Death Rate In Fish Essay

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Introduction
Fishing has been a popular industry, especially in those maritime countries. And of course to make a sustainable future, harvesting has to be restricted somehow to maintain the fish population at certain acceptable levels but at the same time sustains this industry.
To look into the problem, a lot of factors have to be taken into considerations - birth rate, death rate, immigration and emigration rate, etc. Differentiation – one important convention of calculus, is of good practical use in the real life. It provides powerful tools for explaining the behavior of dynamically process. By modeling a fish population system with the basis of differentiation, it can help the government to regulate in order to achieve a fishery optimal
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So the net rate of change in the fish population per year is N. For now we only consider birth rate and death rate, we will have
N’ = Birth rate - Death rate - Immigration rate + Emigration rate
According to a lot of observations of many species for a long-time period, we can make the assumption that birth rate and death rate are both roughly proportional to the size of the population, so we have
Birth rate at time :
Death rate at time :
Where b and d are both positive proportionality constants. If now we assume the birth rate is bigger than the death rate, which is a basic standard for species to live on. That means b is larger than d, then we have

which means that the fish population will keep increasing all the time, and that never happens in the real life. So that we have to bring in a new factor – overcrowding. Since a fixed habitat has its maximum capacity for species, as the population increases, the death rate will accelerate so that it can’t be accounted for by a single constant coefficient d. In order to model a more accurate system, we will then change death rate to
Death rate at time : in which we use term cf(t) to account for overcrowding. Now we

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