Science:Math Exam Resources/Courses/MATH110/April 2014/Question 08 (a)
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Question 08 (a) 

Imagine bushbuck antelopes interacting in a montane forest. Let denote the probability that a given antelope is in contact with another antelope. is modelled by the equation where is a constant denoting the corresponding distances of the species, and is a function denoting the density of antelope in the area. Suppose is determined by field observations. Come up with an expression for the rate of change of the probability that an individual is in contact with another individual at time . (Your answer will involve the term .) 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Start by identifying which symbols in the equations are constants and which ones change over time. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We make the assumption that is the only parameter that changes with time. So we get:
