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## Single first order ODE -- Mathematical Modeling

We begin by considering ODEs related to population dynamics. This is a simple concept that we all have intuition about. Let the function y(t) be a measure of population size at any particular time t. We can consider a large population of people, rabbits, bacteria, mosquitoes, etc. The time rate of change in population size (measured in units of number / second) is proportional to the size of the population at any given time. There are constants of proportionality that characterize the birth rate ( $\beta$ ) and the death rate ( $\delta$ ) that must have units of 1/second. Thus we can express this dynamic process of birth and death and the rate of change of the size of the population using the ODE

$\frac{dy\left(t\right)}{dt}=\beta y\left(t\right)-\delta y\left(t\right)=\left(\beta -\delta \right)y\left(t\right)=\alpha y\left(t\right)$ where $\alpha =\beta -\delta$

You can see that we construct the ODE that describes population dynamics without knowing its solution. All we know is that the function y(t) is the solution; whatever that is. This is called mathematical modeling. The hard part of science and engineering is getting the correct ODE to accurately describe the process that we are modeling. Once we have this ODE, then the second step is to solve it for the unknown function y(t). This is where Maple is useful. In this case we know the solution is

$y\left(t\right)={y}_{0}\mathrm{exp}\left(\alpha t\right)$ where the initial condition is $y\left(0\right)={y}_{0}$

If $\delta >\beta$ , then $\alpha$ is negative and the population declines exponentially. However, if $\beta >\delta$ then $\alpha$ is positive and the population grows exponentially. Exponential growth of mosquitoes or bacteria is a sobering thought. We know that in fact, these creatures can grow in number very rapidly (exponentially) but this cannot go on forever. Yet this mathematical model says that they do. Later we will modify the model to account for the limits to exponential growth.