A population of fish is modeled by the logistic equation modified to account for

Question

A population of fish is modeled by the logistic equation modified to account for harvesting: dp/dt=.25p(1-p/4)-a or equivalently dp/dt=-1/16p^2+1/4p-a What are the equilibrium solutions? When does it have 1 equilibrium? 0 equilibria? Carefully interpret each of these scenarios in the context of this model. Suppose a has the value for which there is 1 equilibrium. What is the value of this equilibrium? Is it an attractor, a repelled, or a node? What does this mean to our population?

A population of fish is modeled by the logistic equation modified to account for harvesting: -16p2+7P-a dp 1 dt 16 4 dp = .25p (1 - -4)-a p2 tip - a --)-a or equivalently = 1. What are the equilibrium solutions? Note: your answer(s) will depend on 'a'. 2. Determine the range of values of a for which this model has 2 different equilibrium values. When does it have 1 equilibrium? 0 equilibria? Carefully interpret each of these scenarios in the context of this model. Suppose a has the value for which there is 1 equilibrium. What is the value of this equilibrium? Is it an attractor, a repellor, or a node? What does this mean to our population? 3.

Explanation / Answer

1.
if we solve the eqauation dp/dt=0 , we'll get the equilibrium points

dp/dt= 0    => -p^2/16 + p/4 - a = 0
                solve for p
                p^2 - 4p + 16a = 0
                p =[4 +- sqrt(16-64)]/2

        P = 2+2sqrt(1-4a) and p=2-2sqrt(1-4a)

2.the range for 'a'
we know that the square root function is defined for all values greater than or equal to 0 under the square root

=> 1-4a>=0
so , a <= 1/4

for 1 equilibrium solution we need a to be = 1/4

this makes dp/dt = p^2 -4p + 4= (p-2)^2
and then the equilibrium solution becomes p = 2
so in this case we get 1 equilibrium solution

the differential equation would have 0 equilibrium solution when dp/dt=0

although p=0 is a trivial solution but it has not much significance.

3. dp/dt has 1 equilibrium solution when a= 1/4
and the value of the solution is p=2

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