BIOS 105: Environmental Biology

Dr Stephen Malcolm

Computer modeling: playing with equations…

(1) Intraspecific Competition:

Questions using Populus software:

by Don Alstad, Dept. Ecology, Evolution and Behavior, University of Minnesota. http://www.cbs.umn.edu/software/populus.html/

Both DOS and java versions of Populus are installed on the computers in the ecology lab. The following questions can be answered using either of these versions in the “Ecology” folder.

A) POPULATION GROWTH – Single-Species Dynamics:

1) Density-Independent Growth

For exponential population increase the rate of population increase is dN/dt and this speed of change is described by:

dN/dt = rN

(where r is the intrinsic rate of natural increase which is lnR or lnR_o/T)

Figure 6.29 from Begon et al. (1996).

Using the Continuous model and the gridding option answer the following questions:

i) At N₀ = 10 and r = 0.1 what is the population size after 10 generations?

ii) At N₀ = 10 and r = 0.5 what is the population size after 10 generations?

iii) At N₀ = 10 and r = 1.0 what is the population size after 10 generations?

2) Density-Dependent, or Logistic Population Growth

Exponential increase can be decreased to a carrying capacity (K) by the logistic term (K - N)/K, so that dN/dt = rN becomes,

dN/dt = rN((K - N)/K)

this is the famous logistic equation as shown in figure 6.29 (above) with a carrying capacity where the recruitment rate is zero because births = deaths (Figure 6.10).

Figure 6.10 (Begon et al. (1996)).

Using the Continuous model, the gridding option and the graph selections, answer the following questions at N₀ = 5, K = 500, r = 0.2:

i) At what “t” does the population reach K?

ii) At what population size is dN/dt highest?

iii) What is the value of lnK?

iv) Using the discrete model is there any difference in the model outcome (apart from the discrete time intervals)?

(2) MULTI-SPECIES INTERACTIONS

1) Lotka-Volterra Competition

Based on a simple expansion of the logistic equation. With the inclusion of the competition coefficients a and b we can represent population size changes for the two competing species as:

dN₁/dt = r₁N₁((K₁-N₁-aN₂)/K₁)

and

dN₂/dt = r₂N₂((K₂-N₂-bN₁)/K₂)

These equations generate 4 predicted outcomes of interspecific competition as shown in Figure 7.8 (a) species 1 wins, (b) species 2 wins, (c) either species 1 or species 2 wins, or (d) coexistence.

Figure 7.8, Begon et al. (1996)

Using the Lotka-Volterra Competition model set to run to a steady state and the gridding option answer the following questions:

At N₁ = 20 r₁ = 0.5 K₁ = 500 a = 0.5

N₂ = 20 r₂ = 0.5 K₂ = 500 b = 0.5

i) What is the competitive outcome?

ii) At what population sizes do the N₁ and N₂ populations end up at when they reach steady state? N₁ =______N₂ =______

iii) Is there a negative impact on the two populations?

iv) Is K₁ larger or smaller than K₂a (give numbers)?

v) Is K₂ larger or smaller than K₁b (give numbers)?

At N₁ = 20 r₁ = 0.7 K₁ = 700 a = 0.7

N₂ = 20 r₂ = 0.5 K₂ = 500 b = 0.5

i) At what population sizes do the N₁ and N₂ populations end up at when they reach steady state? N1 =______N2 =______

ii) Is K₂ > or < than K₁b (give numbers)?

iii) Is K₁ > or < than K₂a (give numbers)?

iv) What are the criteria for coexistence in terms of K₁, K₂, a and b?

Give your own values for N, r, K, a and b to satisfy the criteria in the following questions.

i) if K₁ > K₂a and K₁b > K₂ which species wins?

ii) if K₂a > K₁ and K₂ > K₁b which species wins?

iii) Does changing N or r have any impact on the outcome

of either I) or ii)?

iv) if K₂a > K₁ and K₁b > K₂ which species wins?

v) Does changing N or r have any impact on the outcome

of iv)?