A certain bacterium, given plenty of nutrient and room, is...

A certain bacterium, given plenty of nutrient and room, is known to grow according to the Malthusian model with reproductive rate $r$. Suppose that the biologist working with the culture harvests the bacteria at a constant rate of $h$ bacteria per hour. Use qualitative analysis to discuss the fate of the culture.

Key Concepts

Malthusian Growth Model

This model is a simple exponential model of population dynamics where the rate of change of the population is proportional to its current size. It captures the idea that, in an unrestricted environment, the population will grow (or decline) at a constant relative rate, represented by the parameter r.

Constant Harvesting in Population Dynamics

The concept of constant harvesting involves removing individuals from the population at a fixed numeric rate rather than a rate proportional to the population size. This changes the dynamics because it introduces a constant loss that can overcome the natural growth when the population is small enough, potentially leading to extinction.

Equilibrium Analysis in Differential Equations

Equilibrium analysis involves setting the derivative (or the rate of change) of the population to zero in a differential equation to find steady states. In this case, setting rN - h = 0 yields an equilibrium population size, which represents a threshold: if the population falls below this value, the constant removal may drive it to extinction.

Qualitative Stability and Phase Portraits

By analyzing the sign of the derivative on either side of the equilibrium, one can determine the stability of the equilibrium point. A stable equilibrium means that small deviations will return to the equilibrium, while an unstable one indicates that small perturbations can lead to runaway growth or decline. Phase portraits provide a graphical representation of these dynamics, helping to predict the long?term fate of the population under constant harvesting.

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