CompetitionAssignment(1).doc

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*

competition.

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 tounderstand:**

**1. How differences between the twospecies in each of these 4 variables affects the outcome of competition,and**

**2. How the 4 variables affect the trajectory**

through which N1 and N2 change to reach that outcome.

** **

*Displays:*

For each run of the simulation,

there are two output displays. Look at both displays for

each simulation run, and **understandhow they relate to each other**.

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

*displays as you read the followingexplanation.*

“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:

·

a

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

·

the

zero-isocline for species 1 (red)

·

the

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:

·

For

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.

·

For

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.

·

For

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:

·

Leave

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

predict the trajectory correctly.

·

Change

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

** **

*SIMULATIONS:*

The default values for variables

are:

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.

**QUESTION1: Before running any of the simulations, look at the phase plane. Mentallydraw in the carrying capacity connector and see where it lies in relation tothe equilibrium point. Do you expect the outcome of this simulation to becoexistence 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

Species

2

*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

before.

**QUESTION 2: How do the initialpopulation 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 initialpopulation sizes? Does this match what you would expect from the N2 vs N1graph?**

** **

*B. Effect of intrinsic rate ofincrease (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 inthe intrinsic rate of increase affect the outcome of competition? How does itaffect the trajectory (rate of increase) of population sizes? What happens tothe equilibrium number of individuals when r = 0 for either species (but notboth at the same time)? Why does this happen?**

** C. Effects of competition**α

coefficients (

**β**

*&*

*)*

and carrying capacities (K):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

outcome.

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 haveon the outcome of competition (all else equal)? How small does K1 have to be soswitch the outcome from competitive exclusion to stable coexistence? How smalldoes K1 have to get to switch the outcome to competitive exclusion by species2? Demonstrate this mathematically by using the relationships between K1, K2,a12 and a21.**

4. Set the inputs to values in shown in

below:

Species

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?

**QUESTION 5: ****Whatdoes 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?

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

K2/a21.

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

K2/a21.

**Practice Problems –Do these by handbut you can check your answers in Populus.**

You

have both green sunfish (*Lepomis cyanellus*) and bluegill (*L.macrochirus*) available for stocking in a 10 ha impoundment. These two

species compete to some degree; pertinent population data are:

**Greensunfish **K

_{1}=

600, a

_{12}= 1.50;

**Bluegill**K

_{2}= 600, a

_{21}=

0.90

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: K_{1} = 600, a_{12}=0.50; Bluegill: K_{2}=600,

a_{21}=0.90

·

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 N_{1}=700 and N_{2}=600?