Message Board Assignment $($Ch$5)$

Consider the graph to the right. Grassland butterflies $-$population index $)=(100$ It comes from the article "The European Butterfly Indicator for Grassland Species: $1990-2007."$ The article discusses the causes of this rapid decline, examining how a decrease in biodiversity caused by human interactions with the environment has affected individual species of butterflies.

a$)$ A group of scientists is interesting in developing a mathematical model to better understand the behavior of the grassland butterflies population. Create a list of things that the scientists might want to consider in developing their mathematical model.

b$)$ Suppose one scientist wishes to model this population using a purely exponential model: $E(t)=a*b^t,$ where $E(t)$ represents the population index $t$ years after $1990.$ Use the data points $E(0)=100$ and $E(10)=70$ to find values for a and b in our exponential model $E(t).$

c$)$ Suppose another scientist wishes to improve the model, choosing to use a model of the form $P(t)=E(t)+V(t),$ where $E(t)$ is the exponential function from part b $,$ and $V(t)$ represents the periodic variance, given by the function:

$V(t)=\frac{1}{3}({1\mathrm{.}05}^t-\frac{1}{ln(1\mathrm{.}05)})sin({1\mathrm{.}05}^t-t)+0\mathrm{.}05t^2$

Use the average value formula to find the average population index $P(t)$ from the year $2000$ to the year $2010.$ Round your answer to the nearest thousandth.