BIOS 3010: ECOLOGY

Dr Stephen Malcolm

Laboratory 6: Lotka-Volterra & the logistic

This is a computer-based activity using Populus software (P).

Introduction:

Some white-tailed deer are eating lettuces in your vegetable garden and you want to understand the interaction before you decide what to do to.

Your first step is to build a basic Lotka-Volterra Predator-Prey model for a description of the interaction using Populus:

dN/dt = r₁N - CNP

and

dP/dt = gCNP – d₂P

where:

N = number of lettuce plants (prey),

P = number of deer (predators),

r₁ = the intrinsic rate of increase of lettuce (prey),

C = the feeding rate of deer and CN is the per capita functional response of deer to lettuce density,

g = the efficiency with which deer turn food into progeny, and gCN is the numerical response of deer (rate of new deer born as a function of prey density), and,

d₂ = deer death rate.

which means that lettuce would increase without limit in the absence of deer.

This is the first of the “Continuous Predator-Prey models” in Populus. Starting with the default parameters indicated below please answer the following questions:

Predator: P_o = 20, d₂ = 0.6, g = 0.5

Prey: N_o = 20 r₁ = 0.9 C = 0.1

1. At what predator density does the prey isocline intercept?
2. At what prey density does the predator isocline intercept?
3. What is the maximum amplitude of the prey population?
4. What is the minimum amplitude of the prey population?
5. What is the maximum amplitude of the predator population?
6. What is the minimum amplitude of the predator population?
7. What is the frequency of prey peaks (N/t)?
8. What is the frequency of the predator peaks (N/t)?
9. Does the model change the amplitude or frequency of oscillations through time?
10. Is the model any use for solving your problem?

Given that your answer to 10 is likely to be negative you decide to add more realism to the model and limit the lettuce population with intraspecific competition (you sowed the seeds too densely and foolishly forgot to thin the seedlings as your Granny always told you to do!).

So now you add density-dependent limitation of lettuces with logistic limitation to a carrying capacity as in the Populus theta (q) –logistic predator-prey model in which:

dN/dt = r₁N(1-(N/K)) - CNP

This means that in the absence of predators (P = 0) the lettuces are limited only by intraspecific competition where (1-(N/K)) is the same as (K-N)/K.

You now have the 2 extra parameters of K (the carrying capacity) and q. Set K to 100 and keep q at 1 (this basically ignores the density-dependent feedback provided by q). All other parameters should be the same as above. This model also includes the predator’s functional response (numbers of prey eaten at different prey densities). Set the model to the following:

Predator: P_o = 20, D = 0.6, g = 0.6

Prey: N_o = 20 r₁ = 0.9 K = 100 q = 1

with a type 2 functional response (C = 0.05 and h =1) and answer the following questions:

1. What shape is the prey isocline?
2. Where does the prey isocline intercept with the prey axis at P = 0?
3. Where does the predator isocline intercept the prey isocline – left or right of the hump?
4. Is the interaction stable or unstable?
5. Does the time trajectory spiral inwards or outwards?

Now change D to 0.7:

6. Does the predator isocline intercept to the right or left of the hump in the prey isocline?
7. Is the interaction stable or unstable?
8. Does the time trajectory spiral in or out?

Now change functional response to type 3 (c = 0.05, h = 1):

9. What shape is the prey isocline now?
10. Is the new interaction stable or unstable?

Lastly, you decide to simplify the theta logistic model slightly and build your own logistically limited model of the interaction. So you go to the Interaction Engine in Populus and add your own logistic term to the Lotka-Volterra mass action predator–prey model.

Using the Interaction Engine build the following model of the interaction:

dN₁/dt = (r*N₁*(K – N₁)/K) – f*N₂*N₁

and

dN₂/dt = (c*N₂*N₁)-(m*N₂)

where:

N₁ = number of lettuce plants (prey),

N₂ = number of deer (predators),

r = the intrinsic rate of increase of lettuce (prey),

K = carrying capacity of lettuce

f = the feeding rate of deer,

c = the numerical response of deer (efficiency with which deer turn food into progeny), and,

m = deer death rate.

Model these continuous, differential equations using the POPULUS Interaction Engine and set the plot to N vs T, t = 100 and do not plot isoclines. Enter the equations to build the model (uncheck N3 because you are only interested in 2 species) and set the starting parameters to:

N₁ = 200

N₂ = 20

r = 0.2

m = 0.3

K = 1000

c = 0.001

f = 0.001

Run the model and turn on the gridding function .

Now that you have a functioning, resource-limited, plant-herbivore model examine how changes in r, f, and c influence the stability and final population sizes of lettuce and deer after 100 time intervals.

To do this please fill in the following table of lettuce and deer numbers after 100 time intervals :

Parameter	Numbers of lettuce N1	No's of deer N2
r = 0.2
r = 0.4
r = 1.2
With r = 0.5 examine the following:
f = 0.005
f = 0.001
f = 0.0008
With f = 0.001 examine the following:
c = 0.0005
c = 0.001
c = 0.005

Using this table and your screen answer these questions :

1) What effect does the intrinsic rate of lettuce increase, r, have on the equilibrium population of lettuce plants?

2) What effect does increasing r have on deer abundance?

3) What effect does increasing rhave on the rate of return to equilibrium?

4) What effect does f have on the equilibrium lettuce population?

5) What effect does increasing f have on deer abundance?

6) What effect does increasing c have on the equilibrium lettuce population?

7) What effect does increasing c have on the equilibrium deer population?

8) What effect does increasing c have on the stability of lettuce-deer dynamics?