This chapter deals with very simple dynamics in a population without disease. You have already seen some population dynamics models when you went through the "Getting Started with Stella II" part of the Stella manual. We elaborate a bit more on population dynamics here. This is a first step toward eventually putting population dynamics together with disease dynamics. An introduction to the process of defining model essence and desireable abstractions is provided by constructing a model which at first abstracts across the issue of age and sex and then constructing a model which takes age into account.
We examine first a simple population model with a birth rate and
a death rate. The difference between the two, however, is exactly
the overall growth rate since the rate of increase in the population
(births) and the rate of decrease in the population (deaths) are
made directly proportional to the current population size.
Let us look at this a little more completely and in so doing reiterate
the nature of the equations that the STELLA sets up when you point
and click and enter relationships to set up a model. Set up a
STELLA model of the following form. Note that the population stock
is a reservoir stock. This is the only kind of stock that generates
ordinary, first order, difference equations, as discussed below.
Make sure that you have the skills and knowledge needed to do the following. If you have any questions, do not leave the professor sitting lonely in his office during his office hours.
1 Build the above model. In doing so set the births equal to the population times the birth rate and the deaths equal to the population times the death rate.
a What are the parameters of this model? What are the variables? What are the compartments?
b Set the initial populations size to one. Set the birth rate equal to the death rate first at a value of 0.1 and then at a value of 0.000001. Predict the behavior of the population size over time before you run the simulation. If you got a behavior that was different than the behavior you predicted, find an explanation for your incorrect prediction and please e-mail that explanation to jkoopman@sph.umich.edu. Repeat using a population size of 1,000,000. Explain why the population size behaves as it does in this simulation.
c Set the birth rate equal to 0.1 and the death rate equal to 0.01. Again use initial population sizes of 1 and 1,000,000. Predict the behavior of the population size over time before you run any of the simulations. Again if your prediction was in any way different from what you observed, discuss this in your individual session. Present a graph of your simulation results and explain why the population size behaves as it does in this model.
d Set the birth rate equal to 0.01 and the
death rate equal to 0.1. Use initial population sizes of 1 and
1,000,000. Predict the behavior of the population size over time
before you run any of the simulations. Again if your prediction
was in any way different from what you observed, discuss this
in your individual session. Present a graph of your simulation
results and explain why the population size behaves as it does
in this model.
The number of births in a very short period of time "dt"
is proportional to the number people already in the population,
the birth rate, and the time interval. Using the Euler method
it is exactly proportional and using a Runge-Kutta method it is
almost proportional. The birth rate has the units of the number
of births per person-time. Because this is multiplied by the number
of persons and the time, the two elements of the denominator of
the birth rate are canceled out and the final units of births
per unit time. In the formula for the changing population size
this births per unit time is multiplied by the time "dt"
giving the number of births over the time "dt".
The number of deaths in a very short period of time is proportional
to the number people already in the population, the death rate,
and the time interval. Stella™ writes out the equations as
follows:
Population(t) = Population(t - dt) + (Births - Deaths) * dt
INIT Population = 1000
Births = Population*Birth_Rate
Deaths = Population*Death_Rate
Birth_Rate = .1
Death_Rate = .01
This looks slightly different than on your screen because the
inflow and outflow symbols are not printed. These equations come,
however, from saving the equations from the STELLA® II simulation
in a text file and then importing them into a MSWord™ file.
By substituting the flows into the original equation (and using
P for Population), we can express the first equation as follows:
P(t) = P(t - dt) + {P(t - dt)*Birth_Rate - P(t - dt)*Death_Rate}
*dt
This can be reduced to
P(t) = P(t - dt)*(1 + {Birth rate - Death rate}*dt)
The birth rate minus the death rate we can treat as the population
growth rate. We lable this growth rate with an "a".
In this case this difference equation reduces to the following
simple form where P is used for population:
.....................................Equation
(1).
This is the very simplest difference equation that is possible.
