The model of exponential growth is based on the fact that the larger the stock y (e.g. a population), the more strongly it multiplies the larger the stock itself. This means in the model that the growth rate increases with the size of the population.
The model does not consider limiting factors such as only finite resources. A model that takes this into account leads to the model of logistic growth.
Differential equation of exponential growth:
With the growth function for the general inital values t0 and y0 = y(t0)
The half-life or doubling time T describes the time in which the stock doubles or halves.
Growth of populations
Absorption of light
The continuous exponential growth is described by a linear homogeneous differential equation with the constant factor λ
The equation states that the change in the stock is proportional to the stock itself. λ is the proportionality constant.
Division by y
Applying the chain rule
Dissolving and replacing the initial condition t0, y0 yields the solution for exponential growth. For λ > 0 there is an exponential growth process and for λ < 0 an exponential decay process.
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