Final answer:
The differential equation for the fish population involves separation of variables to find the solution, with the limiting population being the carrying capacity, lessened by harvesting efforts. The equation exemplifies the logistic growth model, which is a realistic representation of population dynamics and considers the impact of harvesting on the carrying capacity.
Step-by-step explanation:
Solution to Given ODE
The differential equation given dx/dt = kx(M - x) - Ex models the population of fish in a lake, taking into account the natural growth of the population and the decrease due to harvesting. To solve this equation for the fish population over time, we implement separation of variables and integrate both sides. However, without specific initial conditions, we can't provide an explicit solution here.
Limiting Population
The limiting population, also known as the carrying capacity, is reached when the population growth rate dx/dt becomes zero. Setting dx/dt = 0, we solve for x to find that the limiting population is x = M - E/k, considering 0 < E < kM. This implies that the influence of harvesting effort reduces the carrying capacity by the fraction of the harvesting rate to the natural growth rate.
Understanding this model reflects how population dynamics work, taking exponential and logistic growth concepts, and applying them to real-world scenarios such as sustainable harvest and extinction risk. The logistic model of population growth limits exponential growth and creates an S-shaped curve as population nears its carrying capacity. Population growth is no longer exponentially influenced but moderated by carrying capacity, represented by the expression (K-N)/K in the logistic growth equation.