Because one of my goals for this course is to increase your capacity
to relate to mathematicians as you consider epidemiologic problems,
let us denominate this equation in the way that mathematicians
do. It is called an autonomous, scalar, homogeneous, dynamically
linear, first order, deterministic, continuous valued, difference
equation. You can probably impress your non-mathematical friends
by telling them that you know about autonomous, scalar, homogeneous,
first order, deterministic, continuous valued, difference equations.
But they are in fact the very simplest equations you will deal
with.
Autonomous refers to the fact that the parameter values affecting
dynamics are not a function of time. All changes in dynamics over
time are "autonomous". They are not set by the modeler.
The modeler only sets parameter values. In this case the which
the population growth rate "a" is independent of time.
If it changed by time, for example if it went up in the winter
and down in the summer, or if it continuously increased or decreased,
we would have a non-autonomous equation. But in this case "a"
is the same now as in the future. Later in the course, when we
build models of disease control programs, we will make parameters
change at different times as control programs are instituted.
Scalar refers to the fact that the entity whose size is being
expressed by the equation has only one dimension. We could have
built a model where not just the overall population size was being
generated but where the size of each of 10 different age groups
was being expressed. The set of 10 numbers would be called a vector.
A set of equations for each of the 10 different age groups could
be expressed in a more compact form called a vector equation.
Homogeneous refers to the fact that the overall rate of change
has only one element and that this is directly multiplied by the
current population size. In a population model, it is possible
that you might have more than one source of growth to the population.
It could be that there is another population that keeps feeding
in 100 people per time unit into the population you are monitoring
through an immigration process. The equation would then be:
Population(t) = Population(t - dt)*(1+a*dt) + 100dt ...................................
Equation (2)
This is a "non-homogenous" equation.
One STELLA® II model of this form is presented below:
STELLA® would write out the equations as follows:
Population(t) = Population(t - dt) + (Births + Migration - Deaths) * dt
INIT Population = 1000
Births = Population*Birth_Rate
Migration = 100
Deaths = Population*Death_Rate
Birth_Rate = .1
Death_Rate = .01
another would be as follows:
whose equations are:
Population(t) = Population(t - dt) + (Births - Deaths) * dt
INIT Population = 100
INFLOWS:
Births = 10
OUTFLOWS:
Deaths = Population*Death_Rate
Death_Rate = .01
Make sure that you have the skills and knowledge needed to do the following. If you have any questions, do not leave the professor sitting lonely in his office during his office hours.
2 Predict the behavior of this model using
the graph below. Run the model and explain the model behavior
and any difference between your prediction and the actual model
behavior. (Note the scale of the following graph and be sure to
draw something on this graph before you run a simulation to discover
the implications for population growth of the model system you
have constructed.)
First order refers to the fact that what happens to a population
depends only upon the state of any variables in the system at
the time step before. In our case of "dt" being small,
that is an instant. A second order equation would be one where
what happens to the system depends not only on the state of the
system at one previous time period, but on an earlier time period
as well. For example, one might be modeling the interaction of
two different populations that have different reproduction cycles.
The number of trees in one year may depend on the number of trees
in the previous year, but the number of Cicadas in one year might
depend upon the number of Cicadas there were 17 years ago. To
get the next step, you would have to know what the state of the
system is now and 16 years ago. That means that you would have
to keep in the computer's memory the state of the system for each
of the past 16 years. That could be a considerable increase in
pieces of data for the computer to keep track of.
With first order equations, the computer can throw away the information
from the previous state as soon as it calculates the values for
the next state. It does not have to keep a lot of data in memory.
This nice characteristic allows one to simulate rather complex
systems on microcomputers. Simulations where the next state of
the system requires getting information on every state of the
system at every moment in the past of course require considerably
more computing power than the first order difference equations
we are dealing with. The simulation of many partial differential
equations does in fact require that information on all past states
of the system be kept in memory. Most of the big partial differential
equation models like weather models or airplane performance simulations
or the global AIDS model are done on supercomputers for that reason.
