I need some help with a maths problem here.
A biologist is researching the population of a species. She tries a number of different models for the rate of growth of the population and solves them to compare with observed data. Her first model is dp/dt=kp(1-(p/m)) where p is the population at time t years, k is a constant and m is the maximum population sustainable by the environment. Find the general solution of the differential equation.
I've separated the variables and integrated both sides of the equation. The general solution is p= mAe^kt/(1+Ae^kt).
Her observations suggest that k= 0.2 and m= 100 000. If the initial population is 30 000, estimate the population after 5 years to 2 sig. fig.
I found the value of A to be 3/7 . Then I substituted it into the general solution to obtain p = 54 000 after 5 years.
Now comes the difficult part.
She decides that the model needs to be refined. She proposes a new model dp/dt= kp(1-(p/m)^α) and investigates suitable values of α. Her observations lead her to the conclusion that the maximum growth rate occurs when the population is 70% of its maximum. Show that (α+1)0.7^α=1 and that an approximate solution of this equation is α=5. How do I prove this?
Express the time that it will take the population to reach 54 000 according to this model as a definite integral, and use the trapezium rule to find this time approximately.
I know how to obtain the definite integral, which is 5/(p(1-((1/100 000)p)^5) dp with the upper and lower limits being 54 000 and 30 000 respectively. The time taken is 3.0 years. My question is how many intervals should I use when I apply the trapezium rule?