Answer:
4,500
Explanation:
The logistic growth model is given by:
[tex]\frac{dP}{dt}=rP(1-\frac{P}{K} )[/tex]
Solving for P(t), we obtain:
[tex]P(t)=\dfrac{KP_0e^{rt}}{(K-P_0)+P_0e^{rt}}$ where:\\P_0=$Initial Population\\K=Carrying Capacity\\r=Growth rate\\P(t)=Population at a time t.[/tex]
We are given that:
[tex]I$nitial Population,P_0=500\\$Carrying Capacity,K=5000\\Population after one year, P(1)=1250 \implies t=1[/tex]
Therefore:
[tex]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[/tex]
We want to determine the population after another three years. i,e When t=4
[tex]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[/tex]
Therefore, after another three years, the population will be 4500.
