are
  those for which the simulation fals to repeatedly
  converge to A and B or to C.
• This is a somewhat difficult view. A clearer
  picture is obtained by filtering points so that
  only those for which the simulation does not
  converge (the boundary between the 2 convergent
  categories) are displayed.

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Displaying only those percentages for which
simulation fails to reliably converge
Points above this boundary curve converge to A or
B while points below the curve converge to C.
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Environment Test 1

• The results are displayed in barycentric
  coordinates on the following sheets, with the N
  percentage as the top vertex, the A percentage as
  the left vertex, and the B percentage as the
  right vertex. Only those points for which the
  simulation did not reliably converge are shown.

75
Environment Test 1
76
Environment Test 2

• This environment test is the exact same as test
  1, except it has a larger directionality power.
• The results are displayed in barycentric
  coordinates on the following sheets, with the N
  percentage as the top vertex, the A percentage as
  the left vertex, and the B percentage as the
  right vertex. Only those points for which the
  simulation did not reliably converge are shown.

77
Environment Test 2
78
Environment Test 3

• This environment test is similar to test 1,
  except it has more short runs and a higher
  directionality power.
• The results are displayed in barycentric
  coordinates on the following sheets, with the N
  percentage as the top vertex, the A percentage as
  the left vertex, and the B percentage as the
  right vertex. Only those points for which the
  simulation did not reliably converge are shown.

79
Environment Test 3
80
Environment Test 4

• This environment test is quite different from the
  previous ones. It has minimum short runs set to
  the same as short runs, so no early exits are
  allowed if the preferred state is reached.
  Moreover, it has a very high directionality
  power, which means that almost all of the C
  populations will end up in the preferred state.
• The results are displayed in barycentric
  coordinates on the following sheets, with the N
  percentage as the top vertex, the A percentage as
  the left vertex, and the B percentage as the
  right vertex. Only those points for which the
  simulation did not reliably converge are shown.

81
Environment Test 4
82
Predicting the Boundary Curve
It is possible to estimate the winning
population. Group A and B populations into one
class and the C, AC, and BC populations into a
second class. Estimate the overall mortality
rate for each class. The predicted winning
population will be that with the lowest mortality
rate. The predicted boundary curve is determined
by setting the mortality rates equal.
83
Predicting the Boundary Curve
The necessary information to do this is (a) the
long run mortality rate for each population in
each environment (b) the percentage of each
environment in the total sequence and (c) the
average proportion of each population type in its
assigned class.
84
Predicting the Boundary Curve
The environmental percentages for (b) are
specified a priori. To obtain the long run
mortality rates and the relative proportions of
each population type the matrices
are computed. For an environment E these will
have the form
The relative proportions of each population type
are estimated from the normalized eigenvectors of
the matrices M(A,B) and M(C). Long run mortality
rates are given in the bottom row.
85
Comparison of Simulation and Prediction
Test 1
Test 2
Test 3
Test 4
86
Mathematical representation of simulation

• The simulation can be modeled Each environment
  has a characteristic transition matrix T used for
  the short runs. From these we calculate a
  transition matrix for an entire long run in a
  particular environment (call these matrices LA,
  LB, and LN).
• If the sequence of environments was AABNAB. ,
  then, neglecting the redistribution of the dead
  individuals, the overall transition matrix of the
  whole simulation would be .LBLALNLBLALA

87
Creating a transition matrix for an entire long
run in A.

• Assume that there are n short runs per long run
  and m minimum short runs per long run. We
  already have a matrix that models the simulation
  for the first m short runs (the TA introduced
  earlier).

88
Matrix for entire long run

• Now we need a matrix for the remaining n-m
  short runs.
• If continued mortality is false, then all of the
  individuals in states A and AC get stuck there.
  This is modeled as follows

89
Matrix for entire long run

• If instead continued mortality is true, then all
  of the individuals in states A and AC cannot move
  to the other live states but still die at the
  normal rate. This could be modeled as follows

90
Combined Matrix for Long Run in A

• If continued mortality is false
• LA (TA)n-m(TA)m
• If continued mortality is true
• LA (TA)n-m(TA)m

91
Combined Matrix for Long Run in B

• If continued mortality is false
• LB (TB)n-m x (TB)m
• If continued mortality is true
• LB (TB)n-m x (TB)m

92
For a long run in N

• There is no preferred state
• TN TN TN, LN (TN)n

93
Using the long run matrices to model the
simulation

• The transition matrices for the long runs can be
  used to model a simulation with a given sequence
  of environments in two ways
• Apply them in the same sequence to a population
  vector.
• Multiply them together in the same sequence to
  get a combined transition matrix for the whole
  simulation.
• The catch is that the redistribution of the dead
  step must also be taken into account.

