Consider a lake with some fish attractive to fishermen. We wish to model the fish-fishermen interaction. Fish Assumptions: i. Fish grow logistically in the absence of fishing. ii. The presence of fishermen depresses fish growth at a rate jointly proportional to the fish and fishermen populations. Fishermen Assumptions: 1\. Fishermen are attracted to the lake at a rate directly proportional to the amount of fish in the lake. i. Fishermen are discouraged from the lake at a rate directly proportional to the number of fishermen already there. (a) Formulate, analyze, and interpret a mathematical model for this situation. (b) Suppose the department of fish and game decides to stock the lake with fish at a constant rate. Formulate, analyze, and interpret a mathematical model for the situation with stocking included. What effect does stocking have on the fishery?

In the study of population dynamics, logistic growth is a fundamental concept. It describes how a population grows rapidly when it is small, slows as it approaches the environment's carrying capacity, and eventually stabilizes. The equation that represents logistic growth is:
\[ \frac{dF}{dt} = rF \bigg(1 - \frac{F}{K}\bigg) \] where:

• \( F \) represents the fish population,


• \( r \) is the intrinsic growth rate,


• \( K \) is the carrying capacity, the maximum population size that the environment can sustain.

This model shows that when the population size \( F \) is far below the carrying capacity \( K \), the growth rate is high. As \( F \) approaches \( K \), the term \( 1 - \frac{F}{K} \) becomes smaller, slowing the growth rate until growth stops altogether when the population size hits the carrying capacity.

Population dynamics explores how and why the numbers of individuals in populations change over time. In this fish-fishermen interaction model, dynamics are influenced by multiple factors:

• Fish population grows logistically without fishing.


• The presence of fishermen negatively impacts fish growth at a rate proportional to both fish and fishermen populations.


• Fishermen are attracted to and discouraged from the lake based on the fish population and the number of other fishermen, respectively.

Combining these, the fish and fishermen populations are described by the following system of differential equations:
\[ \frac{dF}{dt} = rF \bigg(1 - \frac{F}{K}\bigg) - aFM \] \[ \frac{dM}{dt} = bF - dM \] where:

• \( r \) is the fish intrinsic growth rate,


• \( K \) is the carrying capacity,


• \( a \) is the rate at which fishermen depress fish growth,


• \( b \) is the rate at which fishermen are attracted to the lake by fish,


• \( d \) is the rate at which fishermen leave the lake.

This model highlights the dynamic and interdependent nature of the fish and fishermen populations.

Introducing a constant rate of stocking fish into the lake alters the dynamics. Stocking adds a fixed number of fish to the population over time, helping to mitigate the negative effects of fishing.
The modified fish population equation with constant rate stocking \( S \) is:
\[ \frac{dF}{dt} = rF \bigg(1 - \frac{F}{K}\bigg) - aFM + S \] Here, \( S \) represents the number of fish added to the lake per unit time.

The inclusion of stocking can increase the equilibrium fish population compared to when there's no stocking. This means that even with fishing pressures, the fish population can be maintained at a higher level due to the continuous addition of new fish.
The fishermen population remains influenced only by the fish population and their own numbers, hence the equation
\[ \frac{dM}{dt} = bF - dM \] remains unchanged. Stocking compensates for the fish removed by fishermen, which can result in a stabilized fish population that supports ongoing fishing activities.