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## Non-linear first order ODE with steady state solution

Returning to our population dynamics model, the model can be extended to build a more realistic result. There appear to be self-regulating mechanisms in human population growth so that the growth or loss of population is not such a dramatic function of the exponential parameter. A better model is perhaps

$\frac{dy\left(t\right)}{dt}=\alpha y\left(t\right)-\gamma y{\left(t\right)}^{2}$

The extra term in the differential equation serves to decrease the rate of growth as the population size gets larger. For large values of y(t), this term dominates over the first term because $y{\left(t\right)}^{2}$ grows faster than y(t). But now the ODE is non-linear and is more difficult to solve.

The solution to the ODE does not grow exponentially without bound or decline to zero but instead asymptotically approaches a `steady-state' solution meaning that it does not change in time once it reaches that value. We obtain this by considering the solutions to the ODE with the derivative set to zero, which of course means that y(t) is not changing with time.

$0=\frac{dy\left(t\right)}{dt}⇒0=\alpha y\left(t\right)-\gamma y{\left(t\right)}^{2}=y\left(t\right)\left(\alpha -\gamma y\left(t\right)\right)$

The two possible solutions are

$y\left(t\right)=0$ and $\alpha -\gamma y\left(t\right)=0$

The solution $y\left(t\right)=0$ is an obvious steady state (if you have no population to begin with, there will be no reproduction and hence no change in population). This mathematical solution is uninteresting for the physical problem. The other solution,

$y\left(t\right)={y}^{*}=\frac{\alpha }{\gamma }$

is the value of the population for which the two terms on the right hand side of the ODE are equal, and population maintains its size. This solution is very interesting

If y(t) is less than ${y}^{*}$ , then the ODE shows that the rate of change is positive, i.e., the population will grow. If y(t) is greater than ${y}^{*}$ , then the rate of change is negative, i.e., the population will decline. The steady-state solution is an example of a stable solution. As time increases, all solutions tend to the constant solution ${y}^{*}$ .

From a mathematical point of view, this non-linear first order ODE is more difficult to solve than the linear first order ODEs. However, Maple knows how to solve both!