A more realistic model is the logistic equation where C is the carrying capacity

Question

A more realistic model is the logistic equation where C is the carrying capacity of the species in its habitat. A phase portrait of this differential equation will show that there is an attracting equilibrium point at C (assuming R > 0). Are there other equilibrium points? We can also find the population at which the growth rate is the fastest by finding the critical point of f(p), the right-hand side. What is This population? The logistic equation is also separable and can be solved using the method of partial fractions. What is the solution? Suppose there is a particular species that is harvested for food. We want to study two different strategies for harvesting: Continuous harvesting. With this strategy, harvesting is done continuously at a rate k to give the differential equation Periodic harvesting. In this case, harvesting occurs each time the population reaches some value.b, and the population is cut back to the value a. Thus, the harvest is (b-a). Which strategy is better? If you choose to harvest continuously, at what rate k should you harvest? If you choose to harvest periodically, what should a and b be? What else can you add to the model to make it more realistic? How would this change the differential equation?

Explanation / Answer

dp/dt = Rp(1-P/C) - k

dp/dt + Rp P/C = Rp/C - k

now

take integrating factor

= e^ nintegral of Rp/C t

= e^Rpt/C

now

y *e^Rpt/C = integral of e^Rpt/C (Rp/C - k) dt

now solve this.

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