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Answer the Following Questions Related to Ecology




1. Select
“Multi-Species Interactions” from the Model tab on the menu.


2. Select
“Lotka-Volterra Competition” from the pop-up menu.



Variables you control:


In the Lotka-Volterra competition
equations, there are 4 variables controlling the population growth rate (dN/dt):


1. N – the current size of
the population. In Populus, this is labelled N0, the population size at
some arbitrary time zero.


2. r – the intrinsic rate of increase (rmax = per capita birth
rate – per capita death rate under optimal conditions)


3. K – environmental carrying
capacity (which reflects the strength of intraspecific competition)


4. α and β − competition
coefficients. α is the effect on species 2 of
species 1 (α12).
β =
effect on species 1 of species 2 (α21). These reflect the strength of interspecific


The input panel allows you to change
each of these variables, for each of the two competing species.


When you run the simulation “To
steady state”, it will run until population size is no longer changing for
either species. Steady state means dN/dt = 0 for both species. This can occur
with competitive coexistence (both species persist), or competitive exclusion (one
species is driven extinct).



The goal of this exercise is to



1. How differences between the two
species in each of these 4 variables affects the outcome of competition,



2. How the 4 variables affect the trajectory
through which N1 and N2 change to reach that outcome.






For each run of the simulation,
there are two output displays. Look at both displays for


each simulation run, and understand
how they relate to each other



From the input panel, run the
simulation with the default settings and look at the


displays as you read the following


“N vs T” plots population
size for each species against time.


“N2 vs N1” plots
population size for species 2 against population size for species 1 (this is
called a phase-plane). On the N2 vs N1, there are three lines:


trajectory that shows how the numbers of the two species change through time (green),


zero-isocline for species 1 (red)


zero-isocline for species 2 (blue)


Remember that zero-isoclines show
where population growth is zero. An isocline is a line that connects points
with equal growth rates (dN/dt): a zero-isocline connects points with dN/dt =
0. On one side of the isocline, the population grows; on the other side, it shrinks.
With a plot of N2 vs N1:


species 1, population growth is positive LEFT of the (red) isocline, and
negative RIGHT of the isocline. The red isocline refers to the X axis, and
gives information about changes in N1.


species 2, population growth is positive in areas BELOW the (blue) isocline,
and negative in areas ABOVE the isocline. The blue isocline refers to the Y
axis, and gives information about changes in N2.


either species, the arrows are parallel to the axis that plots the numbers of
that species.



Stability Analysis:


This is very useful for testing how
well you understand the Lotka-Volterra competition model!


Select the N2 vs N1 “phase-plane”
plot. Rather than running to steady state, run until time 1, then time 2, 3,
etc, seeing if the trajectory follows the path you predicted ahead of time. Two
valuable ways of going about this are:


everything else the same but change the initial population sizes. See if you can
predict the trajectory correctly.


one variable (r, K, α, β) and see how the trajectory changes.





The default values for variables


Species 1
  Species 2


N0   10   20


r   0.9   0.5


K   500   700


α,β   0.6   0.7



A. Effect of initial population size
(N1 and N2).


1.  Accept the default values for all
variables (if you’ve changed values, the defaults are given above so you can
reset them). Re-set the simulation to run until steady state.


1: Before running any of the simulations, look at the phase plane. Mentally
draw in the carrying capacity connector and see where it lies in relation to
the equilibrium point. Do you expect the outcome of this simulation to be
coexistence or competitive exclusion?


2.  Run the simulation and examine both
output displays. In the N2 vs N1 display, select different pairs of initial
population sizes, and understand the trajectory of changes in population sizes
through time from that starting point.


