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# Fishing with quotas. In view of the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas that vary d...

## Question

###### Fishing with quotas. In view of the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas that vary depending on estimates of the population at the current time. One harvesting model that takes this into account is$$frac{d X}{d t}=r Xleft(1-frac{X}{K}ight)-h X$$(a) Show that the only non-zero equilibrium population is$$X_{e}=Kleft(1-frac{h}{r}ight)$$(b) At what critical harvesting rate can extinction occur?Although extinction can occ

Fishing with quotas. In view of the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas that vary depending on estimates of the population at the current time. One harvesting model that takes this into account is $$frac{d X}{d t}=r Xleft(1-frac{X}{K} ight)-h X$$ (a) Show that the only non-zero equilibrium population is $$X_{e}=Kleft(1-frac{h}{r} ight)$$ (b) At what critical harvesting rate can extinction occur? Although extinction can occur with this model, as the harvesting parameter $h$ increases towards the critical value the equilibrium population tends to zero. This contrasts with the constant harvesting model in Section $3.3$ and Exercise $3.7$, where a sudden population crash (from a large population to extinction) can occur as the harvesting rate increases beyond a critical value. #### Similar Solved Questions

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