94
Model of Simulation

• The population after the ith long run can be
  expressed in a series
• Pi R(LiPi-1)
• where R is the redistribute dead function and Li
  is either LA, LB, or LN depending on what
  environment the ith long run is in.
• If the sequence was AABN, then
• P1 R(LAP0)
• P2 R(LAP1) R(LA R(LAP0))
• P3 R(LBP2) R(LBR(LA R(LAP0)))
• P4 R(LNP3) R(LNR(LBR(LA R(LAP0))))
• .

95
Redistribute the Dead Function

• Where M is the combined matrix up to that point.
• Therefore, the overall combined matrix for the
  simulation with sequence AABN. would be
• S .R(LNR(LBR(LAR(LA))))
• You work from the inside out to calculate it (can
  be done iteratively).

96
A formula to predict a winner

• By predicting the death rates for the A/B
  populations and the AC/BC/C populations, it is
  possible to predict which group will grow to
  dominate.
• There are three things we need to know in order
  to do this is
• The death rate per long run of individuals in
  each of the states for each environment.
• The percentage of long runs that are in each of
  the three environments.
• On average, the relative proportion of A versus B
  and the relative proportions of AC versus BC
  versus C at the start of every long run.
• (ii) is obvious, its one of the simulations
  parameters (so long as the sequence of
  environments is determined by these percentages
  and not by a Markov process).

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(i)

• The death rate per long run for each of the
  states can be found easily from the long run
  matrix for each environment. The last row of the
  matrix gives the death rate for individuals
  starting in each state per long run.

Death rate of A
Death rate of B
Death rate of AC
Death rate of BC
Death rate of C
98
(iii)

• To determine what the death rate for the A/B
  population as a whole in a given environment is
  we need to know, on average, what proportion of
  A/B is A and what proportion is B (similarly for
  the AC/BC/C population).
• It is possible to predict what proportion of the
  A/B is A and what is B after many long runs in
  one environment. By getting a weighted (based on
  (ii)) average of these for the three
  environments, we would then have a fairly good
  estimate of what proportion of A/B is A and what
  is B on average (again similarly for AC/BC/C).
• So how do we find that proportion for many long
  runs in one environment?

99
From the long run matrix

• The linearly normalized eigenvector (for the
  largest eigenvalue) of the shown 2x2 submatrix of
  the long run matrix gives the relative
  proportions of A and B to each other after many
  long runs in this particular environment.
• The linearly normalized eigenvector (for the
  largest eigenvalue) of the shown 3x3 submatrix of
  the long run matrix gives the relative
  proportions of AC, BC, and C to each other after
  many long runs in this particular environment.

100
Calculating those relative proportions

• Let pABi be the eigenvector of the 2x2 sub-matrix
  of Li and pCi be the eigenvector of the 3x3
  submatrix of Li, where i is A, B, or N.
• Let pAB be the vector representing the proportion
  of A/B that is A and that is B and pC be the
  vector representing the proportion of AC/BC/C
  that is AC, BC, and C on average.
• Let A be the percentage on long runs that are in
  environment A, etc.
• Then pAB ApABA BpABB NpABN
• Then pC ApCA BpCB NpCN
• Note that pAB (pA,pB) and pC (pAC,pBC,pC),
  where
• pApB 1 and pACpBCpC 1.

101
Death rates in each environment

• So now we know (i), (ii), and (iii).
• Let DABi be the death rate of the combined A/B
  population in environment i (where i is A, B, or
  N) and DCi be the death rate for the combined
  AC/BC/C population.
• Let dAi, dBi, dACi, dBCi, and dCi be the death
  rates per long run of each of the states in
  environment i (this comes from last row of Li as
  described earlier).
• Then DABi pAdAi pBdBi and
• DCi pACdACi pBCdBCi pCdCi

102
Overall Death Rates

• Let DAB be the overall death rate for the A/B
  population and DC be the overall death rate for
  the AC/BC/C population.
• DAB ADABA BDABB NDABN
• DC ADCA BDCB NDCN
• If DCltDAB, then virtual stability wins, otherwise
  traditional stability wins.
• Note that if DC is very close to DAB, then
  neither will win consistently.