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A differential equation is a mathematical equation that involves functions and their derivatives. In the context of the Logistic Growth Model, the differential equation we are dealing with is:

• \( \frac{dy}{dt} = ky \left( 1 - \frac{y}{M} \right) \)

This equation describes how the biomass of a population, denoted by \( y(t) \), changes over time. Here, \( dy/dt \) represents the rate of change of biomass with respect to time.
The parameter \( k \) is a constant that affects the rate of growth. This type of equation is known as a "logistic differential equation," which models growth that is initially exponential but slows down as the population approaches a maximum size. This represents a more realistic model for growth in constrained environments.

Carrying capacity is a core concept in population biology and ecology that refers to the maximum population size that an environment can sustainably support. In the logistic growth differential equation, carrying capacity is represented by the symbol \( M \).
In our example, the carrying capacity is given as \( M = 8 \times 10^7 \) kg. This value indicates the upper limit of the biomass that can be supported by the environment.

• The carrying capacity is essential because it influences how fast the population grows and stabilizes.
• When the population size \( y \) is much smaller than \( M \), growth is approximately exponential.
• As \( y \) approaches \( M \), growth slows due to limited resources and other ecological factors.

Exponential growth occurs when the rate of population increase is proportional to the existing population. In our logistic growth model, this behavior is most evident at the beginning of population growth.
Initially, when a population is far from its carrying capacity \( M \), the term \( 1 - \frac{y}{M} \) approaches 1, leading the differential equation to approximate exponential growth:

• \( \frac{dy}{dt} \approx ky \)

In this scenario, the population grows at a constant proportional rate \( k \), which in the exercise, is 0.71 per year. However, exponential growth cannot continue indefinitely, as it would soon outstrip available resources.

Initial conditions in differential equations help us determine the particular solution for a given scenario. An initial condition provides specific information about the state of a system at the beginning of observation. In this exercise, the initial condition is \( y(0) = 2 \times 10^7 \) kg.

• This means that at time \( t = 0 \), the biomass of the population starts at \( 2 \times 10^7 \) kg.
• Using this initial condition helps to calculate the constant \( A \) in the logistic growth solution:

The general solution for a logistic growth equation is given as:

• \( y(t) = \frac{M}{1 + Ae^{-kt}} \)

By substituting the initial condition, we find \( A = 3 \), resulting in a specific solution that helps predict the population at any given future time.