Two competing species are modeled by logistic equations with identical initial sizes:…

MATHEMATICS · Single correct · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Two competing species are modeled by logistic equations with identical initial sizes: Model A:

\dfrac{dA}{dt}=0.5A\,(800-A)

with

A(0)=100

Model B:

\dfrac{dB}{dt}=0.3B\,(1200-B)

with

B(0)=100

Which statement is true about their population sizes as

t\to\infty

? (a)

A(t)

approaches

1200

while

B(t)

approaches

800

because

0.5>0.3

so species

A

can sustain the larger population. (b) Both populations grow without bound because the positive coefficients

0.5

and

0.3

indicate exponential growth. (c)

A(t)

approaches

800

while

B(t)

approaches

1200

, each leveling off at its own carrying capacity. (d) Both populations remain near

100

because the population is unchanged when

dP/dt=0

at the initial time.