Modeling Virtual Stability With a Population Simulation

Modeling Virtual Stability With a Population

SimulationBurton 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

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

The Assumption of StabilityThe 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

Discrete State Space ModelsDiscrete 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.

Discrete State Space ModelsRobert 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.

Discrete State Space ModelsStable 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.

Course GrainingState 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.

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.

Control and TransitionControl 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.

Control and TransitionTypically, 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.

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

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 perception—we 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.

Examples such as these illustrate the centrality of stability and instability for every area of complex systems theorizing.

Ashby’s 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.

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.


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.


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.

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.

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.

Virtual StabilityNot 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.

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.

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.

States & Environments

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

State A: high

death rate

State B: low death

rate

State AC: high

death rate

State BC: low death

rate


State A: medium

death rate

State B: medium

death rate

State AC: medium

death rate

State BC: medium

death rate


Stable

Stable

Stable

Virtually Stable

Virtually Stable

Virtually Stable

Env B

Env N

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

Long run in Env. A

New

P

opulation

New

P

opulation

New

P

opulation

New

P

opulation

New

P

opulation

…Long run in Env. A

Long run in Env. N

Long run in Env. B

Long run in Env. B

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 individual’s state will change (or remain the same) according to the probabilities specified in the environment’s transition matrix.

Sample Transition Matrix

• 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.

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

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

T =

A B AC BC C D

Transition Matrix A

• Since state A and state AC are favored, A is less than B and AC is less

than BC. Also, A is greater than B.

TA

AB AB 1 AB A 0 0 0 0

1 AB A ABAB 0 0 0 0

0 0 AC CA 1 C AC A 1 µ C 0

0 0 B 1 C AC CBC B 1 µ C 0

0 0 qB CqA µ 0

A B AC BC C 1

AB 1 B1 A

, BA 1 A1 B

, C 1 BC1 AC

A BCn

ACn

BC

n, B

ACn

ACn

BC

n

qA 1 AC 1 A C A B , qB 1 AC 1 B C A B qA 1 BC 1 A C A B , qB 1 BC 1 B C A B

Transition Matrix N

= B = A and BC = AC, B = A = 0.5.

TN

AB 1 AB A 0 0 0 0

1 AB A AB 0 0 0 0

0 01

2C

1

21 C AC 1

21 µ C 0

0 01

21 C AC 1

2C

1

21 µ C 0

0 01

21 AC 1

21 AC µ 0

AC AC C 1

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.

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

C

BC

0.97753 0.01 0 0 0 00.00998 0.98 0 0 0 0

0 0 0.11568 0.00118 0.08698 00 0 0.00879 0.86403 0.64802 00 0 0.86303 0.12479 0.25 0

0.0125 0.01 0.0125 0.01 0.015 1

TB =

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.

Random #

0.421

Switch states according to

matrix column 1 Stay in state A

Random #

0.623



Random #

0.821



Random #

0.304



Random #

0.502



Random #

0.502



End of short runs

Random #

0.743



Random #

0.523



Random #

0.975


matrix column 1 Switch to state B

Random #

0.314


matrix column 2 Stay in state B

Random #

0.992


matrix column 2Death

}(One short run)

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).

Random #

0.421



Random #

0.623



Random #

0.821



Random #

0.304



Random #

0.502



Random #

0.502



End of short runs

Random #

0.743



Random #

0.523



Random #

0.975


matrix column 1 Switch to state B

Random #

0.314


matrix column 2 Stay in state B

}(One short run)

Entered into preferred state: end of short runs

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

B

AC

C

BC

A

B

AC

C

BC

D

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

B

AC

C

BC

A

B

AC

C

BC

D

Representation of ResultsAfter 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).

Example

0.30.1

0.6

(a=0.6, b=0.3, c=0.1)

A

B C

Some Results

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.

Number of Short Runs = 1A

B C


B C

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.

Minimum Short Runs = 1

Testing the Effect of Directionality Power• The directionality power is used

to determine A and B 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.


Directionality Power = 1.0A

B C


B C

Testing the Effect of • 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 was modified. The values of the other parameters are shown on the screenshot to the right.


