Math Is Fun Forum

Hong Yuan

Guest

Exponential Growth and Differential Equation

a) On January 1 2000, the park estimated that they had 500 deer on their land. Two years later, they estimated that there were 550 deer on the land. Assume that the number of deer was changing exponentially, i.e. P(t)=ae^bt where P is the number of deer at year t, and a and b are parameters.Find the values of a and b.

b) On January 1, 2005, the park management determined that the deer population is growing too quickly. So they decided to cull 20 deer from the herd every year thereafter (starting from January 1, 2006). Assume that the growth rate of the deer population is the same as in (a). Determine the differential equation that governs this situation, and solve it. Assume t=0 for the year 2005.

c) On January 1, 2010, the park management felt that the deer population is still rampant. They decided to remove deer so that the instantaneous rate of decrease is 10% of the deer population. Determine the differential equation that governs this situation, and solve it. Assume t=0 for the year 2010.

Only able to do part (a), I got a = 500, b = 0.4476

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Exponential Growth and Differential Equation

hello Hong Yuan

Welcome to the forum.

I agree with your value for a, but not b.

The model equation is

If you put t = 2 and re-arrange it should be

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

hongyuan94

Member

Registered: 2015-09-02

Posts: 1

Re: Exponential Growth and Differential Equation

hello Hong Yuan

Welcome to the forum.

I agree with your value for a, but not b.

The model equation is

If you put t = 2 and re-arrange it should be

Bob

yes my mistake, the b should be 0.04765

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Exponential Growth and Differential Equation

hi Hong Yuan

I see you have become a member. 

For the second part, you need to find the growth rate.

The 'cull' rate is 20 in each year so we can model this with a term '20t'.  Strictly the cull is not continuous across the year but is all carried out in one go but it is the model you are asked to make.

So you can create a differential equation with this form dP/dt = growth rate minus cull rate.  See if you can use this to do part (b).  You'll need to use your values of a and b, and also the deer population at year 5 from part (a).

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob