Question

Suppose that a lizard species eats only one type of insect and the populations follow Lotka–Volterra dynamics. The intrinsic growth rate of insects in the absence of predators is 0.2 per week, and the mortality rate of the lizards in the absence of insects is 0.05 per week. The capture efficiency rate is 0.002, and the efficiency at which insect biomass is converted into predator biomass is 0.2. The lizard population will increase only if the number of insects is a. above 125. b. above 500. c. above 625. d. below 125. can you show me how to plug in the data into the equation to solve for the answer. Thanks!

Answer 1

Predation relationships consist of the consumption of a living organism, the prey, by the predator. The predatory species produces an immediate effect on the population dynamics of the prey and vice versa. The abundance of the prey will increase the birth rate and survival of the predator and its shortage will mean an increase in mortality. This mutual interdependence suggests that a population of predators can only increase if their prey does and will decrease otherwise. On the other hand, the population of the prey will decrease if the population of predators is high and may increase when it falls. In this way, the populations of predators and prey, instead of remaining stable, oscillate around an average value forming population cycles.

A. Lotka and V. Volterra proposed mathematical equations to determine the relationship between the populations of predators and those of their prey. They applied the following equation for the growth of the prey population:

The population of the prey (P) grows exponentially in the absence of the predator at a constant rate (r). Mortality due to predator ($\alpha$PD) is directly proportional to the number of prey (P) and the number of predators (D), depending on a constant predator capture efficiency ($\alpha$). The population will grow when the growth rate compensate losses due to predation:

dP lt

The proposed equation for the predator population is as follows:

dD dt

Where it is assumed that the predatory population decreases proportionally to its abundance with a constant mortality rate (m) both in the presence and absence of its prey. The predatory population could grow to the extent that it transforms the prey that consumes ($\alpha$PD) into new predatory individuals with a constant transformation efficiency ($\beta$).

So, in the problem we have:

r= 0.2/week

m=0.05/week

$\alpha$= 0.002

$\beta$=0.2

If we replace the values ​​in the equation of the growth of the predator we will obtain the answer. For example=

dD dt

for a) above 125 and assuming an arbitrary number of predators of 40 and prey of 150 (above 125):

dD 0.2 × 0.002 × 150 × D-0.05 × D lt

dD ( 0.2 × 0.002 × 150-0.05) × 40 dt

a) for 125 = 0.4/week

b) for 550 = 6.8/week

c) for 650 = 8.4/week

d) for 100= -0.4/week

I hope help you, Thanks!