3.  Use this approach to check outcomes
from a wide range of initial population sizes.


4.  Now set the input values to:


Species 1


N0   10   10


r   0.5   0.5


K   700     700


α,β   0.7     0.7


5.  Explore the effects of different
initial population sizes for this set of conditions, using the same approach as



QUESTION 2: How do the initial
population sizes for species 1 and species 2 affect the outcome of competition?
Why? When looking at the N vs t graph, what is the final total population size
(for both species combined)? Does this change as you alter the initial
population sizes? Does this match what you would expect from the N2 vs N1



B. Effect of intrinsic rate of
increase (r):


1.  Set the input values to:


Species 1
  Species 2


N0   100   100


r  0.5   0.5


K  700   700


α,β   0.7   0.7


2.  Run the simulation and examine the
outcome. In particular, note the shape of the trajectory in the N2 vs N1 plot
(equivalently, note the shapes of the two population growth curves vs time in
the other plot… understand how the two plots relate).


3.  Run the simulation several times,
varying the intrinsic rate of increase (r) for species 1 (be sure to
include 0 and 1) and leaving everything else constant. Notice that the competition
is symmetric, with only r differing between the species. The carrying capacities
and competition coefficients are the same for each species…so competition itself
is symmetrical in its effects on the two.


4.  Restore the input values from step
1. Now re-run the simulation several times, varying the intrinsic rate of
increase for species 2 (be sure to include 0 and 1)



QUESTION 3: How does variation in
the intrinsic rate of increase affect the outcome of competition? How does it
affect the trajectory (rate of increase) of population sizes? What happens to
the equilibrium number of individuals when r = 0 for either species (but not
both at the same time)? Why does this happen?



C. Effects of competition
coefficients (
α & β)
and carrying capacities (K):


1.  Set the input values to:


Species 1
  Species 2


N0   100   100


r   0.5   0.7


K   500   300


α,β   0.7   0.9


2.  Run the simulation and inspect the


3.  Rerun the simulation, decreasing the
carrying capacity (K) for species 1 by 100 each time (500, 400, 300, 200
& 100), and noting the outcome. Leave everything else constant.



QUESTION 4: What effect does K have
on the outcome of competition (all else equal)? How small does K1 have to be so
switch the outcome from competitive exclusion to stable coexistence? How small
does K1 have to get to switch the outcome to competitive exclusion by species
2? Demonstrate this mathematically by using the relationships between K1, K2,
a12 and a21.


4.  Set the inputs to values in shown in


1   Species 2


N0   100   100


r   0.5   0.5


K   500   500


α,β   0.5   0.5


5.  Rerun the simulation, increasing the
competition coefficient for the effect of species 1 on species 2 (β)
by 0.1 each time, and noting the changes. Run β values from
0.5 to 1.0 Leave everything else constant.


6.  Now run β =
1.1. What happens? Why?


7.  Next set β =
1.0, and change the carrying capacity (K) for species 2 to 600. How does
this result compare with the previous one?



does a competition coefficient greater than one mean? As you increase
β from 0.5 to
1, how does the equilibrium number of each species in the community and the
total number of individuals in the community change? Why?
What effect does changing the carrying
capacity have on equilibrium population values? Growth of species 1 is zero
when N1 = K or N1 = K2/a21. Explain in a sentence or two how K1 is related to



Practice Problems –Do these by hand
but you can check your answers in Populus.



have both green sunfish (Lepomis cyanellus) and bluegill (L.
) available for stocking in a 10 ha impoundment. These two
species compete to some degree; pertinent population data are:


K1 =
600, a12 = 1.50; Bluegill K2 = 600, a21 =


6.  Is there some level of
competition between these 2 species? How do you know?


7.  Would you classify these
species as weak, moderate, or strong competitors? 


8.  Which species appears to
be the stronger competitor? Why?


9.  Draw the phase-plane
diagram for this situation and predict the outcome if you start with 100 of
each species.


10.  What would happen if you
used the same initial population sizes but changed the following data: Green
sunfish: K1 = 600, a12=0.50; Bluegill: K2=600,


Draw the phase plane diagram and describe the outcome.


11.  What would happen if we
took the input parameters from question #10 and changed the initial population
sizes to N1=700 and N2=600?





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