Question

Suppose a population grows according to a logistic model with initial population 500 and carrying capacity 5,000. If the population grows to 1250 after one year, what will the population be after another three years

Answer 1

Answer:

4,500

Explanation:

The logistic growth model is given by:

$\frac{dP}{dt}=rP(1-\frac{P}{K} )$

Solving for P(t), we obtain:

$P(t)=\dfrac{KP_0e^{rt}}{(K-P_0)+P_0e^{rt}} where:\\P_0= Initial Population\\K=Carrying Capacity\\r=Growth rate\\P(t)=Population at a time t.$

We are given that:

$I nitial Population,P_0=500\\ Carrying Capacity,K=5000\\Population after one year, P(1)=1250 \implies t=1$

Therefore:

$1250=\dfrac{500 \times 5000e^{r \times 1}}{(5000-500)+500e^{r\times 1}}\\\\1250[(5000-500)+500e^{r}]=500 \times 5000e^{r}\\4500+500e^{r}=2000e^{r}\\2000e^{r}-500e^{r}=4500\\1500e^{r}=4500\\e^r=3\\ Take the natural logarithm of both sides\\r =\ln 3$

We want to determine the population after another three years. i,e When t=4

$P(4)=\dfrac{500 \times 5000e^{\ln|3| \times 4}}{(5000-500)+500e^{\ln|3| \times 4}}\\=\dfrac{2,500,000e^{\ln|3| \times 4}}{4500+500e^{\ln|3| \times 4}}\\P(4)=4500$

Therefore, after another three years, the population will be 4500.