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← Theoretical B

Q15 — Lotka-Volterra Competition Lines and Coexistence Outcomes

Theoretical B Real exam question — full text reproduced under IBO's CC BY-NC-SA 4.0 license

Ecology

The Lotka-Volterra model is used to predict the dynamics of competing populations. It is based on the following equations and is analyzed by constructing specific lines:

$$\frac{dN_1}{dt} = r_1 N_1 \left(\frac{K_1 - N_1 - \alpha N_2}{K_1}\right), \qquad \frac{dN_2}{dt} = r_2 N_2 \left(\frac{K_2 - N_2 - \beta N_1}{K_2}\right)$$

where K - carrying capacity, N - number of individuals, r - intrinsic rate of increase, α - competition coefficient of species 2 on species 1, β - competition coefficient of species 1 on species 2, and dN/dt = population growth rate.

The Lotka-Volterra line represents the number of individuals in a population at equilibrium (dN/dt = 0). For two species in competition, there are only 4 combinations of the location of these lines. Depending on the initial values of populations, the future changes in populations’ dynamics can be predicted.

Left: a graph of two straight equilibrium lines for species 1 and species 2, from K1/K2 on the axes to K1/K2 on the opposite axis. Right: four small panels showing the four possible combinations of the two lines’ positions, each with arrows showing population trajectories from marked starting points. Figure 1. A Lotka-Volterra lines. B Combinations of line locations and trends — arrows show the further direction of populations’ dynamics for particular initial values, indicated by blue dots.

On your answer sheet, indicate “T” for true statements and “F” for false ones.

A. The y-intercept for line of species 1 is equal to K1/α.
B. This model takes into account the presence of intraspecific competition.
C. Suppose that there is a certain community of elks (K1 = 200, α = 1) and deer (K2 = 100, β = 2) competing with each other. If the current populations are 120 elks and 60 deer, the elks will eventually outcompete the deer.
D. In only 2 scenarios out of 4 described in Figure 1B both species can stably coexist.
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Question reproduced from IBO 2024, Theoretical Exam Part B, licensed under CC BY-NC-SA 4.0 — attributed to the International Biology Olympiad. Open the full exam PDF