Answer to Question #147551 in Differential Equations for fanni

Question

if the population S(t) of a species of mosquitos in the Okavango region Delta at any time t >0 is modeled by the differential equation θ(ds/dt)+γS=〖αe〗^(-φt), whereθ,γ and α are positive constants and φ is a nonnegative constant, and if the initial population of this species is S(0)=S_0,
show that the mosquito population S(t) at any time t >0 is given by S(t)=α/(γ-θφ) e^(-φt)+(S_0-α/(γ-θφ) ) e^(-φt)

show that if φ=0 the species population S(t) approaches α/γ as t →∞ but if φ> the S(t) approaches zero as t→∞

Answer 1

Let us consider the differential equation

$\Theta\frac{dS}{dt}+\gamma S=\alpha e^{-\varphi t}$, $S(0)=S_0$

where $\varphi\ge 0,\ \Theta>0,\ \gamma>0,\ \alpha >0.$

Let us multiply the both part of equation by $\frac{1}{\Theta}e^{\frac{\gamma t}{\Theta}}:$

$\frac{dS}{dt}e^{\frac{\gamma t}{\Theta}}+\frac{\gamma}{\Theta} Se^{\frac{\gamma t}{\Theta}}=\frac{\alpha}{\Theta} e^{-\varphi t}e^{\frac{\gamma t}{\Theta}}$.

The last equation is equivalent to

$\frac{d}{dt}(Se^{\frac{\gamma t}{\Theta}})=\frac{\alpha}{\Theta} e^{(\frac{\gamma }{\Theta}-\varphi) t}$

Therefore,

$Se^{\frac{\gamma t}{\Theta}}=\frac{\alpha}{\gamma -\varphi \Theta} e^{\frac{\gamma-\varphi\Theta }{\Theta} t}+C$

Let us multiply the both part of equation by $e^{-\frac{\gamma t}{\Theta}}:$

$S(t)=\frac{\alpha}{\gamma -\varphi \Theta} e^{-\varphi t}+Ce^{-\frac{\gamma t}{\Theta}}$.

Let $t=0$. Then

$S_0=S(0)=\frac{\alpha}{\gamma -\varphi \Theta} +C,$ and thus $C=S_0-\frac{\alpha}{\gamma -\varphi \Theta}$.

Consequently, the solution of the differential equation is the following:

$S(t)=\frac{\alpha}{\gamma -\varphi \Theta} e^{-\varphi t}+(S_0-\frac{\alpha}{\gamma -\varphi \Theta})e^{-\frac{\gamma t}{\Theta}}$.

If $\varphi =0$, then $S(t)=\frac{\alpha}{\gamma} +(S_0-\frac{\alpha}{\gamma})e^{-\frac{\gamma t}{\Theta}}$, and thus

$\lim_{t\to+\infty}S(t)=\lim_{t\to+\infty}[\frac{\alpha}{\gamma} +(S_0-\frac{\alpha}{\gamma})e^{-\frac{\gamma t}{\Theta}}]=\frac{\alpha}{\gamma}$

So, if $φ=0$ the species population $S(t)$ approaches $\frac{\alpha}{\gamma}$ as $t\to+\infty.$

If $\varphi>0,$ then $\lim_{t\to+\infty}S(t)=\lim_{t\to+\infty}[\frac{\alpha}{\gamma -\varphi \Theta} e^{-\varphi t}+(S_0-\frac{\alpha}{\gamma -\varphi \Theta})e^{-\frac{\gamma t}{\Theta}}]=0$.

Therefore, if $φ>0$ the $S(t)$ approaches zero as $t\to+\infty.$