Title: Calculus differential equation (not sure which type) Post by: yungdumb on Feb 28, 2019 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. Title: Re: Calculus differential equation (not sure which type) Post by: bio_man on Feb 28, 2019 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 Title: Re: Calculus differential equation (not sure which type) Post by: yungdumb on Feb 28, 2019 thank you! I'll try to work through the question
Title: Re: Calculus differential equation (not sure which type) Post by: bio_man on Feb 28, 2019 You're welcome, report back if you need anything else
Title: Re: Calculus differential equation (not sure which type) Post by: yungdumb on Mar 1, 2019 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?
Title: Re: Calculus differential equation (not sure which type) Post by: duddy on Mar 1, 2019 this looks like Bernoulli's equation!
substitute a very large number into t to see what happens https://www.youtube.com/watch?v=BQcFrJt1kx4 https://www.youtube.com/watch?v=kvMXoVxY8rE |