Question

45–46. Harvesting problems Consider the harvesting problem in Example 6.
If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Accepted Answer

Recall the harvesting model differential equation from Example 6, which is typically of the form: $$\frac{dp}{dt} = rp - H,$$ where $$p(t) is $$the amount of the resource at time $$t$$, $$r is $$the growth rate, and $$H is $$the harvesting rate. To determine when the amount of the resource is decreasing, analyze the sign of $$\frac{dp}{dt} = rp - H.$$ The resource decreases when $$\frac{dp}{dt} \u003c 0,$$ which means $$rp - H \u003c 0.$$ Solve this inequality for $$p to $$find the values of $$p_0 ($$initial amount) where the resource decreases. To find when the amount of the resource is constant, set $$\frac{dp}{dt} = 0,$$ which gives $$rp - H = 0.$$ Solve for $$p to $$find the equilibrium value of $$p_0$$ where the resource neither increases nor decreases. Given $$p_0 = 9000$, solve the differential equation $$\frac{dp}{dt} = rp - H$$ with initial condition p(0) = 9000$ to find p(t$$)$$. This is a first-order linear differential equation and can be solved using an integrating factor or separation of variables. Once you have the explicit solution p(t$$)$$, determine the time t$ when the resource vanishes by solving p(t) = 0$ for t$. This will give the time at which the resource is completely depleted.

45–46. Harvesting problems Consider the harvesting problem in Example 6.
If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Differential Equations in Population Dynamics

Equilibrium Points and Stability

Time to Extinction in Resource Models

45–46. Harvesting problems Consider the harvesting problem in Example 6.If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Differential Equations in Population Dynamics

Equilibrium Points and Stability

Time to Extinction in Resource Models

45–46. Harvesting problems Consider the harvesting problem in Example 6.
If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?