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An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the context of population models, it often depicts growth or decay.
For the elk population problem, the function given is \(N(t) = 100(0.9)^t\). Here, the base is 0.9 and represents the decay factor. It implies that the population reduces to 90% of the previous year's number at each step.
Initially, there are 100 elk, but as time progresses, the effect of the exponent \(t\) increases, showing a rapid decline. As the value of \(t\) increases, the population shrinks, demonstrating the decaying nature of the function.
The phrase 'exponential decay' is key here. Unlike linear equations, where changes are uniform, exponential decay leads to increasingly smaller numbers as time passes.

Population models are mathematical descriptions used to predict how populations change over time. The model used here is an exponential decay model suitable for scenarios where each entity (in this case, elk) has a constant probability of surviving or dying.
The elk example uses a decay model considering each year. The idea is that the population does not just decrease in total numbers but does so in a proportional manner. In this model, the term \(0.9\) is pivotal. It tells us not just the survival rate but also the decreasing multiplier for the population from one year to the next.
Such models are useful in ecology, conservation, and resource management. They help predict futures and assist in decision-making processes. Understanding this allows stakeholders to allocate resources effectively and ensure sustainable practices.
Key aspects of any population model include its initial conditions, the growth or decay rate, and the time frame considered.

Percentage rate in exponential models describes the proportion of change from one period to the next. In our elk exercise, the percentage rate provides information about both survival and death rates.
In the equation \(N(t) = 100(0.9)^t\), the percentage of the population that survives each year is 90%. Thus, the decay rate is measured by the complement of this percentage, which is 10%.
To find such rates:

• Identify the base of the exponent (in this case, 0.9).
• Subtract it from 1 to determine the yearly decrease rate (1 - 0.9 = 0.1, or 10%).

This approach helps estimate how factors like disease, predation, or other environmental variables affect the living population on an annual basis. The concept of percentage rates aids in quickly grasping the rate of change over periods, an essential tool in both scientific research and real-world applications.