# Growth Curves for a Mixture of Two Subpopulations

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The growth curves of two-subpopulation mixtures can have shapes that are practically indistinguishable from those of a homogeneous population, but they can also have a variety of distinctly different shapes. This is shown with simulated growth curves generated with a double-stretched exponential model and a two-term non-exponential model. The Demonstration also shows how a minor change in one of the subpopulation's growth parameters can sometimes result in a major change in the shape of the mixture's growth curve.

Contributed by: Mark D. Normand, Przemyslaw Remin, and Micha Peleg (November 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: growth curve having two inflection points generated with the non-exponential model

Snapshot 2: growth curve having two inflection points generated with the double-stretched exponential model

Snapshot 3: oddly shaped highly asymmetric growth curve generated with the double-stretched exponential model

Snapshot 4: non-sigmoid growth curve generated with the non-exponential model

Snapshot 5: typical sigmoid growth curve with a substantial lag time generated with the non-exponential model

Snapshot 6: typical sigmoid growth curve with a short lag time generated with the double-stretched exponential model

Snapshot 7: typical sigmoid growth curve with a long lag time generated with the double-stretched exponential model

The growth curve of a population is frequently sigmoid or concave downward approaching an asymptotic level, primarily determined by the habitat's carrying capacity. Such curves have been described by a variety of mathematical models, which in most cases can be used interchangeably. When the population is a mixture of two subpopulations, these shapes can be maintained. However, depending on the individual subpopulation's growth parameters, the growth curve can assume different shapes that are sometimes quite atypical.

In this Demonstration, the regular and odd shapes are generated with a two-term general growth model , where is either the net growth ratio or the logarithmic growth ratio , where and are the momentary and initial sizes of the population, respectively. The first definition applies to comparatively moderate growth levels and the second to intensive growth resulting in a population rise by several orders of magnitude. According to both definitions, at , where , .

The chosen functions for and , whose complete formulas are displayed above the growth curve's plot, are the three-parameter stretched exponential model or the non-exponential model ), both satisfying the condition that . According to both models, , the subpopulation's asymptotic growth level, is a characteristic time, and is a steepness parameter. You can vary them and the plot's axes maxima. According to both the exponential and non-exponential models, the asymptotic growth level of the mixed population, , is .

The purpose of the Demonstration is to visualize the concept of differential population growth, not to match any experimental observation of a particular organismic or non-organismic population. Consequently, not all parameter combinations necessarily have real-life counterparts.

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