Question

Population Model In one model of the changing population

P(t) of a community, it is assumed that

$\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}\mathbf{=}\frac{\mathbf{d}\mathbf{B}}{\mathbf{d}\mathbf{t}}\mathbf{=}\frac{\mathbf{d}\mathbf{D}}{\mathbf{d}\mathbf{t}}$

where$\frac{\mathbf{dB}}{\mathbf{dt}}$and $\frac{\mathbf{dD}}{\mathbf{dt}}$are the birth and death rates, respectively.

(a) Solve for P(t) if $\frac{\mathbf{d}\mathbf{B}}{\mathbf{d}\mathbf{t}}\mathbf{=}{\mathit{k}}_{\mathbf{1}}\mathit{P}$and $\frac{\mathbf{dD}}{\mathbf{dt}}\mathbf{=}{\mathit{k}}_{\mathbf{2}}\mathit{P}$

(b) Analyze the cases${\mathit{k}}_{\mathbf{1}}\mathbf{>}{\mathit{k}}_{\mathbf{2}}\mathbf{,}{\mathit{k}}_{\mathbf{1}}\mathbf{=}{\mathit{k}}_{\mathbf{2}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\mathit{k}}_{\mathbf{1}}\mathbf{<}{\mathit{k}}_{\mathbf{2}}$

Short answer

$a)P=c{e}^{\left(k_1-k_2\right)t}$

(b) When $k_1>k_2$, the population continues to grow until it reaches infinity. It becomes stable and approaches a constant when $k_1=k_2$. It continues to decrease for$k_1<k_2$ until it reaches zero.

Step-by-step solution

Given Information

$\frac{dP}{dt}=\frac{dB}{dt}=\frac{dD}{dt}$

Determining Population of a community at time t

We have a community population model defined by the differential equation

$\frac{dP}{dt}=\frac{dB}{dt}=\frac{dD}{dt}$

and we must solve it using the following method:

We may write equation(1) as follows because we have the birth rate $\frac{dB}{dt}=k_{1}P$and the death rate $\frac{dD}{dt}=k_{2}P$.

$$\begin{gathered} \frac{dp}{dt}=k_{1}P-k_{2}P \\ \frac{dp}{dt}=\left(k_1-k_2\right)P \end{gathered}$$

Because this differential equation is a first-order, separable D.E., we can solve it as follows:

$\begin{array}{rcl}\frac{dP}{P}& =& \left(k_1-k_2\right)dt\\ \int \frac{1}{P}dP& =& \left(k_1-k_2\right)\int dt\\ ln\left(P\right)& =& \left(k_1-k_2\right)t+c_1\\ e^{ln\left(P\right)}& =& e^{\left(k_1-k_2\right)t+c_1}\end{array}$

Then there's

$\begin{array}{rcl}P& =& e^{c_1}{e}^{\left(k_1-k_2\right)t}\\ & =& c{e}^{\left(k_1-k_2\right)t}\end{array}$

is a community's population at time t, where c is a constant.

Analyzing the population at the given cases

We may examine the population over time for each scenario using equation (3) based on the three cases given.

1. We discover that at $k_1>k_2$the population continues to grow (P>c) at time t until it reaches infinity, P=infinity.
2. We discover that at$k_1=k_2$ the population stabilises and equals a constant, P=c.
3. We discover that at$k_1<k_2$ the population continues to decrease (P<c) at time t until it reaches zero, P=0.

Conclusion

$a)P=c{e}^{\left(k_1-k_2\right)t}$

b) When $k_1>k_2$, the population continues to grow until it reaches infinity. It becomes stable and approaches a constant when $k_1=k_2$. It continues to decrease for$k_1<k_2$ until it reaches zero.