We now almost have supercomputers on our desk tops so I expect
very user friendly programs for numerical soulution of partial
differential equation problems to appear soon.
The software we are using allows us a couple of intermediate steps
between having to keep the state of the entire system at all time
points in memory and throwing away all the information the instant
the next step is calculated. One means is through the delay function.
The other is using a conveyor stock instead of a reservoir stock.
Later in the exercise for this session, we will illustrate the
use of the conveyor stock in a population model.
Dynamically linear refers to the fact that the flows are determined
by the size of only one compartment in the model and the compartment
size in the equation determining the flow is not raised to any
power or multiplied by itself. In the difference equations of
a dynamically linear system, no stock values are multiplied by
other stock values and no stock values are raised to any power.
Many non-infectious disease models are dynamically linear while
models of infection transmission are dynamically non-linear. Models
where the outcome in one individual does not depend upon the state
of any other individuals are usually dynamically linear. In the
population models we have looked at, you can think of the death
of any individual in the population not depending upon how many
people have died in the past.
You can see that here a "dynamically linear" equation
produces a curve that is not a straight line over time. Dynamically
linear refers to a property of the difference or differential
equations, not of the equations relating any two variables or
stocks. If a plot of two variables, such as time and population
produces a straight line, we say they have linear relationships.
Linear or non-linear dynamics refer to the nature of process.
Linear or non-linear relationships refer to the nature of data.
This course focuses upon process rather than data.
In most of the dynamically linear systems of disease development
that we will be dealing with in the early part of this course,
the flows out of a compartment will be determined by the size
of that compartment and the flows in will be determined by the
size of some other compartment. In the first population model
presented in this chapter, the flow in is determined by the compartment
size being flowed into. As long as the flows are directly proportional
to the size of one and only one compartment, the system is dynamically
linear.
Most of the interesting things that happen in the real world arise
from non-linear dynamics. Chaos theory and the new science of
complexity are greatly advancing our understanding of non-linear
systems. In the past most of the abstractions made in science
were dynamically linear models. Almost none of the real world
is really dynamically linear but such abstractions were necessary
in order to achieve manageable mathematics. Almost all of the
statistics you learn in the course of your epidemiology career
assume that the dynamics producing data are linear. For non-infectious
diseases this is reasonable. When one is interested in the transmission
of infections, however, this is not reasonable. The modern age
of computers is helping epidemiologists who are not well trained
in sophisticated mathematical analysis to relate more scientifically
to non-linear systems.
The population size modeled takes on continuous values that can
have any decimal value you can possibly construct. If the population
could only take on discrete values, we would say the model has
a discrete state space.
The fate of any entity in our model is determined by our model.
Every time you run the model, the fate of the entities in the
model will be the same. Their is no chance involved. There are
no probability distributions modeled for the fate of individuals.
Models with chance elements or that describe probability distributions
are stochastic models. As we saw in Chapter 2, under special circumstances
the outcomes of our deterministic simulations can be interpreted
in terms of probability distributions. The special circumstances
are when some initial stock for which there are only outflows
and no inflows is assigned a value of one while all other stocks
are assigned a value of zero. The negative exponential, the Erlang,
and the Weibull probability distributions are distributions that
we can easily generate using this software. But for most of the
uses to which we will be applying our models, we will not be able
to interpret the outcomes in terms of probability distributions.
Difference refers to the fact that these equations express what
happens across discrete time periods. These are the "dt"
in the equations. As "dt" tends toward zero, we enter
the world of calculus and the type of equation that we are dealing
with would change into a differential equation. Whereas we use
discrete time steps, by following the rule that we always check
to see that halving the time step does not meaningfully change
the values of stocks over time, we are essentially making our
simulations correspond to those of continuous time differential
equations.
Note that merely calling a model discrete or continuous is ambiguous
because discrete could refer to the time steps used or to the
entities modeled. We use discrete time steps to model continuous
entities. As discussed in Chapter 2, we use discrete time steps
in a fashion that is equivalent to having a continuous time model.
