BIOS 5970: Plant-Herbivore Interactions
Dr. S. Malcolm, Biological Sciences, WMU
The Logistic Equation & population modeling:
Populus exercise 1 - Population Growth models:
A. Using the density independent exponential model for continuous generations:
1) What values of r (=ln) make the population increase geometrically?
2) What values of r (=ln) make the population decrease geometrically?
3) What values of r (=ln) make the population remain the same?
4) If r = 0.5, what value of  will give exactly the same exponential rate of
population increase (demonstrate this graphically with both the discrete
and continuous models)?
B. Using the Logistic Population Growth model:
5) For the continuous model with N0 = 10 and r = 0.5, what is the population size
after 10 generations?
6) How does the answer to 8) compare with population size after 10 generations
for the same values from the density-independent growth model? What
term in the model has generated this difference?
BIOS 5970: Plant-Herbivore Interactions
Stephen Malcolm
Modelling
page - 1
7) Using time lags in the logistic population growth can you generate population
fluctuations? If so, at what reproductive values (r or ) do you see
monotonic damping, and at what values do you see damped oscillations?
MULTI-SPECIES INTERACTIONS
Lotka-Volterra Competition
Using the Lotka-Volterra Competition model set to run to a steady state and the
gridding function answer the following questions:
At
N1 = 20
N2 = 20
r1 = 0.5
r2 = 0.5
K1 = 500
K2 = 500
α = 0.5
β = 0.5
1) What is the competitive outcome?
__________
2) At what population sizes do the N1 and N2 populations end up when they
reach steady state?
N1 =______
N2 =______
3) Is there a negative impact on the two populations?
__________
4) Is K1 larger or smaller than K2α (give numbers)?
__________
_____________________________________________________________
5) Is K2 larger or smaller than K1β (give numbers)?
__________
_____________________________________________________________
At
N1 = 20
N2 = 20
r1 = 0.7
r2 = 0.5
K1 = 700
K2 = 500
α = 0.7
β = 0.5
6) At what population sizes do the N1 and N2 populations end up when they
reach steady state?
N1 =______
N2 =______
BIOS 5970: Plant-Herbivore Interactions
Stephen Malcolm
Modelling
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7) Is K2 > or < than K1β (give numbers)?
__________
________________________________________________________
8) Is K1 > or < than K2α (give numbers)?
__________
________________________________________________________
9) What are the criteria for coexistence in terms of K 1, K2, α and β?
Give your own values for N, r, K, α and β to satisfy the criteria in the
following questions.
10) if K1 > K2α and K1β > K2 which species wins?
__________
11) if K2α > K1 and K2 > K1β which species wins?
__________
12) Does changing N or r have any impact on the outcome
of either 10) or 11)?
__________
13) if K2α > K1 and K1β > K2 which species wins?
__________
14) Does changing N or r have any impact on the outcome
of 13)?
BIOS 5970: Plant-Herbivore Interactions
__________
Stephen Malcolm
Modelling
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