First off, this is a typed paper project, and I am not asking for anyone to solve my work or to give me the answers. Just please point me in the right direction or help me through any parts I am not understanding. This is also a 7 part project, and I would like help on this first problem to get the ball rolling. I have also done my research on this too.
1. The problem statement, all variables and given/known data
There is a fish population in a lake. By setting up a differential equation, we will investigate the dynamics of this population and show that the population will eventually approach the so called carrying capacity of the environment if the initial population is larger than a "threshold" and become extinct if it is smaller than the threshold. For the purpose of protection of this population, we will set up a scheme for fishing.
1) Denote by P(t) the fish population at time t. Assume the birth rate β(P) and the death rate δ(P) of P(t) per individual per year are given by...
β(P) = a
δ(P) = bP+(c/P)
respectively, where a, b, c are postitive constants such that a2-4bc > 0. Apply the principle
dP/dt = [β(P)-δ(P)]P
to show that the P(t) satisfies the differential equation
dP/dt = k(M-P)(P-m) (1)
where k = b, m = (a-√(a2-4bc))/(2b) and M = (a+√(a2-4bc))/(2b). We observe that the logistic equation is the special form of (1) when m = 0. Let the initial population at t0 be given by
P(t0)=P0
2. Relevant equations
They were stated above, but here they are again.
β(P) = a
δ(P) = bP+(c/P)
dP/dt = [β(P)-δ(P)]P
dP/dt = k(M-P)(P-m) (1)
k = b
m = (a-√(a2-4bc))/(2b)
M = (a+√(a2-4bc))/(2b)
P(t0)=P0
3. The attempt at a solution
Well my attempt in getting is by separation...
dP/([β(P)-δ(P)]P) = dt
and then to integrate it...
∫1/([β(P)-δ(P)]P) dP = ∫ dt
and then partial fractions I thought, but it doesn't come out correct. I have also tried wolfram alpha and to check my work. I throw in the equation to be differentiated and it give me something that is sort of similar, but still far off.
Could someone please help me get on the right track here?
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
There is a fish population in a lake. By setting up a differential equation, we will investigate the dynamics of this population and show that the population will eventually approach the so called carrying capacity of the environment if the initial population is larger than a "threshold" and become extinct if it is smaller than the threshold. For the purpose of protection of this population, we will set up a scheme for fishing.
1) Denote by P(t) the fish population at time t. Assume the birth rate β(P) and the death rate δ(P) of P(t) per individual per year are given by...
β(P) = a
δ(P) = bP+(c/P)
respectively, where a, b, c are postitive constants such that a2-4bc > 0. Apply the principle
dP/dt = [β(P)-δ(P)]P
to show that the P(t) satisfies the differential equation
dP/dt = k(M-P)(P-m) (1)
where k = b, m = (a-√(a2-4bc))/(2b) and M = (a+√(a2-4bc))/(2b). We observe that the logistic equation is the special form of (1) when m = 0. Let the initial population at t0 be given by
P(t0)=P0
2. Relevant equations
They were stated above, but here they are again.
β(P) = a
δ(P) = bP+(c/P)
dP/dt = [β(P)-δ(P)]P
dP/dt = k(M-P)(P-m) (1)
k = b
m = (a-√(a2-4bc))/(2b)
M = (a+√(a2-4bc))/(2b)
P(t0)=P0
3. The attempt at a solution
Well my attempt in getting is by separation...
dP/([β(P)-δ(P)]P) = dt
and then to integrate it...
∫1/([β(P)-δ(P)]P) dP = ∫ dt
and then partial fractions I thought, but it doesn't come out correct. I have also tried wolfram alpha and to check my work. I throw in the equation to be differentiated and it give me something that is sort of similar, but still far off.
Could someone please help me get on the right track here?
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
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