The model we have constructed, like most models we will deal with in this course, is a compartmental model. That means that when a modeled entity flows from one compartment to another it keeps the same units. We are usually modeling populations. In our models individuals in the populations change status and this corresponds to the flow of population from one compartment to another. But the individuals are preserved. Many mathematical models in physics are not compartmental models. Models where energy gets converted into work, for example, are not compartmental models. It is quite possible to construct non-compartmental models using the Stella® software. One way to do this is to make the flow out of one compartment translate through some mathematical formulation into the flow into another compartment.
The Stella® diagram of a compartmental model has all stocks
of the basic entities being modeled connected by flow pipes so
you can think of there being a total quantity of the entity being
modeled, such as population. That total quantity is divided into
compartments defined by the state or location in which the entity
finds itself. A compartmental model does not have transformations
of the entity being modeled which change the units in which it
is measured.
In constructing models of a disease in a population that has births
and deaths, or in migrations and out migrations, we will often
find it convenient to have the total population size at equilibrium.
One way to do that is to set the birth rates equal to the death
rates or the immigration rate equal to the emigration rates. That
gives us a model like the first one in this chapter. We have seen
that models of this type grow explosively (exponentially) or shrink
to nothing unless the birth rate is set exactly equal to the death
rate.
The equilibria achieved in this way is an unstable equilibria.
The slightest movement of the net growth rate (birth rate minus
death rate) away from zero will throw the system into an eventually
explosive growth or a continually contracting situation. When
the net growth rate in this type of model is zero, if you change
the number of people in the population, the population will not
eventually return to its original equilibrium value. It will stay
at whatever size to which it was changed. In this type of model,
the characteristics of the system are not generating the equilibrium.
There are no feedbacks in the system that come into play when
something gets out of bounds. It is a delicate balance between
birth and death rates that is creating the equilibrium. Thus in
this type of population model, a population at equilibrium can
exist -- but not in a stable fashion. If some small influence
should increase the net growth rate, then the population size
will eventually have explosive exponential growth. If something
should decrease the net growth rate, the population will shrink
to zero. Thus a population model like we have built is intrinsically
unstable.
A system with a stable equilibrium is one where if you knock it
off that equilibrium it will return to that equilibrium. If you
change the rates that determine the equilibrium, you may change
the equilibrium value, but you will not change the ability of
the system to return to that equilibrium value if you knock it
off that equilibrium.
If you have a pendulum that can swing through 360 degrees, it
is possible that it could come to rest exactly at the vertical
up position. But any movement will knock it down. That is the
kind of unstable equilibrium we have when we set birth rates equal
to death rates. When a pendulum is hanging down with no motion,
it is at a stable equilibrium. If you move it, it will swing back
and forth but eventually reach the position of no motion from
which you originally disturbed it. Now let us examine some models
with a stable equilibrium such as the pendulum hanging vertically
down.
What it takes to get a stable equilibrium here is feedback from
the population level to affect the birth or death rates. It seems
likely that as the population size increases, there will be forces
that increase the death rate, and or decrease the birth rate.
These might relate to food availability, to increased circulation
of infectious diseases, or the greater stress of having to live
close to other people. The model in the Stella™ getting started
manual made the death fraction a function of the population size.
This changed the model from one with no equilibria to one where
the same equilibria was reached no matter what the starting point
was.
Make sure that you have the skills and knowledge
needed to do the following. If you have any questions, do not
leave the professor sitting lonely in his office during his office
hours.
3a
Construct a model similar to the one in the Stella™ manual that achieves equilibria by changing the death fraction as a function of population size, but do not use a graphical function. Make the death rate equal to some constant "d" plus some other constant "c" times the population size (Death rate = d + c*Population). Make the birth rate equal to some constant "b". Your diagram will thus look like the following:
The equations for this model as set up by Stella™ and copied
into the word processor are:
P(t) = P(t - dt) + (Births - Deaths) * dt
INIT P = 1
INFLOWS:
Births = P*Birth_rate
OUTFLOWS:
Deaths = P*Death_Rate
b = .1
Birth_rate = b
c = .0001
d = .01
Death_Rate = d+c*P
What are the parameters of this model? What
are the variables? How do you classify death rate?
