Question

The per capita growth rate of many species varies temporally for a variety of reasons, including seasonality and habitat destruction. Suppose n(t) represents the population size at time t, where n is measured in individuals and t is measured in years. Solve the differential equation for habitat destruction and describe the predicted population dynamics. n′=(e−t−1)nn(0)=n0 Here the per capita growth rate declines over time, starting at zero and becoming negative. It is modeled by the function e−t−1.

Answer 1

Answer:

$n(t)=n_0e^{(1-e^{-t }-t)}$

Step-by-step explanation:

If n(t) represents the population size at time t, where n is measured in individuals and t is measured in years.

$\frac{dn}{dt}=n(e^{-t }-1), n(0)=n_o$

$\frac{dn}{n}=(e^{-t }-1)dt$

Taking the integral of both sides

$\int\frac{dn}{n}=\int(e^{-t }-1)dt\\\int\frac{dn}{n}= \int e^{-t }dt-\int1dt$

ln |n| = $-e^{-t }-t+C$

Where C is integration constant

Taking the exponential of both sides

$n=e^{(-e^{-t }-t+C)}$

$n=e^{(-e^{-t }-t)}e^C\\n=Ke^{(-e^{-t }-t)}$ whee the exponential of a constant is a constant K.

When t=0, $n(0)=n_o$

$n_0=Ke^{-1$

Therefore:

$n=n_0e^{1}e^{(-e^{-t }-t)}$

$n(t)=n_0e^{(1-e^{-t }-t)}$