Question

Assume a predator and its prey are interacting according to the basic Lotka-Volterra model. The prey population size is 1000, the predator population size is 10, the prey population’s intrinsic rate of growth is 0.1, the predator’s capture efficiency is 0.01, the predator’s assimilation efficiency is 0.2, and the predator’s mortality rate is 0.3.

a. Under these conditions, what are the instantaneous growth rates of the prey and predator populations?

b. If r, c, a, and m remain stable, which population (predator or prey) would you expect to decline over time?

c. If a disease swept through the predator population and reduced each predator’s assimilation efficiency to 0.1, capture efficiency to 0.001, and increased the predator’s mortality rate to 0.6, which population (predator or prey) would you expect to decline over time?

Answer 1

a)

r= intrinsic rate of growth = 0.1

c= predator capture efficiency = 0.01

a= predator assimilation efficiency = 0.2

m= predator mortality rate = 0.3

X = prey population = 1000

Y = predator population = 10

dX/dt = rX - cXY

dY/dt = caXY - mY

dX/dt = (0.1 × 1000) - (0.01 × 1000 × 10)

= 100 - 100

= 0 per unit time

dY/dt = (0.01 × 0.2 × 1000 × 10) - (0.3 × 10)

= 20 - 3

= 17 per unit time

b)

By keeping them all constant (K) the formula would be

dX/dt = K (X - XY) = K (1000 - 10000) = -9000K

dY/dt = K (XY - Y) = K (10000 - 10) = 9990K

So the population of the prey will decline.

c)

The new growth rates would be as follows:

dX/dt = (0.1 × 1000) - (0.001 × 1000 × 10)

= 100 - 10

= 90 per unit time

dY/dt = (0.001 × 0.1 × 1000 × 10) - (0.6 × 10)

= 1 - 6

= -5 per unit time

So the predator population will decline.