3b
Run your simulations at a wide range of
values for b, c, and d. Before every run, predict the shape
of the curve to be generated and be sure you understand why any
differences from what you predicted occured. Present any conclusions
you feel are justified from these explorations. Specifically describe
the shape of the curves as they come to equilibrium from either
initially very high values or initially very low values. When
you start off with a high P, be very careful that your flows are
not a high percentage of the stocks from which they are flowing.
Be sure also to check that changing your dt does not change your
model results.
Send 3c. 3d, and 3e via FTP
3c
Make a similar model but leave the death rate constant at "d" and change the birth rate as a function of population size by again making it equal to some constant "b" minus some constant "c" times the population size (Birth rate = b - c*Population). Your diagram will thus look like the following:
3d
What are the parameters of this model? What
are the variables? How do you classify birth rate and death rate?
3e
Run your simulations at a wide range of values for b, c, and d. Present any conclusions you feel are justified from these explorations. Specifically describe the shape of the curves as they come to equilibrium from either initially very high values or initially very low values. Explain why the curves have the shapes that they do in terms of where they are increasing or decreasing and concave or convex.
The population will be at equilibria when the flows into the population
equal the flows out of it. We saw that this could be achieved
by making the birth rate equal to the death rate or by any of
the three approaches in the last problem. If we set the flows
into the population equal to those out of the population for any
of the above three problems, we can solve the resulting algebraic
equation for the population size at that point where the flows
are equal and the population is not changing. For example, in
the first model where death rate is a function of population size:
For example, in model Q where the population size affects the
death rate but not the birth rate: Flow in = b*P = Flow out =
(d+c*P)*P which after canceling out Population from both sides
is b = d + c*P. So that at equilibrium P = (b - d)÷c.
Send 4 via FTP
4a
Demonstrate that the above formula for the equilibrium value is an equilibrium value and that it is locally stable so that if you knock the Population off equilibrium, it will come back to the same equilibrium. Do this by entering the formula into the starting value for the population rather than a number. Then use an if then statement in the birth and death flows to make one time flows into or out of P when it is at equilibrium. Next solve for the equilibrium values in terms of b, c, and d for the second model where the birth rate is a function of population size. Repeat the demonstration you did for the first model.
4b
Try to solve for the equilibrium in the
very first population model presented where "b" is the
birth rate and "d" is the death rate. Why can't an equilibrium
value be found by this technique in this case?
Either of the first two models in exercise 3 can be recast as
where "a" is "b - d". The "a" parameter
defines the baseline growth rate in a population. This rate will
be decreased as a function of the parameter "c" times
the population size. Note that for model A the c*P term is added
to the death rate but since that is an outflow it enters as a
negative term in the overall difference equation. In model B
the c*P term is entered as a negative term in a positive inflow.
Recast in terms of a differential equation, equation (3) becomes
This is the differential equation formulation of the logistic function. Please make sure you are adept at going between equations (3) and (4). Also you should be come adept at reading either difference or differential equations and constructing the Stella diagrams that correspond to them. The Stella diagram for either (3) or (4) is as follows:
Note that P will not enter into the inflow in the same way that
it will enter into the outflow. In the outflow it will enter as
a squared term. The equations Stella™ wrote for this after
we entered our formulas into the flow regulators are:
P(t) = P(t - dt) + (Inflow - Outflow) * dt
INIT P = 1
INFLOWS:
Inflow = P*a
OUTFLOWS:
Outflow = P^2*c
There are many different ways we might go about choosing a value
for the parameter "c". The way that we use will depend
upon the purposes to which we are applying the model. We are using
this model mainly to introduce some conceptual issues in dynamics
and epidemiology and not for analyzing the behavior of a specific
population. We are not necessarily seeking a biologically or sociologically
meaningful parameter. Therefore our choice of parameter values
is arbitrary. We could define a and c in terms of the equilibrium
population size desired. Setting the inflows equal to the outflows
in equation (2) which is the same as the Stella™ formulas,
we see that
P(at equil) = a÷c.
