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wrote...
2 months ago
A population r(t) of rabbits (at time t) satisfies
dr/dt = kr (1 − (r/r∗)) − αfr                                       (1)
where k > 0 is a constant representing the rabbit breeding rate, r∗ > 0 is the (constant)
maximum sustainable rabbit population size in the absence of predation, f > 0 is the
population of foxes, and α > 0 is the (constant) rate of predation of rabbits by foxes.
1. Suppose that the fox population, f, is constant. Solve the differential equation
(1), and determine
(a) the size of the rabbit population as t → ∞;
(b) the maximum predation rate α for which the rabbit population does not die
out as t → ∞;
(c) the value of α which maximises αfr (the total number of rabbits caught) as
t → ∞, and the corresponding rabbit population.
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wrote...
Educator
2 months ago Edited: 2 months ago, bio_man
Hi there

I found a segment in one of my old Calculus textbooks that likely holds an explanation to your question. Please review it below, and let us know if it helps!

Segment also uploaded here: https://biology-forums.com/index.php?action=downloads;sa=view;down=12400
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wrote...
2 months ago
thank you! I'll try to work through the question
wrote...
Educator
2 months ago
You're welcome, report back if you need anything else
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wrote...
2 months ago
Hi, I attempted to solve this but I’m not sure it’s right. I’ve attached my workings so far, could you have a look? How would I find the size of the population as t tends to infinity?
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wrote...
Staff Member
2 months ago
this looks like Bernoulli's equation!

substitute a very large number into t to see what happens




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