= 0.25

Testing the Effect of AB

• AB 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 AB was modified. The values of the other parameters are shown on the screenshot to the right.


AB = 0.10

AB = 0.60

AB = 0.70

AB = 0.80

AB = 0.90

AB = 0.95

AB = 0.9875

Testing the Effects of Environment Percentages

• The environment percentages determine which environment will be used for a particular long run. If A = 0.4, B=0.4, and N=0.2, 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.

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.

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.

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.

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.

Environment Test #1

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.

Environment Test #2

Environment Test #3• This environment test is

similar to test #1, except it has more short runs and a higher directionality power.


Environment Test #3

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.


Environment Test #4

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.


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.


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 LA TA

m msTAms

LB TBm msTB

ms

LN TNm

are computed. For an environment E these will have the form

LE M E (A,B) 0 0 0 0

0 0 M E (C) 0

A (E) B (E) AC (E) BC (E) C (E) 1

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.

Comparison of Simulation and Prediction

Test 1 Test 2

Test 3 Test 4

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

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).

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:

1 AB(1-AB-A) 0 0 0 0

0 AB

AB 0 0 0 0

0 0 1 C

A(1-C-AC) A(1--C) 0

0 0 0 C

B

C

B(1--C) 0

0 0 0 CqA

0

0 B 0

BC

C 1

TA' =

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:

1-A

AB(1-AB-A) 0 0 0 0

0 AB

AB 0 0 0 0

0 0 1-AC

C

A(1-C-AC) A(1--C) 0

0 0 0 C

B

C

B(1--C) 0

0 0 0 CqA

0

A

B

AC

BC

C 1

TA'' =

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

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

For a long run in N• There is no preferred state:

• TN = TN’ = TN’, LN = (TN)n

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:

i. Apply them in the same sequence to a population vector.

ii. 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.

• The population after the i’th 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 i’th 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))))….

Model of Simulation

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).

1 0 0 0 0 D1

0 1 0 0 0 D2

0 0 1 0 0 D3

0 0 0 1 0 D4

0 0 0 0 1 D5

0 0 0 0 0 0

R(M)= * M

Di = ( Mij )/( ( Mkj))j = 1

6

j = 1

6

k = 1

5

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:i. The death rate per long run of individuals in each of the states

for each environment.ii. The percentage of long runs that are in each of the three

environments.iii. 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, it’s one of the simulation’s parameters (so long as the sequence of environments is determined by these percentages and not by a Markov process).

(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.

0.918586 0.302283 0 0 0 0

0.042432 0.544182 0 0 0 0

0 0 0.9265 0.911885 0.916968 0

0 0 0.005614 0.007066 0.006252 0

0 0 0.006402 0.008058 0.00713 0

0.038982 0.153535 0.061485 0.072991 0.069649 1

LA =

Death rate of A

Death rate of B

Death rate of AC

Death rate of BC

Death rate of C

(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?

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.

0.918586 0.302283 0 0 0 0

0.042432 0.544182 0 0 0 0

0 0 0.9265 0.911885 0.916968 0

0 0 0.005614 0.007066 0.006252 0

0 0 0.006402 0.008058 0.00713 0

0.038982 0.153535 0.061485 0.072991 0.069649 1

LA =

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 = %A*pABA + %B*pAB

B + %N*pABN

• Then pC = %A*pCA + %B*pC

B + %N*pCN

• Note that pAB = (pA,pB) and pC = (pAC,pBC,pC), wherepA+pB = 1 and pAC+pBC+pC = 1.

Death rates in each environment

• So now we know (i), (ii), and (iii).• Let DAB

i be the death rate of the combined A/B population in environment i (where i is A, B, or N) and DC

i be the death rate for the combined AC/BC/C population.

• Let dAi, dB

i, dACi, dBC

i, 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 = pA*dA

i +pB*dBi and

• DCi = pAC*dAC

i + pBC*dBCi + pC*dC

i

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 = %A*DABA + %B*DAB

B + %N*DABN

• DC = %A*DCA + %B*DC

B + %N*DCN

• If DC<DAB, then virtual stability wins, otherwise traditional stability wins.

• Note that if DC is very close to DAB, then neither will win consistently.

Modeling Virtual Stability With a Population Simulation

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