# Modelling Population Growth and Decay

One important application of differential equations arises in modelling population growth or decay within a species. Let $y = y(t)$ denote the population size of a species at time $t$. If the rate of change of this population is proportional by some constant $r$ (where $r > 0$ implies that the population is growing and $r < 0$ implies that the population is decaying) then we can model the change in population with the following differential equation:

(1)Suppose that the initial population at time $t = 0$ is $y_0$, that is $y(0) = y_0$ . We can solve the differential equation above by either using the method of integrating factors or as a separable equation. Let's use integrating factors. We first rewrite the differential equation above in the form:

(2)Then $p(t) = -r$. Let $\mu (t) = e^{\int p(t) \: dt} = e^{\int -r \: dt} = e^{-rt}$ be our integrating factor. Multiplying both sides of this differential equation by $\mu (t)$ and we get that:

(3)With the initial condition $y(0) = y_0$ we see that $C = y_0$ and so the solution to our differential equation is:

(4)As we see, if $r > 0$ then the growth of the population exponentially increases and if $r < 0$ then the decay of the population exponentially decreases. Of course, if $r > 0$ then $\lim_{t \to \infty} y_0e^{rt} = \infty$. Realistically, as time increases, the population $y$ will not increase indefinitely as population dynamics are based off of many other variables that are bounded by limits. For example, a finite amount of space or food may impede on a population from growing indefinitely.

Our original assumption that the population growth/decay rate $r$ was constant is not realistic. Instead, the growth/decay of a population is also dependent on the population itself. We replace the constant $r$ with the function $h(y)$ which depends on the population $y$. Therefore, we have the new differential equation:

(5)The choice of function $h(y)$ can be determined based on various conditions that affect the rate of growth/decay in a population. A generic choice of this function is $h(y) = (r - ay)$, $a > 0$, and where $r$ is the growth/decay rate from earlier. Note that as $y$ gets very large, $h(y)$ does not get too large. Substituting this into the differential equation and letting $K = \frac{r}{a}$ gives us:

(6)The constant $K$ is known as the **Carrying Capacity** for the population. Now, this differential equation is separable since it can be rewritten as:

We we will now solve this separable differential equation by integrating both sides:

(8)If we plug in our initial condition $y(0) = y_0$ we get that:

(9)When we plug this into our differential equation and isolated for $y$ we get that:

(10)The solution to this initial value problem models population growth in constraining the population growth as it approaches the carrying capacity.