Question

Suppose that rabbits are the only prey and food supply of foxes and that the predator-prey interaction follows Lotka‒Volterra dynamics. The mortality rate of foxes in the absence of rabbits is 0.1 per week, and the intrinsic growth rate of rabbits in the absence of predation is 0.2 per week. The capture efficiency is 0.002, and the efficiency at which rabbit biomass is converted into fox biomass is 0.2.

Consider a scenario when the prey population growth is 0 (prey isocline), what will the population of predators be?

Answer 1

The population of predators when the prey population growth is 0 will be P = 50

When the prey population growth is 0 (prey isocline), it means that the rate of birth and death of rabbits are equal. Using the Lotka-Volterra model, we can write the following equations:

Predator equation: dP/dt = k1 * P * R - k2 * P

Prey equation: dR/dt = k3 * R - k4 * P * R

P = Predator population

R = Rabbit population

k1 = Capture efficiency (0.002)

k2 = Mortality rate of foxes (0.1)

k3 = Intrinsic growth rate of rabbits (0.2)

k4 = Efficiency of converting rabbit biomass to fox biomass (0.2)

t dR/dt = 0.

we the Substitute this into the prey equation:

0 = 0.2 * R - 0.2 * P * R

R = P

we can substitute this value of R back into the predator equation:

dP/dt = 0.002 * P * P - 0.1 * P

dP/dt = 0.002 * P² - 0.1 * P

To find the population of predators when the prey population growth is 0 (P = R), we can set dP/dt to 0:

0 = 0.002 * P² - 0.1 * P

0 = P (0.002 * P - 0.1)

P = 0 or P = 50

However, P cannot be 0 because there cannot be any predators without prey. In conclusion, the population of predators when the prey population growth is 0 will be 50.