Similar if using equation (4) we can set the rate of change in
P with respect to time (dP/dt) to zero. When P is not changing
we are at equilibrium. Solving (4) with dP/dt = 0 we get
P(at equil) = a÷c.
Thus to get behavior that comes to a particular equilibrium value,
there are an infinite number of "a" and "c"
parameters that can be used.
For your own edification:
Start from P = 1 and examine the curves
followed reaching equilibrium when a=0.1 and c=0.0001 or when
a= 0.01 and c=0.00001. Make sure you are able to explain the differences
in behavior in these two situations.
With the logistic function, as the population size increases,
the growth rate becomes less and less until it becomes zero or
negative. If the population size starts out very low, then the
growth rate will be positive and the population will grow. If
the population size starts out very large, the growth rate will
be negative and the population will shrink. It will always settle
down to the same size. Let us examine mathematically the equilibrium
to which the system settles.
Equation (4) is the basis of what is called in mathematics the logistic equation. Just as there was a closed form solution for the cumulative risk as a function of the rate and time, there is a closed form solution for the logistic equation. As we let dt tend toward zero, it is possible using calculus to show that
............................. Equation
(5)(The equation editor is screwing up. There is not supposed to be any squigle or infinity sign in the closing parentheses in the denominator. Why the editor is changing the size of some entries and what I can do about it is beyond me.)
When c equals zero you note that we have a simple exponential
growth situation
which is the same as the closed form solution presented for exponential
growth in the last chapter. When c does not equal zero, then as
t goes to infinity, we can see that e-at would
go to zero and the population size would go . The value is the
same equilibrium value you should have calculated by setting the
flows equal to each other in problem 4.
We can also determine equilibrium values using our difference equations. That is essentially what we did earlier when we set the inflows equal to the outflows. We do this again here in a slightly different way that might help your understanding by recognizing that at equilibrium nothing is changing. We can rewrite equation (2) as
..........................Equation
(2alternate)
When the population is at a constant equilibrium, Population(t)
= Population(t - dt) = Population(Equil) and the left hand side
of this equation must equal one.
Solving this equation we see that the constant population size
at equilibrium is
.........................Equation
(6)
You can see that the population will return to this level no matter
whether it starts way above it, or way below it.
For many simulation purposes, we want to start our system out
at equilibrium, let it run at equilibrium for a while to assure
ourselves that it is at equilibrium, and then do something to
it and see how the system behaves. This is the absolute simplest
model in which we can do this. Later we will have more complex
models, some of which may have stable equilibria and some of which
may not. In general you will only seek initial values for equilibria
in first order equations. Higher order equations often have stable
cyclic phenomenon where there is a long term stable pattern of
regular fluctuations, but the mathematical analyses of these is
beyond the scope of this course.
Make sure that you have the skills and knowledge needed to do the following. If you have any questions, do not leave the professor sitting lonely in his office during his office hours.
a) Using the models you constructed in problem
3. set the initial level of the populations to their equilibrium
size by defining a formula for the initial value in terms of the
parameters "b", "c", and "d". Then
using the pulse function (check your manual to learn how to use
this), knock your system off equilibrium a few time steps into
your simulation and see how it behaves. (Include the algebraic
calculations you used to calculate the initial population equilibrium
in your answer.)
b) At two time points before equilibrium
in your above simulations, calculate the expected population size
using the closed form solution of the logistic equation and compare
it to your simulation values at those times.
