Question

6 Harvesting Models We now wish to account for how a fishing industry that provides salmon for the food industry affects the population of salmon. To demonstrate the ideas of harvesting, we assume the salmon population is governed by the logistic growth model (4) with carrying capacity of K = 20 thousand salmon. For simplicity, we assume the the catch of fish per unit effort is independent of the stock level $y$. Thus, the logistic model with constant harvesting at the rate $H > 0$ thousand per year has the form $\frac{dy}{dt} = y \left(1 - \frac{y}{20}\right) - H = f(y)$. Problem: For each of the three cases below find the following: 1. Find equilibrium solutions when when salmon is harvested at the rate of $H$ thousand per year. 2. Sketch the graph of $y$ vs. $f$, phase-line and phase-portraits. 3. Does the population of salmon go extinct? If yes, for what initial population size $y_0$? In that case, do you think it is possible to find the time $t^$ when population goes extinct, that is $y(t^) = 0$?

          6 Harvesting Models
We now wish to account for how a fishing industry that provides salmon for the food industry
affects the population of salmon. To demonstrate the ideas of harvesting, we assume the
salmon population is governed by the logistic growth model (4) with carrying capacity of
K = 20 thousand salmon. For simplicity, we assume the the catch of fish per unit effort is
independent of the stock level $y$. Thus, the logistic model with constant harvesting at the
rate $H > 0$ thousand per year has the form
$\frac{dy}{dt} = y \left(1 - \frac{y}{20}\right) - H = f(y)$.
Problem: For each of the three cases below find the following:
1. Find equilibrium solutions when when salmon is harvested at the rate of $H$ thousand
per year.
2. Sketch the graph of $y$ vs. $f$, phase-line and phase-portraits.
3. Does the population of salmon go extinct? If yes, for what initial population size $y_0$? In
that case, do you think it is possible to find the time $t^*$ when population goes extinct,
that is $y(t^*) = 0$?

6 Harvesting Models
We now wish to account for how a fishing industry that provides salmon for the food industry
affects the population of salmon. To demonstrate the ideas of harvesting, we assume the
salmon population is governed by the logistic growth model (4) with carrying capacity of
K = 20 thousand salmon. For simplicity, we assume the the catch of fish per unit effort is
independent of the stock level y. Thus, the logistic model with constant harvesting at the
rate H > 0 thousand per year has the form
(dy)/(dt) = y (1 - (y)/(20)) - H = f(y).
Problem: For each of the three cases below find the following:
1. Find equilibrium solutions when when salmon is harvested at the rate of H thousand
per year.
2. Sketch the graph of y vs. f, phase-line and phase-portraits.
3. Does the population of salmon go extinct? If yes, for what initial population size y0? In
that case, do you think it is possible to find the time t^* when population goes extinct,
that is y(t^*) = 0?

Added by Kyle L.

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