A fish farmer has 5000 catfish in his pond. The number of...

A fish farmer has 5000 catfish in his pond. The number of catfish increases by $8 \%$ per month and the farmer harvests 300 catfish per month. (a) Show that the catfish population $P_n$ after $n$ months is given recursively by $$ P_n=1.08 P_{n-1}-300 \quad P_0=5000 $$ (b) How many catfish are in the pond after six months?

A fish farmer has 5000 catfish in his pond. The number of catfish increases by $8 \%$ per month and the farmer harvests 300 catfish per month.
(a) Show that the catfish population $P_n$ after $n$ months is given recursively by

$$
P_n=1.08 P_{n-1}-300 \quad P_0=5000
$$

(b) How many catfish are in the pond after six months?

Key Concepts

Difference Equations

Difference equations, which include recurrence relations, are used to model the change in a system over discrete time intervals. They combine the elements of exponential growth and constant harvesting, allowing for the analysis and prediction of population dynamics by calculating the state of the system at each time step.

Constant (Static) Harvesting

Constant harvesting refers to the removal of a fixed number of individuals from a population at regular intervals. This concept is important in modeling real-world scenarios like fishing, where a set number of members (fish) are removed regardless of the current population size, impacting the overall growth dynamics.

Recurrence Relations

A recurrence relation defines each term of a sequence as a function of its previous terms. In the context of population dynamics, it is used to represent how a current population is derived from its prior state, accounting for factors like growth and reduction, making it an essential tool for modeling discrete time processes.

Exponential (Percentage) Growth

Exponential growth describes a process where a quantity increases by a fixed percentage over equal time intervals. This concept is fundamental in population studies, where the addition of a constant percentage, such as an 8% monthly growth rate, leads to multiplicative increases in the population.

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