One way that the population models we have constructed so far
are unrealistic is that they disregard the effect of age. Another
way that they are unrealistic is that they do not take sex into
account in the reproduction process. Sex is always a very complicated
issue so for now we will just consider how to make our population
model more realistic by considering age. The question to be addressed
is whether the conclusions you drew about popultion behavior in
problems 3c would be different if you had taken age into account.
First enumerate each of your conclusions from 3c. The issue now
is which conclusions are dependent upon which age effect assumptions
in the 3c model.
Before proceeding with any further model construction, try to list all the ways that you think your conclusions in 3c are dependent upon the following assumptions about age in 3c
To address either of the above assumptions, we will have to break
our single population into different age group compartments. The
more age groups into which we divide the population, the more
realistic our model is going to be. To be highly realistic we
might divide the life span into weeks and set a reproductive rate
and a death rate for each week of life. But for now, let us just
consider 3 age groups. A1 will be defined as a pre-reproductive
age group. Let the average time spent in this age group be 15
years A2 is a reproductive age group. Let the average time spent
in this age group be 30. A3 is a post-reproductive age group.
Let the average time in this age group before death be 30.
In order to address the first assumption about births independent
of the second assumption about deaths, let us construct a model
where the death rate is the same for all ages but only the size
of A2 affects the number of births. Such a model is presented
below.
A1(t) = A1(t - dt) + (Births - Initiations - A1deaths) * dt ................INIT A1 = 15/75
INFLOWS: Births = A2*BirthRate
OUTFLOWS: Initiations = A1/A1Duration; A1deaths = A1*DeathRate
-----------------
A2(t) = A2(t - dt) + (Initiations - Maturations - A2deaths) * dt .........INIT A2 = 30/75
INFLOWS:Initiations = A1/A1Duration
OUTFLOWS:Maturations = A2/A2Duration; A2deaths = A2*DeathRate
-----------------
A3(t) = A3(t - dt) + (Maturations - A3deaths) * dt .......................INIT A3 = 30/75
INFLOWS:Maturations = A2/A2Duration
OUTFLOWS:A3deaths = A3*DeathRate
-----------------
A1Duration = 15
A2Duration = 30
b = .1*(75/30)
BirthRate = b
c = .0001
d = .01
DeathRate = d+c*(TotalP)
TotalP = A1+A2+A3
FTP your answer to 8
Write the differential equations for the
model in 8 directly in terms of the parameters b, c, and d (don't
use birth rate or death rate). Use 15 and 30 for the durations
of A1 and A2 rather than writing out A1Duration and A2Duration
in your formulas.
To find the equilibrium values for the three compartments in this
model, we will have to set up 3 simultaneous equations and solve
them. The solutions in this case are possible but extremely difficult.
With only a little more complication to the model, it becomes
completely impossible to algebraically find equilibrium solutions.
With many models, there may in fact be no constant equilibrium
that is reached. It is possible that model could continuously
cylce. For example, if you have foxes eating rabbits so that fox
population growth rates depend upon rabbit population sizes and
rabbit population growth rates depend upon fox population size,
the two population sizes might continuous fluctuate. In some models,
the sizes of compartments might fluctuate but in ever dampening
cycles so that eventually after an infinit period of time they
reach a single constant equilibrium.
Answer the following on your own. Check with the professor if you have problems:
Is the total population size in this model
cyclical at equilibrium? Are the age group sizes cyclical at equilibrium?
To answer this question, try to find population sizes for A1,
A2, and A3 and for the parameters b, c, and d where the total
population or any age group will continually oscillate. If
you find some that generate continuous oscillations, FTP the model!
To address the question about whether introducing age structure
will significantly modify population dynamics, we will compare
the model for 3c with the age structured model. For ease in comparison,
put both models on the same sheet.
FTP your answer to 10
10a
How are population dynamics different between the models with and without age groups??
10b
Can the population growth curves starting from a population size of one be made identical for the model with and without age groups by adjusting parameter values?
10c
Specify some issue regarding population
dynamics where the model without age groups could lead to wrong
conclusions.
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